Global Hopf bifurcation analysis on a BAM neural network with delays

NASA Astrophysics Data System (ADS)

Sun, Chengjun; Han, Maoan; Pang, Xiaoming

2007-01-01

A delayed differential equation that models a bidirectional associative memory (BAM) neural network with four neurons is considered. By using a global Hopf bifurcation theorem for FDE and a Bendixon's criterion for high-dimensional ODE, a group of sufficient conditions for the system to have multiple periodic solutions are obtained when the sum of delays is sufficiently large.

Nonlinear Analysis of Mechanical Systems Under Combined Harmonic and Stochastic Excitation

DTIC Science & Technology

1993-05-27

Namachchivaya and Naresh Malhotra Department of Aeronautical and Astronautical Engineering University of Illinois, Urbana-Champaign Urbana, Illinois...Aeronauticai and Astronautical Engineering, University of Illinois, 1991. 2. N. Sri Namachchivaya and N. Malhotra , Parametrically Excited Hopf Bifurcation...Namachchivaya and N. Malhotra , Parametrically Excited Hopf Bifurcation with Non-semisimple 1:1 Resonance, Nonlinear Vibrations, ASME-AMD, Vol. 114, 1992. 3

Stability and bifurcation analysis for the Kaldor-Kalecki model with a discrete delay and a distributed delay

NASA Astrophysics Data System (ADS)

Yu, Jinchen; Peng, Mingshu

2016-10-01

In this paper, a Kaldor-Kalecki model of business cycle with both discrete and distributed delays is considered. With the corresponding characteristic equation analyzed, the local stability of the positive equilibrium is investigated. It is found that there exist Hopf bifurcations when the discrete time delay passes a sequence of critical values. By applying the method of multiple scales, the explicit formulae which determine the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are derived. Finally, numerical simulations are carried out to illustrate our main results.

Hopf and Bautin Bifurcation in a Tritrophic Food Chain Model with Holling Functional Response Types III and IV

NASA Astrophysics Data System (ADS)

Castellanos, Víctor; Castillo-Santos, Francisco Eduardo; Dela-Rosa, Miguel Angel; Loreto-Hernández, Iván

In this paper, we analyze the Hopf and Bautin bifurcation of a given system of differential equations, corresponding to a tritrophic food chain model with Holling functional response types III and IV for the predator and superpredator, respectively. We distinguish two cases, when the prey has linear or logistic growth. In both cases we guarantee the existence of a limit cycle bifurcating from an equilibrium point in the positive octant of ℝ3. In order to do so, for the Hopf bifurcation we compute explicitly the first Lyapunov coefficient, the transversality Hopf condition, and for the Bautin bifurcation we also compute the second Lyapunov coefficient and verify the regularity conditions.

Bursting Types and Bifurcation Analysis in the Pre-Bötzinger Complex Respiratory Rhythm Neuron

NASA Astrophysics Data System (ADS)

Wang, Jing; Lu, Bo; Liu, Shenquan; Jiang, Xiaofang

Many types of neurons and excitable cells could intrinsically generate bursting activity, even in an isolated case, which plays a vital role in neuronal signaling and synaptic plasticity. In this paper, we have mainly investigated bursting types and corresponding bifurcations in the pre-Bötzinger complex respiratory rhythm neuron by using fast-slow dynamical analysis. The numerical simulation results have showed that for some appropriate parameters, the neuron model could exhibit four distinct types of fast-slow bursters. We also explored the bifurcation mechanisms related to these four types of bursters through the analysis of phase plane. Moreover, the first Lyapunov coefficient of the Hopf bifurcation, which can decide whether it is supercritical or subcritical, was calculated with the aid of MAPLE software. In addition, we analyzed the codimension-two bifurcation for equilibria of the whole system and gave a detailed theoretical derivation of the Bogdanov-Takens bifurcation. Finally, we obtained expressions for a fold bifurcation curve, a nondegenerate Hopf bifurcation curve, and a saddle homoclinic bifurcation curve near the Bogdanov-Takens bifurcation point.

Analysis of the stability of nonlinear suspension system with slow-varying sprung mass under dual-excitation

NASA Astrophysics Data System (ADS)

Yao, Jun; Zhang, Jinqiu; Zhao, Mingmei; Li, Xin

2018-07-01

This study investigated the stability of vibration in a nonlinear suspension system with slow-varying sprung mass under dual-excitation. A mathematical model of the system was first established and then solved using the multi-scale method. Finally, the amplitude-frequency curve of vehicle vibration, the solution's stable region and time-domain curve in Hopf bifurcation were derived. The obtained results revealed that an increase in the lower excitation would reduce the system's stability while an increase in the upper excitation can make the system more stable. The slow-varying sprung mass will change the system's damping from negative to positive, leading to the appearance of limit cycle and Hopf bifurcation. As a result, the vehicle's vibration state is forced to change. The stability of this system is extremely fragile under the effect of dynamic Hopf bifurcation as well as static bifurcation.

Stability and Hopf bifurcation for a delayed SLBRS computer virus model.

PubMed

Zhang, Zizhen; Yang, Huizhong

2014-01-01

By incorporating the time delay due to the period that computers use antivirus software to clean the virus into the SLBRS model a delayed SLBRS computer virus model is proposed in this paper. The dynamical behaviors which include local stability and Hopf bifurcation are investigated by regarding the delay as bifurcating parameter. Specially, direction and stability of the Hopf bifurcation are derived by applying the normal form method and center manifold theory. Finally, an illustrative example is also presented to testify our analytical results.

Hopf Bifurcation Analysis of a Gene Regulatory Network Mediated by Small Noncoding RNA with Time Delays and Diffusion

NASA Astrophysics Data System (ADS)

Li, Chengxian; Liu, Haihong; Zhang, Tonghua; Yan, Fang

2017-12-01

In this paper, a gene regulatory network mediated by small noncoding RNA involving two time delays and diffusion under the Neumann boundary conditions is studied. Choosing the sum of delays as the bifurcation parameter, the stability of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous periodic solutions are investigated by analyzing the corresponding characteristic equation. It is shown that the sum of delays can induce Hopf bifurcation and the diffusion incorporated into the system can effect the amplitude of periodic solutions. Furthermore, the spatially homogeneous periodic solution always exists and the spatially inhomogeneous periodic solution will arise when the diffusion coefficients of protein and mRNA are suitably small. Particularly, the small RNA diffusion coefficient is more robust and its effect on model is much less than protein and mRNA. Finally, the explicit formulae for determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are derived by employing the normal form theory and center manifold theorem for partial functional differential equations. Finally, numerical simulations are carried out to illustrate our theoretical analysis.

Bifurcation and chaos analysis of a nonlinear electromechanical coupling relative rotation system

NASA Astrophysics Data System (ADS)

Liu, Shuang; Zhao, Shuang-Shuang; Sun, Bao-Ping; Zhang, Wen-Ming

2014-09-01

Hopf bifurcation and chaos of a nonlinear electromechanical coupling relative rotation system are studied in this paper. Considering the energy in air-gap field of AC motor, the dynamical equation of nonlinear electromechanical coupling relative rotation system is deduced by using the dissipation Lagrange equation. Choosing the electromagnetic stiffness as a bifurcation parameter, the necessary and sufficient conditions of Hopf bifurcation are given, and the bifurcation characteristics are studied. The mechanism and conditions of system parameters for chaotic motions are investigated rigorously based on the Silnikov method, and the homoclinic orbit is found by using the undetermined coefficient method. Therefore, Smale horseshoe chaos occurs when electromagnetic stiffness changes. Numerical simulations are also given, which confirm the analytical results.

Stability and Hopf Bifurcation for a Delayed SLBRS Computer Virus Model

PubMed Central

Yang, Huizhong

2014-01-01

By incorporating the time delay due to the period that computers use antivirus software to clean the virus into the SLBRS model a delayed SLBRS computer virus model is proposed in this paper. The dynamical behaviors which include local stability and Hopf bifurcation are investigated by regarding the delay as bifurcating parameter. Specially, direction and stability of the Hopf bifurcation are derived by applying the normal form method and center manifold theory. Finally, an illustrative example is also presented to testify our analytical results. PMID:25202722

A proof of Wright's conjecture

NASA Astrophysics Data System (ADS)

van den Berg, Jan Bouwe; Jaquette, Jonathan

2018-06-01

Wright's conjecture states that the origin is the global attractor for the delay differential equation y‧ (t) = - αy (t - 1) [ 1 + y (t) ] for all α ∈ (0, π/2 ] when y (t) > - 1. This has been proven to be true for a subset of parameter values α. We extend the result to the full parameter range α ∈ (0, π/2 ], and thus prove Wright's conjecture to be true. Our approach relies on a careful investigation of the neighborhood of the Hopf bifurcation occurring at α = π/2. This analysis fills the gap left by complementary work on Wright's conjecture, which covers parameter values further away from the bifurcation point. Furthermore, we show that the branch of (slowly oscillating) periodic orbits originating from this Hopf bifurcation does not have any subsequent bifurcations (and in particular no folds) for α ∈ (π/2, π/2 + 6.830 ×10-3 ]. When combined with other results, this proves that the branch of slowly oscillating solutions that originates from the Hopf bifurcation at α = π/2 is globally parametrized by α > π/2.

Straight ahead running of a nonlinear car and driver model - new nonlinear behaviours highlighted

NASA Astrophysics Data System (ADS)

Della Rossa, Fabio; Mastinu, Giampiero

2018-05-01

The paper deals with the bifurcation analysis of a validated simple model describing a vehicle+driver running straight ahead. The mechanical model of the car has two degrees of freedom and the related equations of motion contain the nonlinear tyre characteristics. The driver is described by a very simple model. Bifurcation analysis is adopted for characterising straight ahead motion at different speeds for different drivers. A nonlinear sensitivity analysis is performed as a function of the driver's parameters and forward vehicle speed. A wealth of unreferenced bifurcations is discovered both for the understeering (UN) and for the oversteering (OV) vehicle. For the UN vehicle, a supercritical Hopf bifurcation may occur as the forward speed is increased. Also tangent (fold) bifurcations (saddle-node bifurcation of limit cycles) occur as the speed (or disturbance) is further increased. For the OV vehicle, a subcritical Hopf bifurcation occurs as the speed reaches a critical value. The preview distance (a driver's control parameter) plays a fundamental role in straight ahead driving. Either too short or too long preview distances are negative for straight ahead running.

Nonlinear analysis of a closed-loop tractor-semitrailer vehicle system with time delay

NASA Astrophysics Data System (ADS)

Liu, Zhaoheng; Hu, Kun; Chung, Kwok-wai

2016-08-01

In this paper, a nonlinear analysis is performed on a closed-loop system of articulated heavy vehicles with driver steering control. The nonlinearity arises from the nonlinear cubic tire force model. An integration method is employed to derive an analytical periodic solution of the system in the neighbourhood of the critical speed. The results show that excellent accuracy can be achieved for the calculation of periodic solutions arising from Hopf bifurcation of the vehicle motion. A criterion is obtained for detecting the Bautin bifurcation which separates branches of supercritical and subcritical Hopf bifurcations. The integration method is compared to the incremental harmonic balance method in both supercritical and subcritical scenarios.

Complexity dynamics and Hopf bifurcation analysis based on the first Lyapunov coefficient about 3D IS-LM macroeconomics system

NASA Astrophysics Data System (ADS)

Ma, Junhai; Ren, Wenbo; Zhan, Xueli

2017-04-01

Based on the study of scholars at home and abroad, this paper improves the three-dimensional IS-LM model in macroeconomics, analyzes the equilibrium point of the system and stability conditions, focuses on the parameters and complex dynamic characteristics when Hopf bifurcation occurs in the three-dimensional IS-LM macroeconomics system. In order to analyze the stability of limit cycles when Hopf bifurcation occurs, this paper further introduces the first Lyapunov coefficient to judge the limit cycles, i.e. from a practical view of the business cycle. Numerical simulation results show that within the range of most of the parameters, the limit cycle of 3D IS-LM macroeconomics is stable, that is, the business cycle is stable; with the increase of the parameters, limit cycles becomes unstable, and the value range of the parameters in this situation is small. The research results of this paper have good guide significance for the analysis of macroeconomics system.

Controlling the onset of Hopf bifurcation in the Hodgkin-Huxley model

NASA Astrophysics Data System (ADS)

Xie, Yong; Chen, Luonan; Kang, Yan Mei; Aihara, Kazuyuki

2008-06-01

It is a challenging problem to establish safe and simple therapeutic methods for various complicated diseases of the nervous system, particularly dynamical diseases such as epilepsy, Alzheimer’s disease, and Parkinson’s disease. From the viewpoint of nonlinear dynamical systems, a dynamical disease can be considered to be caused by a bifurcation induced by a change in the values of one or more regulating parameter. Therefore, the theory of bifurcation control may have potential applications in the diagnosis and therapy of dynamical diseases. In this study, we employ a washout filter-aided dynamic feedback controller to control the onset of Hopf bifurcation in the Hodgkin-Huxley (HH) model. Specifically, by the control scheme, we can move the Hopf bifurcation to a desired point irrespective of whether the corresponding steady state is stable or unstable. In other words, we are able to advance or delay the Hopf bifurcation, so as to prevent it from occurring in a certain range of the externally applied current. Moreover, we can control the criticality of the bifurcation and regulate the oscillation amplitude of the bifurcated limit cycle. In the controller, there are only two terms: the linear term and the nonlinear cubic term. We show that while the former determines the location of the Hopf bifurcation, the latter regulates the criticality of the Hopf bifurcation. According to the conditions of the occurrence of Hopf bifurcation and the bifurcation stability coefficient, we can analytically deduce the linear term and the nonlinear cubic term, respectively. In addition, we also show that mixed-mode oscillations (MMOs), featuring slow action potential generation, which are frequently observed in both experiments and models of chemical and biological systems, appear in the controlled HH model. It is well known that slow firing rates in single neuron models could be achieved only by type-I neurons. However, the controlled HH model is still classified as a type-II neuron, as is the original HH model. We explain that the occurrence of MMOs can be related to the presence of the canard explosion where a small oscillation grows through a sequence of canard cycles to a relaxation oscillation as the control parameter moves through an interval of exponentially small width.

Small aspect ratio Taylor-Couette flow: onset of a very-low-frequency three-torus state.

PubMed

Lopez, Juan M; Marques, Francisco

2003-09-01

The nonlinear dynamics of Taylor-Couette flow in a small aspect ratio annulus (where the length of the cylinders is half of the annular gap between them) is investigated by numerically solving the full three-dimensional Navier-Stokes equations. The system is invariant to arbitrary rotations about the annulus axis and to a reflection about the annulus half-height, so that the symmetry group is SO(2)xZ2. In this paper, we systematically investigate primary and subsequent bifurcations of the basic state, concentrating on a parameter regime where the basic state becomes unstable via Hopf bifurcations. We derive the four distinct cases for the symmetries of the bifurcated orbit, and numerically find two of these. In the parameter regime considered, we also locate the codimension-two double Hopf bifurcation where these two Hopf bifurcations coincide. Secondary Hopf bifurcations (Neimark-Sacker bifurcations), leading to modulated rotating waves, are subsequently found and a saddle-node-infinite-period bifurcation between a stable (node) and an unstable (saddle) modulated rotating wave is located, which gives rise to a very-low-frequency three-torus. This paper provides the computed example of such a state, along with a comprehensive bifurcation sequence leading to its onset.

Qualitative dynamical analysis of chaotic plasma perturbations model

NASA Astrophysics Data System (ADS)

Elsadany, A. A.; Elsonbaty, Amr; Agiza, H. N.

2018-06-01

In this work, an analytical framework to understand nonlinear dynamics of plasma perturbations model is introduced. In particular, we analyze the model presented by Constantinescu et al. [20] which consists of three coupled ODEs and contains three parameters. The basic dynamical properties of the system are first investigated by the ways of bifurcation diagrams, phase portraits and Lyapunov exponents. Then, the normal form technique and perturbation methods are applied so as to the different types of bifurcations that exist in the model are investigated. It is proved that pitcfork, Bogdanov-Takens, Andronov-Hopf bifurcations, degenerate Hopf and homoclinic bifurcation can occur in phase space of the model. Also, the model can exhibit quasiperiodicity and chaotic behavior. Numerical simulations confirm our theoretical analytical results.

Bifurcation analysis in SIR epidemic model with treatment

NASA Astrophysics Data System (ADS)

Balamuralitharan, S.; Radha, M.

2018-04-01

We investigated the bifurcation analysis of nonlinear system of SIR epidemic model with treatment. It is accepted that the treatment is corresponding to the quantity of infective which is below the limit and steady when the quantity of infective achieves the limit. We analyze about the Transcritical bifurcation which occurs at the disease free equilibrium point and Hopf bifurcation which occurs at endemic equilibrium point. Using MATLAB we show the picture of bifurcation at the disease free equilibrium point.

Hopf bifurcation and chaos in a third-order phase-locked loop

NASA Astrophysics Data System (ADS)

Piqueira, José Roberto C.

2017-01-01

Phase-locked loops (PLLs) are devices able to recover time signals in several engineering applications. The literature regarding their dynamical behavior is vast, specifically considering that the process of synchronization between the input signal, coming from a remote source, and the PLL local oscillation is robust. For high-frequency applications it is usual to increase the PLL order by increasing the order of the internal filter, for guarantying good transient responses; however local parameter variations imply structural instability, thus provoking a Hopf bifurcation and a route to chaos for the phase error. Here, one usual architecture for a third-order PLL is studied and a range of permitted parameters is derived, providing a rule of thumb for designers. Out of this range, a Hopf bifurcation appears and, by increasing parameters, the periodic solution originated by the Hopf bifurcation degenerates into a chaotic attractor, therefore, preventing synchronization.

Non-smooth Hopf-type bifurcations arising from impact–friction contact events in rotating machinery

PubMed Central

Mora, Karin; Budd, Chris; Glendinning, Paul; Keogh, Patrick

2014-01-01

We analyse the novel dynamics arising in a nonlinear rotor dynamic system by investigating the discontinuity-induced bifurcations corresponding to collisions with the rotor housing (touchdown bearing surface interactions). The simplified Föppl/Jeffcott rotor with clearance and mass unbalance is modelled by a two degree of freedom impact–friction oscillator, as appropriate for a rigid rotor levitated by magnetic bearings. Two types of motion observed in experiments are of interest in this paper: no contact and repeated instantaneous contact. We study how these are affected by damping and stiffness present in the system using analytical and numerical piecewise-smooth dynamical systems methods. By studying the impact map, we show that these types of motion arise at a novel non-smooth Hopf-type bifurcation from a boundary equilibrium bifurcation point for certain parameter values. A local analysis of this bifurcation point allows us a complete understanding of this behaviour in a general setting. The analysis identifies criteria for the existence of such smooth and non-smooth bifurcations, which is an essential step towards achieving reliable and robust controllers that can take compensating action. PMID:25383034

Backward bifurcations, turning points and rich dynamics in simple disease models.

PubMed

Zhang, Wenjing; Wahl, Lindi M; Yu, Pei

2016-10-01

In this paper, dynamical systems theory and bifurcation theory are applied to investigate the rich dynamical behaviours observed in three simple disease models. The 2- and 3-dimensional models we investigate have arisen in previous investigations of epidemiology, in-host disease, and autoimmunity. These closely related models display interesting dynamical behaviors including bistability, recurrence, and regular oscillations, each of which has possible clinical or public health implications. In this contribution we elucidate the key role of backward bifurcations in the parameter regimes leading to the behaviors of interest. We demonstrate that backward bifurcations with varied positions of turning points facilitate the appearance of Hopf bifurcations, and the varied dynamical behaviors are then determined by the properties of the Hopf bifurcation(s), including their location and direction. A Maple program developed earlier is implemented to determine the stability of limit cycles bifurcating from the Hopf bifurcation. Numerical simulations are presented to illustrate phenomena of interest such as bistability, recurrence and oscillation. We also discuss the physical motivations for the models and the clinical implications of the resulting dynamics.

Dynamic bifurcation and strange nonchaos in a two-frequency parametrically driven nonlinear oscillator

NASA Astrophysics Data System (ADS)

Premraj, D.; Suresh, K.; Palanivel, J.; Thamilmaran, K.

2017-09-01

A periodically forced series LCR circuit with Chua's diode as a nonlinear element exhibits slow passage through Hopf bifurcation. This slow passage leads to a delay in the Hopf bifurcation. The delay in this bifurcation is a unique quantity and it can be predicted using various numerical analysis. We find that when an additional periodic force is added to the system, the delay in bifurcation becomes chaotic which leads to an unpredictability in bifurcation delay. Further, we study the bifurcation of the periodic delay to chaotic delay in the slow passage effect through strange nonchaotic delay. We also report the occurrence of strange nonchaotic dynamics while varying the parameter of the additional force included in the system. We observe that the system exhibits a hitherto unknown dynamical transition to a strange nonchaotic attractor. With the help of Lyapunov exponent, we explain the new transition to strange nonchaotic attractor and its mechanism is studied by making use of rational approximation theory. The birth of SNA has also been confirmed numerically, using Poincaré maps, phase sensitivity exponent, the distribution of finite-time Lyapunov exponents and singular continuous spectrum analysis.

Characterization of spiraling patterns in spatial rock-paper-scissors games.

PubMed

Szczesny, Bartosz; Mobilia, Mauro; Rucklidge, Alastair M

2014-09-01

The spatiotemporal arrangement of interacting populations often influences the maintenance of species diversity and is a subject of intense research. Here, we study the spatiotemporal patterns arising from the cyclic competition between three species in two dimensions. Inspired by recent experiments, we consider a generic metapopulation model comprising "rock-paper-scissors" interactions via dominance removal and replacement, reproduction, mutations, pair exchange, and hopping of individuals. By combining analytical and numerical methods, we obtain the model's phase diagram near its Hopf bifurcation and quantitatively characterize the properties of the spiraling patterns arising in each phase. The phases characterizing the cyclic competition away from the Hopf bifurcation (at low mutation rate) are also investigated. Our analytical approach relies on the careful analysis of the properties of the complex Ginzburg-Landau equation derived through a controlled (perturbative) multiscale expansion around the model's Hopf bifurcation. Our results allow us to clarify when spatial "rock-paper-scissors" competition leads to stable spiral waves and under which circumstances they are influenced by nonlinear mobility.

Dynamic transitions in a model of the hypothalamic-pituitary-adrenal axis

NASA Astrophysics Data System (ADS)

Čupić, Željko; Marković, Vladimir M.; Maćešić, Stevan; Stanojević, Ana; Damjanović, Svetozar; Vukojević, Vladana; Kolar-Anić, Ljiljana

2016-03-01

Dynamic properties of a nonlinear five-dimensional stoichiometric model of the hypothalamic-pituitary-adrenal (HPA) axis were systematically investigated. Conditions under which qualitative transitions between dynamic states occur are determined by independently varying the rate constants of all reactions that constitute the model. Bifurcation types were further characterized using continuation algorithms and scale factor methods. Regions of bistability and transitions through supercritical Andronov-Hopf and saddle loop bifurcations were identified. Dynamic state analysis predicts that the HPA axis operates under basal (healthy) physiological conditions close to an Andronov-Hopf bifurcation. Dynamic properties of the stress-control axis have not been characterized experimentally, but modelling suggests that the proximity to a supercritical Andronov-Hopf bifurcation can give the HPA axis both, flexibility to respond to external stimuli and adjust to new conditions and stability, i.e., the capacity to return to the original dynamic state afterwards, which is essential for maintaining homeostasis. The analysis presented here reflects the properties of a low-dimensional model that succinctly describes neurochemical transformations underlying the HPA axis. However, the model accounts correctly for a number of experimentally observed properties of the stress-response axis. We therefore regard that the presented analysis is meaningful, showing how in silico investigations can be used to guide the experimentalists in understanding how the HPA axis activity changes under chronic disease and/or specific pharmacological manipulations.

Stability and bifurcation analysis of three-species predator-prey model with non-monotonic delayed predator response

NASA Astrophysics Data System (ADS)

Balilo, Aldrin T.; Collera, Juancho A.

2018-03-01

In this paper, we consider delayed three-species predator-prey model with non-monotonic functional response where two predator populations feed on a single prey population. Response function in both predator populations includes a time delay which represents the gestation period of the predator populations. We call a positive equlibrium solution of the form E*S=(x*,y*,y*) as a symmetric equilibrium. The goal of this paper is to determine the effect of the difference in gestation periods of predator populations to the local dynamics of symmetric equilibria. Our results include conditions on the existence of equilibrium solutions, and stability and bifurcations of symmetric equilibria as the gestation periods of predator populations are varied. A numerical bifurcation analysis tool is also used to illustrate our results. Stability switch occurs at a Hopf bifurcation. Moreover, a branch of stable periodic solutions, obtained using numerical continuation, emerges from the Hopf bifurcation. This shows that the predator population with longer gestation period oscillates higher than the predator population with shorter gestation period.

Multiple-parameter bifurcation analysis in a Kuramoto model with time delay and distributed shear

NASA Astrophysics Data System (ADS)

Niu, Ben; Zhang, Jiaming; Wei, Junjie

2018-05-01

In this paper, time delay effect and distributed shear are considered in the Kuramoto model. On the Ott-Antonsen's manifold, through analyzing the associated characteristic equation of the reduced functional differential equation, the stability boundary of the incoherent state is derived in multiple-parameter space. Moreover, very rich dynamical behavior such as stability switches inducing synchronization switches can occur in this equation. With the loss of stability, Hopf bifurcating coherent states arise, and the criticality of Hopf bifurcations is determined by applying the normal form theory and the center manifold theorem. On one hand, theoretical analysis indicates that the width of shear distribution and time delay can both eliminate the synchronization then lead the Kuramoto model to incoherence. On the other, time delay can induce several coexisting coherent states. Finally, some numerical simulations are given to support the obtained results where several bifurcation diagrams are drawn, and the effect of time delay and shear is discussed.

Stability and Hopf Bifurcation for Two Advertising Systems, Coupled with Delay

NASA Astrophysics Data System (ADS)

Sterpu, Mihaela; Rocşoreanu, Carmen

2007-09-01

Two advertising systems were linearly coupled via the first variable, with time delay. The stability and the Hopf bifurcation corresponding to the symmetric equilibrium point (the origin) in the 4D system are analyzed. Different types of oscillations corresponding to the limit cycles are compared.

Stability Switches, Hopf Bifurcations, and Spatio-temporal Patterns in a Delayed Neural Model with Bidirectional Coupling

NASA Astrophysics Data System (ADS)

Song, Yongli; Zhang, Tonghua; Tadé, Moses O.

2009-12-01

The dynamical behavior of a delayed neural network with bi-directional coupling is investigated by taking the delay as the bifurcating parameter. Some parameter regions are given for conditional/absolute stability and Hopf bifurcations by using the theory of functional differential equations. As the propagation time delay in the coupling varies, stability switches for the trivial solution are found. Conditions ensuring the stability and direction of the Hopf bifurcation are determined by applying the normal form theory and the center manifold theorem. We also discuss the spatio-temporal patterns of bifurcating periodic oscillations by using the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. In particular, we obtain that the spatio-temporal patterns of bifurcating periodic oscillations will alternate according to the change of the propagation time delay in the coupling, i.e., different ranges of delays correspond to different patterns of neural activities. Numerical simulations are given to illustrate the obtained results and show the existence of bursts in some interval of the time for large enough delay.

D3-Equivariant coupled advertising oscillators model

NASA Astrophysics Data System (ADS)

Zhang, Chunrui; Zheng, Huifeng

2011-04-01

A ring of three coupled advertising oscillators with delay is considered. Using the symmetric functional differential equation theories, the multiple Hopf bifurcations of the equilibrium at the origin are demonstrated. The existence of multiple branches of bifurcating periodic solution is obtained. Numerical simulation supports our analysis results.

Subcritical Hopf Bifurcation and Stochastic Resonance of Electrical Activities in Neuron under Electromagnetic Induction

PubMed Central

Fu, Yu-Xuan; Kang, Yan-Mei; Xie, Yong

2018-01-01

The FitzHugh–Nagumo model is improved to consider the effect of the electromagnetic induction on single neuron. On the basis of investigating the Hopf bifurcation behavior of the improved model, stochastic resonance in the stochastic version is captured near the bifurcation point. It is revealed that a weak harmonic oscillation in the electromagnetic disturbance can be amplified through stochastic resonance, and it is the cooperative effect of random transition between the resting state and the large amplitude oscillating state that results in the resonant phenomenon. Using the noise dependence of the mean of interburst intervals, we essentially suggest a biologically feasible clue for detecting weak signal by means of neuron model with subcritical Hopf bifurcation. These observations should be helpful in understanding the influence of the magnetic field to neural electrical activity. PMID:29467642

Subcritical Hopf Bifurcation and Stochastic Resonance of Electrical Activities in Neuron under Electromagnetic Induction.

PubMed

Fu, Yu-Xuan; Kang, Yan-Mei; Xie, Yong

2018-01-01

The FitzHugh-Nagumo model is improved to consider the effect of the electromagnetic induction on single neuron. On the basis of investigating the Hopf bifurcation behavior of the improved model, stochastic resonance in the stochastic version is captured near the bifurcation point. It is revealed that a weak harmonic oscillation in the electromagnetic disturbance can be amplified through stochastic resonance, and it is the cooperative effect of random transition between the resting state and the large amplitude oscillating state that results in the resonant phenomenon. Using the noise dependence of the mean of interburst intervals, we essentially suggest a biologically feasible clue for detecting weak signal by means of neuron model with subcritical Hopf bifurcation. These observations should be helpful in understanding the influence of the magnetic field to neural electrical activity.

Bifurcation theory for finitely smooth planar autonomous differential systems

NASA Astrophysics Data System (ADS)

Han, Maoan; Sheng, Lijuan; Zhang, Xiang

2018-03-01

In this paper we establish bifurcation theory of limit cycles for planar Ck smooth autonomous differential systems, with k ∈ N. The key point is to study the smoothness of bifurcation functions which are basic and important tool on the study of Hopf bifurcation at a fine focus or a center, and of Poincaré bifurcation in a period annulus. We especially study the smoothness of the first order Melnikov function in degenerate Hopf bifurcation at an elementary center. As we know, the smoothness problem was solved for analytic and C∞ differential systems, but it was not tackled for finitely smooth differential systems. Here, we present their optimal regularity of these bifurcation functions and their asymptotic expressions in the finite smooth case.

Stability switches, Hopf bifurcation and chaos of a neuron model with delay-dependent parameters

NASA Astrophysics Data System (ADS)

Xu, X.; Hu, H. Y.; Wang, H. L.

2006-05-01

It is very common that neural network systems usually involve time delays since the transmission of information between neurons is not instantaneous. Because memory intensity of the biological neuron usually depends on time history, some of the parameters may be delay dependent. Yet, little attention has been paid to the dynamics of such systems. In this Letter, a detailed analysis on the stability switches, Hopf bifurcation and chaos of a neuron model with delay-dependent parameters is given. Moreover, the direction and the stability of the bifurcating periodic solutions are obtained by the normal form theory and the center manifold theorem. It shows that the dynamics of the neuron model with delay-dependent parameters is quite different from that of systems with delay-independent parameters only.

Bifurcation theory and cardiac arrhythmias

PubMed Central

Karagueuzian, Hrayr S; Stepanyan, Hayk; Mandel, William J

2013-01-01

In this paper we review two types of dynamic behaviors defined by the bifurcation theory that are found to be particularly useful in describing two forms of cardiac electrical instabilities that are of considerable importance in cardiac arrhythmogenesis. The first is action potential duration (APD) alternans with an underlying dynamics consistent with the period doubling bifurcation theory. This form of electrical instability could lead to spatially discordant APD alternans leading to wavebreak and reentrant form of tachyarrhythmias. Factors that modulate the APD alternans are discussed. The second form of bifurcation of importance to cardiac arrhythmogenesis is the Hopf-homoclinic bifurcation that adequately describes the dynamics of the onset of early afterdepolarization (EAD)-mediated triggered activity (Hopf) that may cause ventricular tachycardia and ventricular fibrillation (VT/VF respectively). The self-termination of the triggered activity is compatible with the homoclinic bifurcation. Ionic and intracellular calcium dynamics underlying these dynamics are discussed using available experimental and simulation data. The dynamic analysis provides novel insights into the mechanisms of VT/VF, a major cause of sudden cardiac death in the US. PMID:23459417

Effects of additional food in a delayed predator-prey model.

PubMed

Sahoo, Banshidhar; Poria, Swarup

2015-03-01

We examine the effects of supplying additional food to predator in a gestation delay induced predator-prey system with habitat complexity. Additional food works in favor of predator growth in our model. Presence of additional food reduces the predatory attack rate to prey in the model. Supplying additional food we can control predator population. Taking time delay as bifurcation parameter the stability of the coexisting equilibrium point is analyzed. Hopf bifurcation analysis is done with respect to time delay in presence of additional food. The direction of Hopf bifurcations and the stability of bifurcated periodic solutions are determined by applying the normal form theory and the center manifold theorem. The qualitative dynamical behavior of the model is simulated using experimental parameter values. It is observed that fluctuations of the population size can be controlled either by supplying additional food suitably or by increasing the degree of habitat complexity. It is pointed out that Hopf bifurcation occurs in the system when the delay crosses some critical value. This critical value of delay strongly depends on quality and quantity of supplied additional food. Therefore, the variation of predator population significantly effects the dynamics of the model. Model results are compared with experimental results and biological implications of the analytical findings are discussed in the conclusion section. Copyright © 2015 Elsevier Inc. All rights reserved.

Symmetry-breaking Hopf bifurcations to 1-, 2-, and 3-tori in small-aspect-ratio counterrotating Taylor-Couette flow.

PubMed

Altmeyer, S; Do, Y; Marques, F; Lopez, J M

2012-10-01

The nonlinear dynamics of Taylor-Couette flow in a small-aspect-ratio wide-gap annulus in the counterrotating regime is investigated by solving the full three-dimensional Navier-Stokes equations. The system is invariant under arbitrary rotations about the axis, reflection about the annulus midplane, and time translations. A systematic investigation is presented both in terms of the flow physics elucidated from the numerical simulations and from a dynamical system perspective provided by equivariant normal form theory. The dynamics are primarily associated with the behavior of the jet of angular momentum that emerges from the inner cylinder boundary layer at about the midplane. The sequence of bifurcations as the differential rotation is increased consists of an axisymmetric Hopf bifurcation breaking the reflection symmetry of the basic state leading to an axisymmetric limit cycle with a half-period-flip spatiotemporal symmetry. This undergoes a Hopf bifurcation breaking axisymmetry, leading to quasiperiodic solutions evolving on a 2-torus that is setwise symmetric. These undergo a further Hopf bifurcation, introducing a third incommensurate frequency leading to a 3-torus that is also setwise symmetric. On the 3-torus, as the differential rotation is further increased, a saddle-node-invariant-circle bifurcation takes place, destroying the 3-torus and leaving a pair of symmetrically related 2-tori states on which all symmetries of the system have been broken.

Effects of stressor characteristics on early warning signs of critical transitions and "critical coupling" in complex dynamical systems.

PubMed

Blume, Steffen O P; Sansavini, Giovanni

2017-12-01

Complex dynamical systems face abrupt transitions into unstable and catastrophic regimes. These critical transitions are triggered by gradual modifications in stressors, which push the dynamical system towards unstable regimes. Bifurcation analysis can characterize such critical thresholds, beyond which systems become unstable. Moreover, the stochasticity of the external stressors causes small-scale fluctuations in the system response. In some systems, the decomposition of these signal fluctuations into precursor signals can reveal early warning signs prior to the critical transition. Here, we present a dynamical analysis of a power system subjected to an increasing load level and small-scale stochastic load perturbations. We show that the auto- and cross-correlations of bus voltage magnitudes increase, leading up to a Hopf bifurcation point, and further grow until the system collapses. This evidences a gradual transition into a state of "critical coupling," which is complementary to the established concept of "critical slowing down." Furthermore, we analyze the effects of the type of load perturbation and load characteristics on early warning signs and find that gradient changes in the autocorrelation provide early warning signs of the imminent critical transition under white-noise but not for auto-correlated load perturbations. Furthermore, the cross-correlation between all voltage magnitude pairs generally increases prior to and beyond the Hopf bifurcation point, indicating "critical coupling," but cannot provide early warning indications. Finally, we show that the established early warning indicators are oblivious to limit-induced bifurcations and, in the case of the power system model considered here, only react to an approaching Hopf bifurcation.

Bifurcations of a periodically forced microbial continuous culture model with restrained growth rate

NASA Astrophysics Data System (ADS)

Ren, Jingli; Yuan, Qigang

2017-08-01

A three dimensional microbial continuous culture model with a restrained microbial growth rate is studied in this paper. Two types of dilution rates are considered to investigate the dynamic behaviors of the model. For the unforced system, fold bifurcation and Hopf bifurcation are detected, and numerical simulations reveal that the system undergoes degenerate Hopf bifurcation. When the system is periodically forced, bifurcation diagrams for periodic solutions of period-one and period-two are given by researching the Poincaré map, corresponding to different bifurcation cases in the unforced system. Stable and unstable quasiperiodic solutions are obtained by Neimark-Sacker bifurcation with different parameter values. Periodic solutions of various periods can occur or disappear and even change their stability, when the Poincaré map of the forced system undergoes Neimark-Sacker bifurcation, flip bifurcation, and fold bifurcation. Chaotic attractors generated by a cascade of period doublings and some phase portraits are given at last.

Dynamics analysis of a predator-prey system with harvesting prey and disease in prey species.

PubMed

Meng, Xin-You; Qin, Ni-Ni; Huo, Hai-Feng

2018-12-01

In this paper, a predator-prey system with harvesting prey and disease in prey species is given. In the absence of time delay, the existence and stability of all equilibria are investigated. In the presence of time delay, some sufficient conditions of the local stability of the positive equilibrium and the existence of Hopf bifurcation are obtained by analysing the corresponding characteristic equation, and the properties of Hopf bifurcation are given by using the normal form theory and centre manifold theorem. Furthermore, an optimal harvesting policy is investigated by applying the Pontryagin's Maximum Principle. Numerical simulations are performed to support our analytic results.

Anticontrol of Hopf bifurcation and control of chaos for a finance system through washout filters with time delay.

PubMed

Zhao, Huitao; Lu, Mengxia; Zuo, Junmei

2014-01-01

A controlled model for a financial system through washout-filter-aided dynamical feedback control laws is developed, the problem of anticontrol of Hopf bifurcation from the steady state is studied, and the existence, stability, and direction of bifurcated periodic solutions are discussed in detail. The obtained results show that the delay on price index has great influences on the financial system, which can be applied to suppress or avoid the chaos phenomenon appearing in the financial system.

Bifurcation analysis of a discrete SIS model with bilinear incidence depending on new infection.

PubMed

Cao, Hui; Zhou, Yicang; Ma, Zhien

2013-01-01

A discrete SIS epidemic model with the bilinear incidence depending on the new infection is formulated and studied. The condition for the global stability of the disease free equilibrium is obtained. The existence of the endemic equilibrium and its stability are investigated. More attention is paid to the existence of the saddle-node bifurcation, the flip bifurcation, and the Hopf bifurcation. Sufficient conditions for those bifurcations have been obtained. Numerical simulations are conducted to demonstrate our theoretical results and the complexity of the model.

Modelling the fear effect in predator-prey interactions.

PubMed

Wang, Xiaoying; Zanette, Liana; Zou, Xingfu

2016-11-01

A recent field manipulation on a terrestrial vertebrate showed that the fear of predators alone altered anti-predator defences to such an extent that it greatly reduced the reproduction of prey. Because fear can evidently affect the populations of terrestrial vertebrates, we proposed a predator-prey model incorporating the cost of fear into prey reproduction. Our mathematical analyses show that high levels of fear (or equivalently strong anti-predator responses) can stabilize the predator-prey system by excluding the existence of periodic solutions. However, relatively low levels of fear can induce multiple limit cycles via subcritical Hopf bifurcations, leading to a bi-stability phenomenon. Compared to classic predator-prey models which ignore the cost of fear where Hopf bifurcations are typically supercritical, Hopf bifurcations in our model can be both supercritical and subcritical by choosing different sets of parameters. We conducted numerical simulations to explore the relationships between fear effects and other biologically related parameters (e.g. birth/death rate of adult prey), which further demonstrate the impact that fear can have in predator-prey interactions. For example, we found that under the conditions of a Hopf bifurcation, an increase in the level of fear may alter the direction of Hopf bifurcation from supercritical to subcritical when the birth rate of prey increases accordingly. Our simulations also show that the prey is less sensitive in perceiving predation risk with increasing birth rate of prey or increasing death rate of predators, but demonstrate that animals will mount stronger anti-predator defences as the attack rate of predators increases.

A dynamic IS-LM business cycle model with two time delays in capital accumulation equation

NASA Astrophysics Data System (ADS)

Zhou, Lujun; Li, Yaqiong

2009-06-01

In this paper, we analyze a augmented IS-LM business cycle model with the capital accumulation equation that two time delays are considered in investment processes according to Kalecki's idea. Applying stability switch criteria and Hopf bifurcation theory, we prove that time delays cause the equilibrium to lose or gain stability and Hopf bifurcation occurs.

Stability switches and multistability coexistence in a delay-coupled neural oscillators system.

PubMed

Song, Zigen; Xu, Jian

2012-11-21

In this paper, we present a neural network system composed of two delay-coupled neural oscillators, where each of these can be regarded as the dynamical system describing the average activity of neural population. Analyzing the corresponding characteristic equation, the local stability of rest state is studied. The system exhibits the switch phenomenon between the rest state and periodic activity. Furthermore, the Hopf bifurcation is analyzed and the bifurcation curve is given in the parameters plane. The stability of the bifurcating periodic solutions and direction of the Hopf bifurcation are exhibited. Regarding time delay and coupled weight as the bifurcation parameters, the Fold-Hopf bifurcation is investigated in detail in terms of the central manifold reduction and normal form method. The neural system demonstrates the coexistence of the rest states and periodic activities in the different parameter regions. Employing the normal form of the original system, the coexistence regions are illustrated approximately near the Fold-Hopf singularity point. Finally, numerical simulations are performed to display more complex dynamics. The results illustrate that system may exhibit the rich coexistence of the different neuro-computational properties, such as the rest states, periodic activities, and quasi-periodic behavior. In particular, some periodic activities can evolve into the bursting-type behaviors with the varying time delay. It implies that the coexistence of the quasi-periodic activity and bursting-type behavior can be obtained if the suitable value of system parameter is chosen. Copyright © 2012 Elsevier Ltd. All rights reserved.

Symmetry, Hopf bifurcation, and the emergence of cluster solutions in time delayed neural networks.

PubMed

Wang, Zhen; Campbell, Sue Ann

2017-11-01

We consider the networks of N identical oscillators with time delayed, global circulant coupling, modeled by a system of delay differential equations with Z N symmetry. We first study the existence of Hopf bifurcations induced by the coupling time delay and then use symmetric Hopf bifurcation theory to determine how these bifurcations lead to different patterns of symmetric cluster oscillations. We apply our results to a case study: a network of FitzHugh-Nagumo neurons with diffusive coupling. For this model, we derive the asymptotic stability, global asymptotic stability, absolute instability, and stability switches of the equilibrium point in the plane of coupling time delay (τ) and excitability parameter (a). We investigate the patterns of cluster oscillations induced by the time delay and determine the direction and stability of the bifurcating periodic orbits by employing the multiple timescales method and normal form theory. We find that in the region where stability switching occurs, the dynamics of the system can be switched from the equilibrium point to any symmetric cluster oscillation, and back to equilibrium point as the time delay is increased.

Negative Feedback Mediated by Fast Inhibitory Autapse Enhances Neuronal Oscillations Near a Hopf Bifurcation Point

NASA Astrophysics Data System (ADS)

Jia, Bing

One-parameter and two-parameter bifurcations of the Morris-Lecar (ML) neuron model with and without the fast inhibitory autapse, which is a synapse from a neuron onto itself, are investigated. The ML neuron model without autapse manifests an inverse Hopf bifurcation point from firing to a depolarized resting state with high level of membrane potential, with increasing depolarization current. When a fast inhibitory autapse is introduced, a negative feedback or inhibitory current is applied to the ML model. With increasing conductance of the autapse to middle level, the depolarized resting state near the inverse Hopf bifurcation point can change to oscillation and the parameter region of the oscillation becomes wide, which can be well interpreted by the dynamic responses of the depolarized resting state to the inhibitory current stimulus mediated by the autapse. The enlargement of the parameter region of the oscillation induced by the negative feedback presents a novel viewpoint different from the traditional one that inhibitory synapse often suppresses the neuronal oscillation activities. Furthermore, complex nonlinear dynamics such as the coexisting behaviors and codimension-2 bifurcations including the Bautin and cusp bifurcations are acquired. The relationship between the bifurcations and the depolarization block, a physiological concept that indicates a neuron can enter resting state when receiving the depolarization current, is discussed.

A modified Leslie-Gower predator-prey interaction model and parameter identifiability

NASA Astrophysics Data System (ADS)

Tripathi, Jai Prakash; Meghwani, Suraj S.; Thakur, Manoj; Abbas, Syed

2018-01-01

In this work, bifurcation and a systematic approach for estimation of identifiable parameters of a modified Leslie-Gower predator-prey system with Crowley-Martin functional response and prey refuge is discussed. Global asymptotic stability is discussed by applying fluctuation lemma. The system undergoes into Hopf bifurcation with respect to parameters intrinsic growth rate of predators (s) and prey reserve (m). The stability of Hopf bifurcation is also discussed by calculating Lyapunov number. The sensitivity analysis of the considered model system with respect to all variables is performed which also supports our theoretical study. To estimate the unknown parameter from the data, an optimization procedure (pseudo-random search algorithm) is adopted. System responses and phase plots for estimated parameters are also compared with true noise free data. It is found that the system dynamics with true set of parametric values is similar to the estimated parametric values. Numerical simulations are presented to substantiate the analytical findings.

Bifurcation Analysis and Optimal Harvesting of a Delayed Predator-Prey Model

NASA Astrophysics Data System (ADS)

Tchinda Mouofo, P.; Djidjou Demasse, R.; Tewa, J. J.; Aziz-Alaoui, M. A.

A delay predator-prey model is formulated with continuous threshold prey harvesting and Holling response function of type III. Global qualitative and bifurcation analyses are combined to determine the global dynamics of the model. The positive invariance of the non-negative orthant is proved and the uniform boundedness of the trajectories. Stability of equilibria is investigated and the existence of some local bifurcations is established: saddle-node bifurcation, Hopf bifurcation. We use optimal control theory to provide the correct approach to natural resource management. Results are also obtained for optimal harvesting. Numerical simulations are given to illustrate the results.

Hopf bifurcation of an (n + 1) -neuron bidirectional associative memory neural network model with delays.

PubMed

Xiao, Min; Zheng, Wei Xing; Cao, Jinde

2013-01-01

Recent studies on Hopf bifurcations of neural networks with delays are confined to simplified neural network models consisting of only two, three, four, five, or six neurons. It is well known that neural networks are complex and large-scale nonlinear dynamical systems, so the dynamics of the delayed neural networks are very rich and complicated. Although discussing the dynamics of networks with a few neurons may help us to understand large-scale networks, there are inevitably some complicated problems that may be overlooked if simplified networks are carried over to large-scale networks. In this paper, a general delayed bidirectional associative memory neural network model with n + 1 neurons is considered. By analyzing the associated characteristic equation, the local stability of the trivial steady state is examined, and then the existence of the Hopf bifurcation at the trivial steady state is established. By applying the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction and stability of the bifurcating periodic solution. Furthermore, the paper highlights situations where the Hopf bifurcations are particularly critical, in the sense that the amplitude and the period of oscillations are very sensitive to errors due to tolerances in the implementation of neuron interconnections. It is shown that the sensitivity is crucially dependent on the delay and also significantly influenced by the feature of the number of neurons. Numerical simulations are carried out to illustrate the main results.

Hopf-Pitchfork Bifurcation in a Symmetrically Conservative Two-Mass System with Delay

NASA Astrophysics Data System (ADS)

Sun, Ye; Zhang, Chunrui; Cai, Yuting

2018-06-01

A symmetrically conservative two-mass system with time delay is considered here. We analyse the influence of interaction coefficient and time delay on the Hopf-pitchfork bifurcation. The bifurcation diagrams and phase portraits are then obtained by computing the normal forms for the system in which, particularly, the unfolding form for case III is seldom given in delayed differential equations. Furthermore, we also find some interesting dynamical behaviours of the original system, such as the coexistence of two stable non-trivial equilibria and a pair of stable periodic orbits, which are verified both theoretically and numerically.

Third-Order Memristive Morris-Lecar Model of Barnacle Muscle Fiber

NASA Astrophysics Data System (ADS)

Rajamani, Vetriveeran; Sah, Maheshwar Pd.; Mannan, Zubaer Ibna; Kim, Hyongsuk; Chua, Leon

This paper presents a detailed analysis of various oscillatory behaviors observed in relation to the calcium and potassium ions in the third-order Morris-Lecar model of giant barnacle muscle fiber. Since, both the calcium and potassium ions exhibit all of the characteristics of memristor fingerprints, we claim that the time-varying calcium and potassium ions in the third-order Morris-Lecar model are actually time-invariant calcium and potassium memristors in the third-order memristive Morris-Lecar model. We confirmed the existence of a small unstable limit cycle oscillation in both the second-order and the third-order Morris-Lecar model by numerically calculating the basin of attraction of the asymptotically stable equilibrium point associated with two subcritical Hopf bifurcation points. We also describe a comprehensive analysis of the generation of oscillations in third-order memristive Morris-Lecar model via small-signal circuit analysis and a subcritical Hopf bifurcation phenomenon.

Bifurcation analysis of a delay reaction-diffusion malware propagation model with feedback control

NASA Astrophysics Data System (ADS)

Zhu, Linhe; Zhao, Hongyong; Wang, Xiaoming

2015-05-01

With the rapid development of network information technology, information networks security has become a very critical issue in our work and daily life. This paper attempts to develop a delay reaction-diffusion model with a state feedback controller to describe the process of malware propagation in mobile wireless sensor networks (MWSNs). By analyzing the stability and Hopf bifurcation, we show that the state feedback method can successfully be used to control unstable steady states or periodic oscillations. Moreover, formulas for determining the properties of the bifurcating periodic oscillations are derived by applying the normal form method and center manifold theorem. Finally, we conduct extensive simulations on large-scale MWSNs to evaluate the proposed model. Numerical evidences show that the linear term of the controller is enough to delay the onset of the Hopf bifurcation and the properties of the bifurcation can be regulated to achieve some desirable behaviors by choosing the appropriate higher terms of the controller. Furthermore, we obtain that the spatial-temporal dynamic characteristics of malware propagation are closely related to the rate constant for nodes leaving the infective class for recovered class and the mobile behavior of nodes.

The relationship between two fast/slow analysis techniques for bursting oscillations

PubMed Central

Teka, Wondimu; Tabak, Joël; Bertram, Richard

2012-01-01

Bursting oscillations in excitable systems reflect multi-timescale dynamics. These oscillations have often been studied in mathematical models by splitting the equations into fast and slow subsystems. Typically, one treats the slow variables as parameters of the fast subsystem and studies the bifurcation structure of this subsystem. This has key features such as a z-curve (stationary branch) and a Hopf bifurcation that gives rise to a branch of periodic spiking solutions. In models of bursting in pituitary cells, we have recently used a different approach that focuses on the dynamics of the slow subsystem. Characteristic features of this approach are folded node singularities and a critical manifold. In this article, we investigate the relationships between the key structures of the two analysis techniques. We find that the z-curve and Hopf bifurcation of the two-fast/one-slow decomposition are closely related to the voltage nullcline and folded node singularity of the one-fast/two-slow decomposition, respectively. They become identical in the double singular limit in which voltage is infinitely fast and calcium is infinitely slow. PMID:23278052

Spatiotemporal Symmetry in Rings of Coupled Biological Oscillators of Physarum Plasmodial Slime Mold

NASA Astrophysics Data System (ADS)

Takamatsu, Atsuko; Tanaka, Reiko; Yamada, Hiroyasu; Nakagaki, Toshiyuki; Fujii, Teruo; Endo, Isao

2001-08-01

Spatiotemporal patterns in rings of coupled biological oscillators of the plasmodial slime mold, Physarum polycephalum, were investigated by comparing with results analyzed by the symmetric Hopf bifurcation theory based on group theory. In three-, four-, and five-oscillator systems, all types of oscillation modes predicted by the theory were observed including a novel oscillation mode, a half period oscillation, which has not been reported anywhere in practical systems. Our results support the effectiveness of the symmetric Hopf bifurcation theory in practical systems.

Spatiotemporal symmetry in rings of coupled biological oscillators of Physarum plasmodial slime mold.

PubMed

Takamatsu, A; Tanaka, R; Yamada, H; Nakagaki, T; Fujii, T; Endo, I

2001-08-13

Spatiotemporal patterns in rings of coupled biological oscillators of the plasmodial slime mold, Physarum polycephalum, were investigated by comparing with results analyzed by the symmetric Hopf bifurcation theory based on group theory. In three-, four-, and five-oscillator systems, all types of oscillation modes predicted by the theory were observed including a novel oscillation mode, a half period oscillation, which has not been reported anywhere in practical systems. Our results support the effectiveness of the symmetric Hopf bifurcation theory in practical systems.

Bifurcation and Stability in a Delayed Predator-Prey Model with Mixed Functional Responses

NASA Astrophysics Data System (ADS)

Yafia, R.; Aziz-Alaoui, M. A.; Merdan, H.; Tewa, J. J.

2015-06-01

The model analyzed in this paper is based on the model set forth by Aziz Alaoui et al. [Aziz Alaoui & Daher Okiye, 2003; Nindjin et al., 2006] with time delay, which describes the competition between the predator and prey. This model incorporates a modified version of the Leslie-Gower functional response as well as that of Beddington-DeAngelis. In this paper, we consider the model with one delay consisting of a unique nontrivial equilibrium E* and three others which are trivial. Their dynamics are studied in terms of local and global stabilities and of the description of Hopf bifurcation at E*. At the third trivial equilibrium, the existence of the Hopf bifurcation is proven as the delay (taken as a parameter of bifurcation) that crosses some critical values.

Effects of time delays on stability and Hopf bifurcation in a fractional ring-structured network with arbitrary neurons

NASA Astrophysics Data System (ADS)

Huang, Chengdai; Cao, Jinde; Xiao, Min; Alsaedi, Ahmed; Hayat, Tasawar

2018-04-01

This paper is comprehensively concerned with the dynamics of a class of high-dimension fractional ring-structured neural networks with multiple time delays. Based on the associated characteristic equation, the sum of time delays is regarded as the bifurcation parameter, and some explicit conditions for describing delay-dependent stability and emergence of Hopf bifurcation of such networks are derived. It reveals that the stability and bifurcation heavily relies on the sum of time delays for the proposed networks, and the stability performance of such networks can be markedly improved by selecting carefully the sum of time delays. Moreover, it is further displayed that both the order and the number of neurons can extremely influence the stability and bifurcation of such networks. The obtained criteria enormously generalize and improve the existing work. Finally, numerical examples are presented to verify the efficiency of the theoretical results.

The Dynamical Behaviors for a Class of Immunogenic Tumor Model with Delay

PubMed Central

Muthoni, Mutei Damaris; Pang, Jianhua

2017-01-01

This paper aims at studying the model proposed by Kuznetsov and Taylor in 1994. Inspired by Mayer et al., time delay is introduced in the general model. The dynamic behaviors of this model are studied, which include the existence and stability of the equilibria and Hopf bifurcation of the model with discrete delays. The properties of the bifurcated periodic solutions are studied by using the normal form on the center manifold. Numerical examples and simulations are given to illustrate the bifurcation analysis and the obtained results. PMID:29312457

A heterogenous Cournot duopoly with delay dynamics: Hopf bifurcations and stability switching curves

NASA Astrophysics Data System (ADS)

Pecora, Nicolò; Sodini, Mauro

2018-05-01

This article considers a Cournot duopoly model in a continuous-time framework and analyze its dynamic behavior when the competitors are heterogeneous in determining their output decision. Specifically the model is expressed in the form of differential equations with discrete delays. The stability conditions of the unique Nash equilibrium of the system are determined and the emergence of Hopf bifurcations is shown. Applying some recent mathematical techniques (stability switching curves) and performing numerical simulations, the paper confirms how different time delays affect the stability of the economy.

Defect chaos and bursts: hexagonal rotating convection and the complex Ginzburg-Landau equation.

PubMed

Madruga, Santiago; Riecke, Hermann; Pesch, Werner

2006-02-24

We employ numerical computations of the full Navier-Stokes equations to investigate non-Boussinesq convection in a rotating system using water as the working fluid. We identify two regimes. For weak non-Boussinesq effects the Hopf bifurcation from steady to oscillating (whirling) hexagons is supercritical and typical states exhibit defect chaos that is systematically described by the cubic complex Ginzburg-Landau equation. For stronger non-Boussinesq effects the Hopf bifurcation becomes subcritical and the oscillations exhibit localized chaotic bursting, which is modeled by a quintic complex Ginzburg-Landau equation.

Stratified rotating Boussinesq equations in geophysical fluid dynamics: Dynamic bifurcation and periodic solutions

NASA Astrophysics Data System (ADS)

Hsia, Chun-Hsiung; Ma, Tian; Wang, Shouhong

2007-06-01

The main objective of this article is to study the dynamics of the stratified rotating Boussinesq equations, which are a basic model in geophysical fluid dynamics. First, for the case where the Prandtl number is greater than 1, a complete stability and bifurcation analysis near the first critical Rayleigh number is carried out. Second, for the case where the Prandtl number is smaller than 1, the onset of the Hopf bifurcation near the first critical Rayleigh number is established, leading to the existence of nontrivial periodic solutions. The analysis is based on a newly developed bifurcation and stability theory for nonlinear dynamical systems (both finite and infinite dimensional) by two of the authors [T. Ma and S. Wang, Bifurcation Theory and Applications, World Scientific Series on Nonlinear Sciences Vol. 53 (World Scientific, Singapore, 2005)].

Bifurcation and Firing Patterns of the Pancreatic β-Cell

NASA Astrophysics Data System (ADS)

Wang, Jing; Liu, Shenquan; Liu, Xuanliang; Zeng, Yanjun

Using a model of individual isolated pancreatic β-cells, we investigated bifurcation diagrams of interspike intervals (ISIs) and largest Lyapunov exponents (LLE), which clearly demonstrated a wide range of transitions between different firing patterns. The numerical simulation results revealed the effect of different time constants and ion channels on the neuronal discharge rhythm. Furthermore, an individual cell exhibited tonic spiking, square-wave bursting, and tapered bursting. Additionally, several bifurcation phenomena can be observed in this paper, such as period-doubling, period-adding, inverse period-doubling and inverse period-adding scenarios. In addition, we researched the mechanisms underlying two kinds of bursting (tapered and square-wave bursting) by use of fast-slow dynamics analysis. Finally, we analyzed the codimension-two bifurcation of the fast subsystem and studied cusp bifurcation, generalized Hopf (or Bautin) bifurcation and Bogdanov-Takens bifurcation.

A model of a fishery with fish stock involving delay equations.

PubMed

Auger, P; Ducrot, Arnaud

2009-12-13

The aim of this paper is to provide a new mathematical model for a fishery by including a stock variable for the resource. This model takes the form of an infinite delay differential equation. It is mathematically studied and a bifurcation analysis of the steady states is fulfilled. Depending on the different parameters of the problem, we show that Hopf bifurcation may occur leading to oscillating behaviours of the system. The mathematical results are finally discussed.

Stability and bifurcation for an SEIS epidemic model with the impact of media

NASA Astrophysics Data System (ADS)

Huo, Hai-Feng; Yang, Peng; Xiang, Hong

2018-01-01

A novel SEIS epidemic model with the impact of media is introduced. By analyzing the characteristic equation of equilibrium, the basic reproduction number is obtained and the stability of the steady states is proved. The occurrence of a forward, backward and Hopf bifurcation is derived. Numerical simulations and sensitivity analysis are performed. Our results manifest that media can regard as a good indicator in controlling the emergence and spread of the epidemic disease.

Tangential acceleration feedback control of friction induced vibration

NASA Astrophysics Data System (ADS)

Nath, Jyayasi; Chatterjee, S.

2016-09-01

Tangential control action is studied on a phenomenological mass-on-belt model exhibiting friction-induced self-excited vibration attributed to the low-velocity drooping characteristics of friction which is also known as Stribeck effect. The friction phenomenon is modelled by the exponential model. Linear stability analysis is carried out near the equilibrium point and local stability boundary is delineated in the plane of control parameters. The system is observed to undergo a Hopf bifurcation as the eigenvalues determined from the linear stability analysis are found to cross the imaginary axis transversally from RHS s-plane to LHS s-plane or vice-versa as one varies the control parameters, namely non-dimensional belt velocity and the control gain. A nonlinear stability analysis by the method of Averaging reveals the subcritical nature of the Hopf bifurcation. Thus, a global stability boundary is constructed so that any choice of control parameters from the globally stable region leads to a stable equilibrium. Numerical simulations in a MATLAB SIMULINK model and bifurcation diagrams obtained in AUTO validate these analytically obtained results. Pole crossover design is implemented to optimize the filter parameters with an independent choice of belt velocity and control gain. The efficacy of this optimization (based on numerical results) in the delicate low velocity region is also enclosed.

Signal Processing in Periodically Forced Gradient Frequency Neural Networks

PubMed Central

Kim, Ji Chul; Large, Edward W.

2015-01-01

Oscillatory instability at the Hopf bifurcation is a dynamical phenomenon that has been suggested to characterize active non-linear processes observed in the auditory system. Networks of oscillators poised near Hopf bifurcation points and tuned to tonotopically distributed frequencies have been used as models of auditory processing at various levels, but systematic investigation of the dynamical properties of such oscillatory networks is still lacking. Here we provide a dynamical systems analysis of a canonical model for gradient frequency neural networks driven by a periodic signal. We use linear stability analysis to identify various driven behaviors of canonical oscillators for all possible ranges of model and forcing parameters. The analysis shows that canonical oscillators exhibit qualitatively different sets of driven states and transitions for different regimes of model parameters. We classify the parameter regimes into four main categories based on their distinct signal processing capabilities. This analysis will lead to deeper understanding of the diverse behaviors of neural systems under periodic forcing and can inform the design of oscillatory network models of auditory signal processing. PMID:26733858

On the analysis of the double Hopf bifurcation in machining processes via centre manifold reduction

NASA Astrophysics Data System (ADS)

Molnar, T. G.; Dombovari, Z.; Insperger, T.; Stepan, G.

2017-11-01

The single-degree-of-freedom model of orthogonal cutting is investigated to study machine tool vibrations in the vicinity of a double Hopf bifurcation point. Centre manifold reduction and normal form calculations are performed to investigate the long-term dynamics of the cutting process. The normal form of the four-dimensional centre subsystem is derived analytically, and the possible topologies in the infinite-dimensional phase space of the system are revealed. It is shown that bistable parameter regions exist where unstable periodic and, in certain cases, unstable quasi-periodic motions coexist with the equilibrium. Taking into account the non-smoothness caused by loss of contact between the tool and the workpiece, the boundary of the bistable region is also derived analytically. The results are verified by numerical continuation. The possibility of (transient) chaotic motions in the global non-smooth dynamics is shown.

Critical Fluctuations in Cortical Models Near Instability

PubMed Central

Aburn, Matthew J.; Holmes, C. A.; Roberts, James A.; Boonstra, Tjeerd W.; Breakspear, Michael

2012-01-01

Computational studies often proceed from the premise that cortical dynamics operate in a linearly stable domain, where fluctuations dissipate quickly and show only short memory. Studies of human electroencephalography (EEG), however, have shown significant autocorrelation at time lags on the scale of minutes, indicating the need to consider regimes where non-linearities influence the dynamics. Statistical properties such as increased autocorrelation length, increased variance, power law scaling, and bistable switching have been suggested as generic indicators of the approach to bifurcation in non-linear dynamical systems. We study temporal fluctuations in a widely-employed computational model (the Jansen–Rit model) of cortical activity, examining the statistical signatures that accompany bifurcations. Approaching supercritical Hopf bifurcations through tuning of the background excitatory input, we find a dramatic increase in the autocorrelation length that depends sensitively on the direction in phase space of the input fluctuations and hence on which neuronal subpopulation is stochastically perturbed. Similar dependence on the input direction is found in the distribution of fluctuation size and duration, which show power law scaling that extends over four orders of magnitude at the Hopf bifurcation. We conjecture that the alignment in phase space between the input noise vector and the center manifold of the Hopf bifurcation is directly linked to these changes. These results are consistent with the possibility of statistical indicators of linear instability being detectable in real EEG time series. However, even in a simple cortical model, we find that these indicators may not necessarily be visible even when bifurcations are present because their expression can depend sensitively on the neuronal pathway of incoming fluctuations. PMID:22952464

Hopf bifurcation with dihedral group symmetry - Coupled nonlinear oscillators

NASA Technical Reports Server (NTRS)

Golubitsky, Martin; Stewart, Ian

1986-01-01

The theory of Hopf bifurcation with symmetry developed by Golubitsky and Stewart (1985) is applied to systems of ODEs having the symmetries of a regular polygon, that is, whose symmetry group is dihedral. The existence and stability of symmetry-breaking branches of periodic solutions are considered. In particular, these results are applied to a general system of n nonlinear oscillators coupled symmetrically in a ring, and the generic oscillation patterns are described. It is found that the symmetry can force some oscillators to have twice the frequency of others. The case of four oscillators has exceptional features.

Stability and Hopf Bifurcation in a Reaction-Diffusion Model with Chemotaxis and Nonlocal Delay Effect

NASA Astrophysics Data System (ADS)

Li, Dong; Guo, Shangjiang

Chemotaxis is an observed phenomenon in which a biological individual moves preferentially toward a relatively high concentration, which is contrary to the process of natural diffusion. In this paper, we study a reaction-diffusion model with chemotaxis and nonlocal delay effect under Dirichlet boundary condition by using Lyapunov-Schmidt reduction and the implicit function theorem. The existence, multiplicity, stability and Hopf bifurcation of spatially nonhomogeneous steady state solutions are investigated. Moreover, our results are illustrated by an application to the model with a logistic source, homogeneous kernel and one-dimensional spatial domain.

Stability and Hopf bifurcation of a delayed ratio-dependent predator-prey system

NASA Astrophysics Data System (ADS)

Wang, Wan-Yong; Pei, Li-Jun

2011-04-01

Since the ratio-dependent theory reflects the fact that predators must share and compete for food, it is suitable for describing the relationship between predators and their preys and has recently become a very important theory put forward by biologists. In order to investigate the dynamical relationship between predators and their preys, a so-called Michaelis-Menten ratio-dependent predator-prey model is studied in this paper with gestation time delays of predators and preys taken into consideration. The stability of the positive equilibrium is investigated by the Nyquist criteria, and the existence of the local Hopf bifurcation is analyzed by employing the theory of Hopf bifurcation. By means of the center manifold and the normal form theories, explicit formulae are derived to determine the stability, direction and other properties of bifurcating periodic solutions. The above theoretical results are validated by numerical simulations with the help of dynamical software WinPP. The results show that if both the gestation delays are small enough, their sizes will keep stable in the long run, but if the gestation delays of predators are big enough, their sizes will periodically fluctuate in the long term. In order to reveal the effects of time delays on the ratio-dependent predator-prey model, a ratio-dependent predator-prey model without time delays is considered. By Hurwitz criteria, the local stability of positive equilibrium of this model is investigated. The conditions under which the positive equilibrium is locally asymptotically stable are obtained. By comparing the results with those of the model with time delays, it shows that the dynamical behaviors of ratio-dependent predator-prey model with time delays are more complicated. Under the same conditions, namely, with the same parameters, the stability of positive equilibrium of ratio-dependent predator-prey model would change due to the introduction of gestation time delays for predators and preys. Moreover, with the variation of time delays, the positive equilibrium of the ratio-dependent predator-prey model subjects to Hopf bifurcation.

Dynamical investigation and parameter stability region analysis of a flywheel energy storage system in charging mode

NASA Astrophysics Data System (ADS)

Zhang, Wei-Ya; Li, Yong-Li; Chang, Xiao-Yong; Wang, Nan

2013-09-01

In this paper, the dynamic behavior analysis of the electromechanical coupling characteristics of a flywheel energy storage system (FESS) with a permanent magnet (PM) brushless direct-current (DC) motor (BLDCM) is studied. The Hopf bifurcation theory and nonlinear methods are used to investigate the generation process and mechanism of the coupled dynamic behavior for the average current controlled FESS in the charging mode. First, the universal nonlinear dynamic model of the FESS based on the BLDCM is derived. Then, for a 0.01 kWh/1.6 kW FESS platform in the Key Laboratory of the Smart Grid at Tianjin University, the phase trajectory of the FESS from a stable state towards chaos is presented using numerical and stroboscopic methods, and all dynamic behaviors of the system in this process are captured. The characteristics of the low-frequency oscillation and the mechanism of the Hopf bifurcation are investigated based on the Routh stability criterion and nonlinear dynamic theory. It is shown that the Hopf bifurcation is directly due to the loss of control over the inductor current, which is caused by the system control parameters exceeding certain ranges. This coupling nonlinear process of the FESS affects the stability of the motor running and the efficiency of energy transfer. In this paper, we investigate into the effects of control parameter change on the stability and the stability regions of these parameters based on the averaged-model approach. Furthermore, the effect of the quantization error in the digital control system is considered to modify the stability regions of the control parameters. Finally, these theoretical results are verified through platform experiments.

Turing-Hopf bifurcations in a predator-prey model with herd behavior, quadratic mortality and prey-taxis

NASA Astrophysics Data System (ADS)

Liu, Xia; Zhang, Tonghua; Meng, Xinzhu; Zhang, Tongqian

2018-04-01

In this paper, we propose a predator-prey model with herd behavior and prey-taxis. Then, we analyze the stability and bifurcation of the positive equilibrium of the model subject to the homogeneous Neumann boundary condition. By using an abstract bifurcation theory and taking prey-tactic sensitivity coefficient as the bifurcation parameter, we obtain a branch of stable nonconstant solutions bifurcating from the positive equilibrium. Our results show that prey-taxis can yield the occurrence of spatial patterns.

Transition to chaos of natural convection between two infinite differentially heated vertical plates

NASA Astrophysics Data System (ADS)

Gao, Zhenlan; Sergent, Anne; Podvin, Berengere; Xin, Shihe; Le Quéré, Patrick; Tuckerman, Laurette S.

2013-08-01

Natural convection of air between two infinite vertical differentially heated plates is studied analytically in two dimensions (2D) and numerically in two and three dimensions (3D) for Rayleigh numbers Ra up to 3 times the critical value Rac=5708. The first instability is a supercritical circle pitchfork bifurcation leading to steady 2D corotating rolls. A Ginzburg-Landau equation is derived analytically for the flow around this first bifurcation and compared with results from direct numerical simulation (DNS). In two dimensions, DNS shows that the rolls become unstable via a Hopf bifurcation. As Ra is further increased, the flow becomes quasiperiodic, and then temporally chaotic for a limited range of Rayleigh numbers, beyond which the flow returns to a steady state through a spatial modulation instability. In three dimensions, the rolls instead undergo another pitchfork bifurcation to 3D structures, which consist of transverse rolls connected by counter-rotating vorticity braids. The flow then becomes time dependent through a Hopf bifurcation, as exchanges of energy occur between the rolls and the braids. Chaotic behavior subsequently occurs through two competing mechanisms: a sequence of period-doubling bifurcations leading to intermittency or a spatial pattern modulation reminiscent of the Eckhaus instability.

Rank One Strange Attractors in Periodically Kicked Predator-Prey System with Time-Delay

NASA Astrophysics Data System (ADS)

Yang, Wenjie; Lin, Yiping; Dai, Yunxian; Zhao, Huitao

2016-06-01

This paper is devoted to the study of the problem of rank one strange attractor in a periodically kicked predator-prey system with time-delay. Our discussion is based on the theory of rank one maps formulated by Wang and Young. Firstly, we develop the rank one chaotic theory to delayed systems. It is shown that strange attractors occur when the delayed system undergoes a Hopf bifurcation and encounters an external periodic force. Then we use the theory to the periodically kicked predator-prey system with delay, deriving the conditions for Hopf bifurcation and rank one chaos along with the results of numerical simulations.

Stochastic mixed-mode oscillations in a three-species predator-prey model

NASA Astrophysics Data System (ADS)

Sadhu, Susmita; Kuehn, Christian

2018-03-01

The effect of demographic stochasticity, in the form of Gaussian white noise, in a predator-prey model with one fast and two slow variables is studied. We derive the stochastic differential equations (SDEs) from a discrete model. For suitable parameter values, the deterministic drift part of the model admits a folded node singularity and exhibits a singular Hopf bifurcation. We focus on the parameter regime near the Hopf bifurcation, where small amplitude oscillations exist as stable dynamics in the absence of noise. In this regime, the stochastic model admits noise-driven mixed-mode oscillations (MMOs), which capture the intermediate dynamics between two cycles of population outbreaks. We perform numerical simulations to calculate the distribution of the random number of small oscillations between successive spikes for varying noise intensities and distance to the Hopf bifurcation. We also study the effect of noise on a suitable Poincaré map. Finally, we prove that the stochastic model can be transformed into a normal form near the folded node, which can be linked to recent results on the interplay between deterministic and stochastic small amplitude oscillations. The normal form can also be used to study the parameter influence on the noise level near folded singularities.

Numerical results on noise-induced dynamics in the subthreshold regime for thermoacoustic systems

NASA Astrophysics Data System (ADS)

Gupta, Vikrant; Saurabh, Aditya; Paschereit, Christian Oliver; Kabiraj, Lipika

2017-03-01

Thermoacoustic instability is a serious issue in practical combustion systems. Such systems are inherently noisy, and hence the influence of noise on the dynamics of thermoacoustic instability is an aspect of practical importance. The present work is motivated by a recent report on the experimental observation of coherence resonance, or noise-induced coherence with a resonance-like dependence on the noise intensity as the system approaches the stability margin, for a prototypical premixed laminar flame combustor (Kabiraj et al., Phys. Rev. E, 4 (2015)). We numerically investigate representative thermoacoustic models for such noise-induced dynamics. Similar to the experiments, we study variation in system dynamics in response to variations in the noise intensity and in a critical control parameter as the systems approach their stability margins. The qualitative match identified between experimental results and observations in the representative models investigated here confirms that coherence resonance is a feature of thermoacoustic systems. We also extend the experimental results, which were limited to the case of subcritical Hopf bifurcation, to the case of supercritical Hopf bifurcation. We identify that the phenomenon has qualitative differences for the systems undergoing transition via subcritical and supercritical Hopf bifurcations. Two important practical implications are associated with the findings. Firstly, the increase in noise-induced coherence as the system approaches the onset of thermoacoustic instability can be considered as a precursor to the instability. Secondly, the dependence of noise-induced dynamics on the bifurcation type can be utilised to distinguish between subcritical and supercritical bifurcation prior to the onset of the instability.

Stability and bifurcation analysis on a ratio-dependent predator-prey model with time delay

NASA Astrophysics Data System (ADS)

Xu, Rui; Gan, Qintao; Ma, Zhien

2009-08-01

A ratio-dependent predator-prey model with time delay due to the gestation of the predator is investigated. By analyzing the corresponding characteristic equations, the local stability of a positive equilibrium and a semi-trivial boundary equilibrium is discussed, respectively. Further, it is proved that the system undergoes a Hopf bifurcation at the positive equilibrium. Using the normal form theory and the center manifold reduction, explicit formulae are derived to determine the direction of bifurcations and the stability and other properties of bifurcating periodic solutions. By means of an iteration technique, sufficient conditions are obtained for the global attractiveness of the positive equilibrium. By comparison arguments, the global stability of the semi-trivial equilibrium is also addressed. Numerical simulations are carried out to illustrate the main results.

Bifurcation analysis of nephron pressure and flow regulation

NASA Astrophysics Data System (ADS)

Barfred, Mikael; Mosekilde, Erik; Holstein-Rathlou, Niels-Henrik

1996-09-01

One- and two-dimensional continuation techniques are applied to study the bifurcation structure of a model of renal flow and pressure control. Integrating the main physiological mechanisms by which the individual nephron regulates the incoming blood flow, the model describes the interaction between the tubuloglomerular feedback and the response of the afferent arteriole. It is shown how a Hopf bifurcation leads the system to perform self-sustained oscillations if the feedback gain becomes sufficiently strong, and how a further increase of this parameter produces a folded structure of overlapping period-doubling cascades. Similar phenomena arise in response to increasing blood pressure. The numerical analyses are supported by existing experimental results on anesthetized rats.

Bursting patterns and mixed-mode oscillations in reduced Purkinje model

NASA Astrophysics Data System (ADS)

Zhan, Feibiao; Liu, Shenquan; Wang, Jing; Lu, Bo

2018-02-01

Bursting discharge is a ubiquitous behavior in neurons, and abundant bursting patterns imply many physiological information. There exists a closely potential link between bifurcation phenomenon and the number of spikes per burst as well as mixed-mode oscillations (MMOs). In this paper, we have mainly explored the dynamical behavior of the reduced Purkinje cell and the existence of MMOs. First, we adopted the codimension-one bifurcation to illustrate the generation mechanism of bursting in the reduced Purkinje cell model via slow-fast dynamics analysis and demonstrate the process of spike-adding. Furthermore, we have computed the first Lyapunov coefficient of Hopf bifurcation to determine whether it is subcritical or supercritical and depicted the diagrams of inter-spike intervals (ISIs) to examine the chaos. Moreover, the bifurcation diagram near the cusp point is obtained by making the codimension-two bifurcation analysis for the fast subsystem. Finally, we have a discussion on mixed-mode oscillations and it is further investigated using the characteristic index that is Devil’s staircase.

Stability and Bifurcation of a Fishery Model with Crowley-Martin Functional Response

NASA Astrophysics Data System (ADS)

Maiti, Atasi Patra; Dubey, B.

To understand the dynamics of a fishery system, a nonlinear mathematical model is proposed and analyzed. In an aquatic environment, we considered two populations: one is prey and another is predator. Here both the fish populations grow logistically and interaction between them is of Crowley-Martin type functional response. It is assumed that both the populations are harvested and the harvesting effort is assumed to be dynamical variable and tax is considered as a control variable. The existence of equilibrium points and their local stability are examined. The existence of Hopf-bifurcation, stability and direction of Hopf-bifurcation are also analyzed with the help of Center Manifold theorem and normal form theory. The global stability behavior of the positive equilibrium point is also discussed. In order to find the value of optimal tax, the optimal harvesting policy is used. To verify our analytical findings, an extensive numerical simulation is carried out for this model system.

The Effect of Symmetry on the Hydrodynamic Stability of and Bifurcation from Planar Shear Flows

DTIC Science & Technology

1990-12-01

Effect of Symmetry on the Hydrodynamic Stability of ant Bifurcation from Planar Shear Flows AFOSR-88-0196 6. AUTHOR(S) 61102F 2304/A4 Thomas J. Bridges 7...December 1990 The Effect of Symmetry on the Hydrodynamic Stability of and Bifurcation from Planar Shear Flows TIIhOMAS J. BIUDGES MATl EM ATIc(AL...spatial stabili’.y into the nonlinear regime and a theory for spa- tial Hopf bifurcation , spatial Floquet theory, wavelength doubling and spatially quasi

Convection in colloidal suspensions with particle-concentration-dependent viscosity.

PubMed

Glässl, M; Hilt, M; Zimmermann, W

2010-07-01

The onset of thermal convection in a horizontal layer of a colloidal suspension is investigated in terms of a continuum model for binary-fluid mixtures where the viscosity depends on the local concentration of colloidal particles. With an increasing difference between the viscosity at the warmer and the colder boundary the threshold of convection is reduced in the range of positive values of the separation ratio psi with the onset of stationary convection as well as in the range of negative values of psi with an oscillatory Hopf bifurcation. Additionally the convection rolls are shifted downwards with respect to the center of the horizontal layer for stationary convection psi>0 and upwards for the Hopf bifurcation (psi<0.

Transitions of stationary to pulsating solutions in the complex cubic-quintic Ginzburg-Landau equation under the influence of nonlinear gain and higher-order effects

NASA Astrophysics Data System (ADS)

Uzunov, Ivan M.; Georgiev, Zhivko D.; Arabadzhiev, Todor N.

2018-05-01

In this paper we study the transitions of stationary to pulsating solutions in the complex cubic-quintic Ginzburg-Landau equation (CCQGLE) under the influence of nonlinear gain, its saturation, and higher-order effects: self-steepening, third-order of dispersion, and intrapulse Raman scattering in the anomalous dispersion region. The variation method and the method of moments are applied in order to obtain the dynamic models with finite degrees of freedom for the description of stationary and pulsating solutions. Having applied the first model and its bifurcation analysis we have discovered the existence of families of subcritical Poincaré-Andronov-Hopf bifurcations due to the intrapulse Raman scattering, as well as some small nonlinear gain and the saturation of the nonlinear gain. A phenomenon of nonlinear stability has been studied and it has been shown that long living pulsating solutions with relatively small fluctuations of amplitude and frequencies exist at the bifurcation point. The numerical analysis of the second model has revealed the existence of Poincaré-Andronov-Hopf bifurcations of Raman dissipative soliton under the influence of the self-steepening effect and large nonlinear gain. All our theoretical predictions have been confirmed by the direct numerical solution of the full perturbed CCQGLE. The detailed comparison between the results obtained by both dynamic models and the direct numerical solution of the perturbed CCQGLE has proved the applicability of the proposed models in the investigation of the solutions of the perturbed CCQGLE.

Rich Global Dynamics in a Prey-Predator Model with Allee Effect and Density Dependent Death Rate of Predator

NASA Astrophysics Data System (ADS)

Sen, Moitri; Banerjee, Malay

In this work we have considered a prey-predator model with strong Allee effect in the prey growth function, Holling type-II functional response and density dependent death rate for predators. It presents a comprehensive study of the complete global dynamics for the considered system. Especially to see the effect of the density dependent death rate of predator on the system behavior, we have presented the two parametric bifurcation diagrams taking it as one of the bifurcation parameters. In course of that we have explored all possible local and global bifurcations that the system could undergo, namely the existence of transcritical bifurcation, saddle node bifurcation, cusp bifurcation, Hopf-bifurcation, Bogdanov-Takens bifurcation and Bautin bifurcation respectively.

Symmetry-breaking oscillations in membrane optomechanics

NASA Astrophysics Data System (ADS)

Wurl, C.; Alvermann, A.; Fehske, H.

2016-12-01

We study the classical dynamics of a membrane inside a cavity in the situation where this optomechanical system possesses a reflection symmetry. Symmetry breaking occurs through supercritical and subcritical pitchfork bifurcations of the static fixed-point solutions. Both bifurcations can be observed through variation of the laser-cavity detuning, which gives rise to a boomerang-like fixed-point pattern with hysteresis. The symmetry-breaking fixed points evolve into self-sustained oscillations when the laser intensity is increased. In addition to the analysis of the accompanying Hopf bifurcations we describe these oscillations at finite amplitudes with an ansatz that fully accounts for the frequency shift relative to the natural membrane frequency. We complete our study by following the route to chaos for the membrane dynamics.

Detecting malicious chaotic signals in wireless sensor network

NASA Astrophysics Data System (ADS)

Upadhyay, Ranjit Kumar; Kumari, Sangeeta

2018-02-01

In this paper, an e-epidemic Susceptible-Infected-Vaccinated (SIV) model has been proposed to analyze the effect of node immunization and worms attacking dynamics in wireless sensor network. A modified nonlinear incidence rate with cyrtoid type functional response has been considered using sleep and active mode approach. Detailed stability analysis and the sufficient criteria for the persistence of the model system have been established. We also established different types of bifurcation analysis for different equilibria at different critical points of the control parameters. We performed a detailed Hopf bifurcation analysis and determine the direction and stability of the bifurcating periodic solutions using center manifold theorem. Numerical simulations are carried out to confirm the theoretical results. The impact of the control parameters on the dynamics of the model system has been investigated and malicious chaotic signals are detected. Finally, we have analyzed the effect of time delay on the dynamics of the model system.

Breathing pulses in singularly perturbed reaction-diffusion systems

NASA Astrophysics Data System (ADS)

Veerman, Frits

2015-07-01

The weakly nonlinear stability of pulses in general singularly perturbed reaction-diffusion systems near a Hopf bifurcation is determined using a centre manifold expansion. A general framework to obtain leading order expressions for the (Hopf) centre manifold expansion for scale separated, localised structures is presented. Using the scale separated structure of the underlying pulse, directly calculable expressions for the Hopf normal form coefficients are obtained in terms of solutions to classical Sturm-Liouville problems. The developed theory is used to establish the existence of breathing pulses in a slowly nonlinear Gierer-Meinhardt system, and is confirmed by direct numerical simulation.

Local Bifurcations and Optimal Theory in a Delayed Predator-Prey Model with Threshold Prey Harvesting

NASA Astrophysics Data System (ADS)

Tankam, Israel; Tchinda Mouofo, Plaire; Mendy, Abdoulaye; Lam, Mountaga; Tewa, Jean Jules; Bowong, Samuel

2015-06-01

We investigate the effects of time delay and piecewise-linear threshold policy harvesting for a delayed predator-prey model. It is the first time that Holling response function of type III and the present threshold policy harvesting are associated with time delay. The trajectories of our delayed system are bounded; the stability of each equilibrium is analyzed with and without delay; there are local bifurcations as saddle-node bifurcation and Hopf bifurcation; optimal harvesting is also investigated. Numerical simulations are provided in order to illustrate each result.

The dynamics of phase locking and points of resonance in a forced magnetic oscillator

NASA Astrophysics Data System (ADS)

Bryant, Paul; Jeffries, Carson

1987-03-01

We report data on an experimental system: a forced symmetric oscillator containing a saturable inductor with magnetic hysteresis. It displays a Hopf bifurcation to quasiperiodicity, entrainment horns, and chaos. We study in detail the bifurcations and hysteresis occurring near points of resonance (particularly “ strong resonance”) and show how the observed behavior can be understood using Arnold's theory. Much of the behavior relating to the entrainment horns is explored: period doubling and symmetry breaking bifurcations; homoclinic bifurcations; and crises and other bifurcations taking place at the horn boundaries. Important features of the behavior related to symmetry properties of the oscillator are studied and explained through the concept of a half-cycle map. The system is shown to exhibit a Hopf bifurcation from a phase-locked state to periodic “islands”, similar to those found in Hamiltonian systems. An initialization technique is used to observe the manifolds of saddle orbits and other hidden structure. An unusual differential equation model is developed which is irreversible and generates a noninvertible Poincaré map of the plane. Noninvertibility of this planar map has important effects on the behavior observed. The Poincaré map may also be approximated through experimental measurements, resulting in a planar map with parameter dependence. This model gives good correspondence with the system in a region of the parameter space.

Synchronization properties of networks of electrically coupled neurons in the presence of noise and heterogeneities.

PubMed

Ostojic, Srdjan; Brunel, Nicolas; Hakim, Vincent

2009-06-01

We investigate how synchrony can be generated or induced in networks of electrically coupled integrate-and-fire neurons subject to noisy and heterogeneous inputs. Using analytical tools, we find that in a network under constant external inputs, synchrony can appear via a Hopf bifurcation from the asynchronous state to an oscillatory state. In a homogeneous net work, in the oscillatory state all neurons fire in synchrony, while in a heterogeneous network synchrony is looser, many neurons skipping cycles of the oscillation. If the transmission of action potentials via the electrical synapses is effectively excitatory, the Hopf bifurcation is supercritical, while effectively inhibitory transmission due to pronounced hyperpolarization leads to a subcritical bifurcation. In the latter case, the network exhibits bistability between an asynchronous state and an oscillatory state where all the neurons fire in synchrony. Finally we show that for time-varying external inputs, electrical coupling enhances the synchronization in an asynchronous network via a resonance at the firing-rate frequency.

Stability of soliton families in nonlinear Schrödinger equations with non-parity-time-symmetric complex potentials

NASA Astrophysics Data System (ADS)

Yang, Jianke; Nixon, Sean

2016-11-01

Stability of soliton families in one-dimensional nonlinear Schrödinger equations with non-parity-time (PT)-symmetric complex potentials is investigated numerically. It is shown that these solitons can be linearly stable in a wide range of parameter values both below and above phase transition. In addition, a pseudo-Hamiltonian-Hopf bifurcation is revealed, where pairs of purely-imaginary eigenvalues in the linear-stability spectra of solitons collide and bifurcate off the imaginary axis, creating oscillatory instability, which resembles Hamiltonian-Hopf bifurcations of solitons in Hamiltonian systems even though the present system is dissipative and non-Hamiltonian. The most important numerical finding is that, eigenvalues of linear-stability operators of these solitons appear in quartets (λ , - λ ,λ* , -λ*), similar to conservative systems and PT-symmetric systems. This quartet eigenvalue symmetry is very surprising for non- PT-symmetric systems, and it has far-reaching consequences on the stability behaviors of solitons.

Modeling and Analysis of a Fractional-Order Generalized Memristor-Based Chaotic System and Circuit Implementation

NASA Astrophysics Data System (ADS)

Yang, Ningning; Xu, Cheng; Wu, Chaojun; Jia, Rong; Liu, Chongxin

2017-12-01

Memristor is a nonlinear “missing circuit element”, that can easily achieve chaotic oscillation. Memristor-based chaotic systems have received more and more attention. Research shows that fractional-order systems are more close to real systems. As an important parameter, the order can increase the flexibility and degree of freedom of the system. In this paper, a fractional-order generalized memristor, which consists of a diode bridge and a parallel circuit with an equivalent unit circuit and a linear resistance, is proposed. Frequency and electrical characteristics of the fractional-order memristor are analyzed. A chain structure circuit is used to implement the fractional-order unit circuit. Then replacing the conventional Chua’s diode by the fractional-order generalized memristor, a fractional-order memristor-based chaotic circuit is proposed. A large amount of research work has been done to investigate the influence of the order on the dynamical behaviors of the fractional-order memristor-based chaotic circuit. Varying with the order, the system enters the chaotic state from the periodic state through the Hopf bifurcation and period-doubling bifurcation. The chaotic state of the system has two types of attractors: single-scroll and double-scroll attractor. The stability theory of fractional-order systems is used to determine the minimum order occurring Hopf bifurcation. And the influence of the initial value on the system is analyzed. Circuit simulations are designed to verify the results of theoretical analysis and numerical simulation.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Rizwan-uddin

Recently, various branches of engineering and science have seen a rapid increase in the number of dynamical analyses undertaken. This modern phenomenon often obscures the fact that such analyses were sometimes carried out even before the current trend began. Moreover, these earlier analyses, which even now seem very ingenuous, were carried out at a time when the available information about dynamical systems was not as well disseminated as it is today. One such analysis, carried out in the early 1960s, showed the existence of stable limit cycles in a simple model for space-independent xenon dynamics in nuclear reactors. The authors,more » apparently unaware of the now well-known bifurcation theorem by Hopf, could not numerically discover unstable limit cycles, though they did find regions in parameter space where the fixed points are stable for small perturbations but unstable for very large perturbations. The analysis was carried out both analytically and numerically. As a tribute to these early nonlinear dynamicists in the field of nuclear engineering, in this paper, the Hopf theorem and its conclusions are briefly described, and then the solution of the space-independent xenon oscillation problem is presented, which was obtained using the bifurcation analysis BIFDD code. These solutions are presented along with a discussion of the earlier results.« less

Convection in a nematic liquid crystal with homeotropic alignment and heated from below

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ahlers, G.

Experimental results for convection in a thin horizontal layer of a homeotropically aligned nematic liquid crystal heated from below and in a vertical magnetic field are presented. A subcritical Hopf bifurcation leads to the convecting state. There is quantitative agreement between the measured and the predicted bifurcation line as a function of magnetic field. The nonlinear state near the bifurcation is one of spatio-temporal chaos which seems to be the result of a zig-zag instability of the straight-roll state.

Bistability and delay-induced stability switches in a cancer network with the regulation of microRNA

NASA Astrophysics Data System (ADS)

Song, Yongli; Cao, Xin; Zhang, Tonghua

2018-01-01

In this paper, we are concerned with a cancer network including a protein module and a corresponding microRNA cluster that inhibits the synthesis of proteins. The existence of multiple steady states and their stability depending on the parameters are firstly determined. Bistability and dependency on the parameters, Hopf bifurcations and the corresponding properties like direction and stability of Hopf bifurcations are determined by computing the normal form on the center manifold. Then, the role of the delay in the process of synthesis of the protein is investigated. We show that the delay can stabilize the unstable equilibrium and destabilize the stable equilibrium. Some simulations are carried out to numerically illustrate the obtained theoretical results. Finally, the biological interpretation of the theoretical results is discussed.

The dynamics of a harvested predator-prey system with Holling type IV functional response.

PubMed

Liu, Xinxin; Huang, Qingdao

2018-05-31

The paper aims to investigate the dynamical behavior of a predator-prey system with Holling type IV functional response in which both the species are subject to capturing. We mainly consider how the harvesting affects equilibria, stability, limit cycles and bifurcations in this system. We adopt the method of qualitative and quantitative analysis, which is based on the dynamical theory, bifurcation theory and numerical simulation. The boundedness of solutions, the existence and stability of equilibrium points of the system are further studied. Based on the Sotomayor's theorem, the existence of transcritical bifurcation and saddle-node bifurcation are derived. We use the normal form theorem to analyze the Hopf bifurcation. Simulation results show that the first Lyapunov coefficient is negative and a stable limit cycle may bifurcate. Numerical simulations are performed to make analytical studies more complete. This work illustrates that using the harvesting effort as control parameter can change the behaviors of the system, which may be useful for the biological management. Copyright © 2018 Elsevier B.V. All rights reserved.

Analysis of the interaction among rice, weeds, inorganic fertilizer, and a herbivore in a composite farming paddy ecosystem.

PubMed

Wu, Zhaohua; Wang, Yi; Zhou, Xiaoli; Zhou, Tiejun

2018-06-01

As one of the Globally Important Agricultural Heritage Systems (GIAHS), rice field composite farming is an ecological measure in rice production, which can reduce the amount of chemical fertilizers, pesticides and herbicides. This research studies the interaction among rice, weed, inorganic fertilizer and herbivore in a composite farming paddy ecosystem. We develop a differential equation model to analyze the relations and interactions among those components. Results show the existence of an equilibrium for paddy and weed extinction, one or two equilibria for rice extinction, an equilibrium for weed extinction, and an equilibrium for rice and weed coexistence. Based on the obtained stability conditions of these equilibria, measures are proposed to avoid the existence or the stability of equilibria for rice extinction. Other measures are proposed to lead to a stable equilibrium for weed extinction, which is the most desirable result in rice production. Conditions for maximizing the yield of rice are also obtained by taking the relative mortality of rice as variable. In addition, we discover the existence of Hopf bifurcation phenomenon in the system, and develop the critical value of Hopf bifurcation by taking the artificial fertilizer rate as the bifurcation parameter. Our findings provide effective guidance and insights for rice production in a composite farming paddy ecosystem. Copyright © 2018 Elsevier Inc. All rights reserved.

Wavenumber distribution in Hopf-wave instability: the reversible Selkov model of glycolytic oscillation.

PubMed

Dutt, Arun K

2005-09-22

We have investigated the short-wave instability due to Hopf bifurcation in a reaction-diffusion model of glycolytic oscillations. Very low values of the ratio d of the diffusion coefficient of the inhibitor (ATP) and that of the activator (ADP) do help to create short waves, whereas high values of the ratio d and the complexing reaction of the activator ADP reduces drastically the wave-instability domain, generating much longer wavelengths.

Hybrid control of the Neimark-Sacker bifurcation in a delayed Nicholson's blowflies equation.

PubMed

Wang, Yuanyuan; Wang, Lisha

In this article, for delayed Nicholson's blowflies equation, we propose a hybrid control nonstandard finite-difference (NSFD) scheme in which state feedback and parameter perturbation are used to control the Neimark-Sacker bifurcation. Firstly, the local stability of the positive equilibria for hybrid control delay differential equation is discussed according to Hopf bifurcation theory. Then, for any step-size, a hybrid control numerical algorithm is introduced to generate the Neimark-Sacker bifurcation at a desired point. Finally, numerical simulation results confirm that the control strategy is efficient in controlling the Neimark-Sacker bifurcation. At the same time, the results show that the NSFD control scheme is better than the Euler control method.

Global asymptotic stability and hopf bifurcation for a blood cell production model.

PubMed

Crauste, Fabien

2006-04-01

We analyze the asymptotic stability of a nonlinear system of two differential equations with delay, describing the dynamics of blood cell produc- tion. This process takes place in the bone marrow, where stem cells differen- tiate throughout division in blood cells. Taking into account an explicit role of the total population of hematopoietic stem cells in the introduction of cells in cycle, we are led to study a characteristic equation with delay-dependent coefficients. We determine a necessary and sufficient condition for the global stability of the first steady state of our model, which describes the popula- tion's dying out, and we obtain the existence of a Hopf bifurcation for the only nontrivial positive steady state, leading to the existence of periodic solutions. These latter are related to dynamical diseases affecting blood cells known for their cyclic nature.

Stationary and oscillatory bound states of dissipative solitons created by third-order dispersion

NASA Astrophysics Data System (ADS)

Sakaguchi, Hidetsugu; Skryabin, Dmitry V.; Malomed, Boris A.

2018-06-01

We consider the model of fiber-laser cavities near the zero-dispersion point, based on the complex Ginzburg-Landau equation with the cubic-quintic nonlinearity, including the third-order dispersion (TOD) term. It is well known that this model supports stable dissipative solitons. We demonstrate that the same model gives rise to several families of robust bound states of the solitons, which exists only in the presence of the TOD. There are both stationary and dynamical bound states, with oscillating separation between the bound solitons. Stationary states are multistable, corresponding to different values of the separation. With the increase of the TOD coefficient, the bound state with the smallest separation gives rise the oscillatory state through the Hopf bifurcation. Further growth of TOD leads to a bifurcation transforming the oscillatory limit cycle into a strange attractor, which represents a chaotically oscillating dynamical bound state. Families of multistable three- and four-soliton complexes are found too, the ones with the smallest separation between the solitons again ending by a transition to oscillatory states through the Hopf bifurcation.

Analysis of precision in chemical oscillators: implications for circadian clocks

NASA Astrophysics Data System (ADS)

d'Eysmond, Thomas; De Simone, Alessandro; Naef, Felix

2013-10-01

Biochemical reaction networks often exhibit spontaneous self-sustained oscillations. An example is the circadian oscillator that lies at the heart of daily rhythms in behavior and physiology in most organisms including humans. While the period of these oscillators evolved so that it resonates with the 24 h daily environmental cycles, the precision of the oscillator (quantified via the Q factor) is another relevant property of these cell-autonomous oscillators. Since this quantity can be measured in individual cells, it is of interest to better understand how this property behaves across mathematical models of these oscillators. Current theoretical schemes for computing the Q factors show limitations for both high-dimensional models and in the vicinity of Hopf bifurcations. Here, we derive low-noise approximations that lead to numerically stable schemes also in high-dimensional models. In addition, we generalize normal form reductions that are appropriate near Hopf bifurcations. Applying our approximations to two models of circadian clocks, we show that while the low-noise regime is faithfully recapitulated, increasing the level of noise leads to species-dependent precision. We emphasize that subcomponents of the oscillator gradually decouple from the core oscillator as noise increases, which allows us to identify the subnetworks responsible for robust rhythms.

Dynamical analysis of cigarette smoking model with a saturated incidence rate

NASA Astrophysics Data System (ADS)

Zeb, Anwar; Bano, Ayesha; Alzahrani, Ebraheem; Zaman, Gul

2018-04-01

In this paper, we consider a delayed smoking model in which the potential smokers are assumed to satisfy the logistic equation. We discuss the dynamical behavior of our proposed model in the form of Delayed Differential Equations (DDEs) and show conditions for asymptotic stability of the model in steady state. We also discuss the Hopf bifurcation analysis of considered model. Finally, we use the nonstandard finite difference (NSFD) scheme to show the results graphically with help of MATLAB.

Synchronization transition of a coupled system composed of neurons with coexisting behaviors near a Hopf bifurcation

NASA Astrophysics Data System (ADS)

Jia, Bing

2014-05-01

The coexistence of a resting condition and period-1 firing near a subcritical Hopf bifurcation point, lying between the monostable resting condition and period-1 firing, is often observed in neurons of the central nervous systems. Near such a bifurcation point in the Morris—Lecar (ML) model, the attraction domain of the resting condition decreases while that of the coexisting period-1 firing increases as the bifurcation parameter value increases. With the increase of the coupling strength, and parameter and initial value dependent synchronization transition processes from non-synchronization to compete synchronization are simulated in two coupled ML neurons with coexisting behaviors: one neuron chosen as the resting condition and the other the coexisting period-1 firing. The complete synchronization is either a resting condition or period-1 firing dependent on the initial values of period-1 firing when the bifurcation parameter value is small or middle and is period-1 firing when the parameter value is large. As the bifurcation parameter value increases, the probability of the initial values of a period-1 firing neuron that lead to complete synchronization of period-1 firing increases, while that leading to complete synchronization of the resting condition decreases. It shows that the attraction domain of a coexisting behavior is larger, the probability of initial values leading to complete synchronization of this behavior is higher. The bifurcations of the coupled system are investigated and discussed. The results reveal the complex dynamics of synchronization behaviors of the coupled system composed of neurons with the coexisting resting condition and period-1 firing, and are helpful to further identify the dynamics of the spatiotemporal behaviors of the central nervous system.

Simplest bifurcation diagrams for monotone families of vector fields on a torus

NASA Astrophysics Data System (ADS)

Baesens, C.; MacKay, R. S.

2018-06-01

In part 1, we prove that the bifurcation diagram for a monotone two-parameter family of vector fields on a torus has to be at least as complicated as the conjectured simplest one proposed in Baesens et al (1991 Physica D 49 387–475). To achieve this, we define ‘simplest’ by sequentially minimising the numbers of equilibria, Bogdanov–Takens points, closed curves of centre and of neutral saddle, intersections of curves of centre and neutral saddle, Reeb components, other invariant annuli, arcs of rotational homoclinic bifurcation of horizontal homotopy type, necklace points, contractible periodic orbits, points of neutral horizontal homoclinic bifurcation and half-plane fan points. We obtain two types of simplest case, including that initially proposed. In part 2, we analyse the bifurcation diagram for an explicit monotone family of vector fields on a torus and prove that it has at most two equilibria, precisely four Bogdanov–Takens points, no closed curves of centre nor closed curves of neutral saddle, at most two Reeb components, precisely four arcs of rotational homoclinic connection of ‘horizontal’ homotopy type, eight horizontal saddle-node loop points, two necklace points, four points of neutral horizontal homoclinic connection, and two half-plane fan points, and there is no simultaneous existence of centre and neutral saddle, nor contractible homoclinic connection to a neutral saddle. Furthermore, we prove that all saddle-nodes, Bogdanov–Takens points, non-neutral and neutral horizontal homoclinic bifurcations are non-degenerate and the Hopf condition is satisfied for all centres. We also find it has four points of degenerate Hopf bifurcation. It thus provides an example of a family satisfying all the assumptions of part 1 except the one of at most one contractible periodic orbit.

Bifurcation Analysis of a Predator-Prey System with Ratio-Dependent Functional Response

NASA Astrophysics Data System (ADS)

Jiang, Xin; She, Zhikun; Feng, Zhaosheng; Zheng, Xiuliang

2017-12-01

In this paper, we are concerned with the structural stability of a density dependent predator-prey system with ratio-dependent functional response. Starting with the geometrical analysis of hyperbolic curves, we obtain that the system has one or two positive equilibria under various conditions. Inspired by the S-procedure and semi-definite programming, we use the sum of squares decomposition based method to ensure the global asymptotic stability of the positive equilibrium through the associated polynomial Lyapunov functions. By exploring the monotonic property of the trace of the Jacobian matrix with respect to r under the given different conditions, we analytically verify that there is a corresponding unique r∗ such that the trace is equal to zero and prove the existence of Hopf bifurcation, respectively.

Cannibalistic Predator-Prey Model with Disease in Predator — A Delay Model

NASA Astrophysics Data System (ADS)

Biswas, Santosh; Samanta, Sudip; Chattopadhyay, Joydev

In this paper, we propose and analyze a cannibalistic predator-prey model with a transmissible disease in the predator population. The disease can be transmitted through contacts with infected individuals as well as the cannibalism of an infected predator. We also consider incubation delay in disease transmission, where the incubation period represents the time in which the infectious agent develops in the host. Local stability analysis of the system around the biologically feasible equilibria is studied. Bifurcation analysis of the system around interior equilibrium is also studied. Applying the normal form theory and central manifold theorem, the direction of Hopf bifurcation, the stability and the period of bifurcating periodic solutions are derived. Under appropriate conditions, the permanence of the system with time delay is proved. Our results suggest that incubation delay destabilizes the system and can produce chaos. We also observe that cannibalism can control disease and population oscillations. Extensive numerical simulations are performed to support our analytical results.

Spread of a disease and its effect on population dynamics in an eco-epidemiological system

NASA Astrophysics Data System (ADS)

Upadhyay, Ranjit Kumar; Roy, Parimita

2014-12-01

In this paper, an eco-epidemiological model with simple law of mass action and modified Holling type II functional response has been proposed and analyzed to understand how a disease may spread among natural populations. The proposed model is a modification of the model presented by Upadhyay et al. (2008) [1]. Existence of the equilibria and their stability analysis (linear and nonlinear) has been studied. The dynamical transitions in the model have been studied by identifying the existence of backward Hopf-bifurcations and demonstrated the period-doubling route to chaos when the death rate of predator (μ1) and the growth rate of susceptible prey population (r) are treated as bifurcation parameters. Our studies show that the system exhibits deterministic chaos when some control parameters attain their critical values. Chaotic dynamics is depicted using the 2D parameter scans and bifurcation analysis. Possible implications of the results for disease eradication or its control are discussed.

Arnold tongues in a billiard problem in nonlinear and nonequilibrium systems

NASA Astrophysics Data System (ADS)

Miyaji, Tomoyuki

2017-02-01

We study a billiard problem in nonlinear and nonequilibrium systems. This is motivated by the motions of a traveling spot in a reaction-diffusion system (RDS) in a rectangular domain. We consider a four-dimensional dynamical system, defined by ordinary differential equations. This was first derived by S.-I. Ei et al. (2006), based on a reduced system on the center manifold in a neighborhood of a pitchfork bifurcation of a stationary spot for the RDS. In contrast to the classical billiard problem, this defines a dynamical system that is dissipative rather than conservative, and has an attractor. According to previous numerical studies, the attractor of the system changes depending on parameters such as the aspect ratio of the domain. It may be periodic, quasi-periodic, or chaotic. In this paper, we elucidate that it results from parameters crossing Arnold tongues and that the organizing center is a Hopf-Hopf bifurcation of the trivial equilibrium.

Periodic and chaotic oscillations in a tumor and immune system interaction model with three delays

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bi, Ping; Center for Partial Differential Equations, East China Normal University, 500 Dongchuan Rd., Shanghai 200241; Ruan, Shigui, E-mail: ruan@math.miami.edu

2014-06-15

In this paper, a tumor and immune system interaction model consisted of two differential equations with three time delays is considered in which the delays describe the proliferation of tumor cells, the process of effector cells growth stimulated by tumor cells, and the differentiation of immune effector cells, respectively. Conditions for the asymptotic stability of equilibria and existence of Hopf bifurcations are obtained by analyzing the roots of a second degree exponential polynomial characteristic equation with delay dependent coefficients. It is shown that the positive equilibrium is asymptotically stable if all three delays are less than their corresponding critical valuesmore » and Hopf bifurcations occur if any one of these delays passes through its critical value. Numerical simulations are carried out to illustrate the rich dynamical behavior of the model with different delay values including the existence of regular and irregular long periodic oscillations.« less

Stability and Hopf bifurcation for a business cycle model with expectation and delay

NASA Astrophysics Data System (ADS)

Liu, Xiangdong; Cai, Wenli; Lu, Jiajun; Wang, Yangyang

2015-08-01

According to rational expectation hypothesis, the government will take into account the future capital stock in the process of investment decision. By introducing anticipated capital stock into an economic model with investment delay, we construct a mixed functional differential system including delay and advanced variables. The system is converted to the one containing only delay by variable substitution. The equilibrium point of the system is obtained and its dynamical characteristics such as stability, Hopf bifurcation and its stability and direction are investigated by using the related theories of nonlinear dynamics. We carry out some numerical simulations to confirm these theoretical conclusions. The results indicate that both capital stock's anticipation and investment lag are the certain factors leading to the occurrence of cyclical fluctuations in the macroeconomic system. Moreover, the level of economic fluctuation can be dampened to some extent if investment decisions are made by the reasonable short-term forecast on capital stock.

Stability diagram for the forced Kuramoto model.

PubMed

Childs, Lauren M; Strogatz, Steven H

2008-12-01

We analyze the periodically forced Kuramoto model. This system consists of an infinite population of phase oscillators with random intrinsic frequencies, global sinusoidal coupling, and external sinusoidal forcing. It represents an idealization of many phenomena in physics, chemistry, and biology in which mutual synchronization competes with forced synchronization. In other words, the oscillators in the population try to synchronize with one another while also trying to lock onto an external drive. Previous work on the forced Kuramoto model uncovered two main types of attractors, called forced entrainment and mutual entrainment, but the details of the bifurcations between them were unclear. Here we present a complete bifurcation analysis of the model for a special case in which the infinite-dimensional dynamics collapse to a two-dimensional system. Exact results are obtained for the locations of Hopf, saddle-node, and Takens-Bogdanov bifurcations. The resulting stability diagram bears a striking resemblance to that for the weakly nonlinear forced van der Pol oscillator.

Dynamic behavior of the bray-liebhafsky oscillatory reaction controlled by sulfuric acid and temperature

NASA Astrophysics Data System (ADS)

Pejić, N.; Vujković, M.; Maksimović, J.; Ivanović, A.; Anić, S.; Čupić, Ž.; Kolar-Anić, Lj.

2011-12-01

The non-periodic, periodic and chaotic regimes in the Bray-Liebhafsky (BL) oscillatory reaction observed in a continuously fed well stirred tank reactor (CSTR) under isothermal conditions at various inflow concentrations of the sulfuric acid were experimentally studied. In each series (at any fixed temperature), termination of oscillatory behavior via saddle loop infinite period bifurcation (SNIPER) as well as some kind of the Andronov-Hopf bifurcation is presented. In addition, it was found that an increase of temperature, in different series of experiments resulted in the shift of bifurcation point towards higher values of sulfuric acid concentration.

Hopf bifurcation in a nonlocal nonlinear transport equation stemming from stochastic neural dynamics

NASA Astrophysics Data System (ADS)

Drogoul, Audric; Veltz, Romain

2017-02-01

In this work, we provide three different numerical evidences for the occurrence of a Hopf bifurcation in a recently derived [De Masi et al., J. Stat. Phys. 158, 866-902 (2015) and Fournier and löcherbach, Ann. Inst. H. Poincaré Probab. Stat. 52, 1844-1876 (2016)] mean field limit of a stochastic network of excitatory spiking neurons. The mean field limit is a challenging nonlocal nonlinear transport equation with boundary conditions. The first evidence relies on the computation of the spectrum of the linearized equation. The second stems from the simulation of the full mean field. Finally, the last evidence comes from the simulation of the network for a large number of neurons. We provide a "recipe" to find such bifurcation which nicely complements the works in De Masi et al. [J. Stat. Phys. 158, 866-902 (2015)] and Fournier and löcherbach [Ann. Inst. H. Poincaré Probab. Stat. 52, 1844-1876 (2016)]. This suggests in return to revisit theoretically these mean field equations from a dynamical point of view. Finally, this work shows how the noise level impacts the transition from asynchronous activity to partial synchronization in excitatory globally pulse-coupled networks.

Spontaneous symmetry breaking due to the trade-off between attractive and repulsive couplings.

PubMed

Sathiyadevi, K; Karthiga, S; Chandrasekar, V K; Senthilkumar, D V; Lakshmanan, M

2017-04-01

Spontaneous symmetry breaking is an important phenomenon observed in various fields including physics and biology. In this connection, we here show that the trade-off between attractive and repulsive couplings can induce spontaneous symmetry breaking in a homogeneous system of coupled oscillators. With a simple model of a system of two coupled Stuart-Landau oscillators, we demonstrate how the tendency of attractive coupling in inducing in-phase synchronized (IPS) oscillations and the tendency of repulsive coupling in inducing out-of-phase synchronized oscillations compete with each other and give rise to symmetry breaking oscillatory states and interesting multistabilities. Further, we provide explicit expressions for synchronized and antisynchronized oscillatory states as well as the so called oscillation death (OD) state and study their stability. If the Hopf bifurcation parameter (λ) is greater than the natural frequency (ω) of the system, the attractive coupling favors the emergence of an antisymmetric OD state via a Hopf bifurcation whereas the repulsive coupling favors the emergence of a similar state through a saddle-node bifurcation. We show that an increase in the repulsive coupling not only destabilizes the IPS state but also facilitates the reentrance of the IPS state.

Solvable model for chimera states of coupled oscillators.

PubMed

Abrams, Daniel M; Mirollo, Rennie; Strogatz, Steven H; Wiley, Daniel A

2008-08-22

Networks of identical, symmetrically coupled oscillators can spontaneously split into synchronized and desynchronized subpopulations. Such chimera states were discovered in 2002, but are not well understood theoretically. Here we obtain the first exact results about the stability, dynamics, and bifurcations of chimera states by analyzing a minimal model consisting of two interacting populations of oscillators. Along with a completely synchronous state, the system displays stable chimeras, breathing chimeras, and saddle-node, Hopf, and homoclinic bifurcations of chimeras.

INTERDISCIPLINARY PHYSICS AND RELATED AREAS OF SCIENCE AND TECHNOLOGY: Branch Processes of Vortex Filaments and Hopf Invariant Constraint on Scroll Wave

NASA Astrophysics Data System (ADS)

Zhu, Tao; Ren, Ji-Rong; Mo, Shu-Fan

2009-12-01

In this paper, by making use of Duan's topological current theory, the evolution of the vortex filaments in excitable media is discussed in detail. The vortex filaments are found generating or annihilating at the limit points and encountering, splitting, or merging at the bifurcation points of a complex function Z(vec x, t). It is also shown that the Hopf invariant of knotted scroll wave filaments is preserved in the branch processes (splitting, merging, or encountering) during the evolution of these knotted scroll wave filaments. Furthermore, it also revealed that the “exclusion principle" in some chemical media is just the special case of the Hopf invariant constraint, and during the branch processes the “exclusion principle" is also protected by topology.

Application of taxonomy theory, Volume 1: Computing a Hopf bifurcation-related segment of the feasibility boundary. Final report

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zaborszky, J.; Venkatasubramanian, V.

1995-10-01

Taxonomy Theory is the first precise comprehensive theory for large power system dynamics modeled in any detail. The motivation for this project is to show that it can be used, practically, for analyzing a disturbance that actually occurred on a large system, which affected a sizable portion of the Midwest with supercritical Hopf type oscillations. This event is well documented and studied. The report first summarizes Taxonomy Theory with an engineering flavor. Then various computational approaches are sighted and analyzed for desirability to use with Taxonomy Theory. Then working equations are developed for computing a segment of the feasibility boundarymore » that bounds the region of (operating) parameters throughout which the operating point can be moved without losing stability. Then experimental software incorporating large EPRI software packages PSAPAC is developed. After a summary of the events during the subject disturbance, numerous large scale computations, up to 7600 buses, are reported. These results are reduced into graphical and tabular forms, which then are analyzed and discussed. The report is divided into two volumes. This volume illustrates the use of the Taxonomy Theory for computing the feasibility boundary and presents evidence that the event indeed led to a Hopf type oscillation on the system. Furthermore it proves that the Feasibility Theory can indeed be used for practical computation work with very large systems. Volume 2, a separate volume, will show that the disturbance has led to a supercritical (that is stable oscillation) Hopf bifurcation.« less

Numerical exploration of Kaldorian interregional macrodynamics: stability and the trade threshold for business cycles under fixed exchange rates.

PubMed

Asada, Toichiro; Douskos, Christos; Markellos, Panagiotis

2011-01-01

The stability of equilibrium and the possibility of generation of business cycles in a discrete interregional Kaldorian macrodynamic model with fixed exchange rates are explored using numerical methods. One of the aims is to illustrate the feasibility and effectiveness of the numerical approach for dynamical systems of moderately high dimensionality and several parameters. The model considered is five-dimensional with four parameters, the speeds of adjustment of the goods markets and the degrees of economic interactions between the regions through trade and capital movement. Using a grid search method for the determination of the region of stability of equilibrium in two-dimensional parameter subspaces, and coefficient criteria for the flip bifurcation - and Hopf bifurcation - curve, we determine the stability region in several parameter ranges and identify Hopf bifurcation curves when they exist. It is found that interregional cycles emerge only for sufficient interregional trade. The relevant threshold is predicted by the model at 14 - 16 % of trade transactions. By contrast, no minimum level of capital mobility exists in a global sense as a requirement for the emergence of interregional cycles; the main conclusion being, therefore, that cycles may occur for very low levels of capital mobility if trade is sufficient. Examples of bifurcation and Lyapunov exponent diagrams illustrating the occurrence of cycles or period doubling, and examples of the development of the occurring cycles, are given. Both supercritical and subcritical bifurcations are found to occur, the latter type indicating coexistence of a point and a cyclical attractor.

Dynamics of eco-epidemiological model with harvesting

NASA Astrophysics Data System (ADS)

Purnomo, Anna Silvia; Darti, Isnani; Suryanto, Agus

2017-12-01

In this paper, we study an eco-epidemiology model which is derived from S I epidemic model with bilinear incidence rate and modified Leslie Gower predator-prey model with harvesting on susceptible prey. Existence condition and stability of all equilibrium points are discussed for the proposed model. Furthermore, we show that the model exhibits a Hopf bifurcation around interior equilibrium point which is driven by the rate of infection. Our numerical simulations using some different value of parameters confirm our analytical analysis.

Architecture and Robust Networks

DTIC Science & Technology

2011-08-18

supercritical Hopf bifurcation). Thus at least in this model, oscillations have no direct purpose but are side effects of hard tradeoffs crucial to...The HRV tradeoffs already involve more complex mechanisms in (9) than in (1), but both models need expansion to include control of redox (and CO2

Singular Hopf bifurcation in a differential equation with large state-dependent delay

PubMed Central

Kozyreff, G.; Erneux, T.

2014-01-01

We study the onset of sustained oscillations in a classical state-dependent delay (SDD) differential equation inspired by control theory. Owing to the large delays considered, the Hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold. Using asymptotic techniques, we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations. The dependence of the delay on the solution can be either linear or nonlinear, with at least quadratic dependence. In the former case, an asymptotic connection is made with the Rayleigh oscillator. In the latter, van der Pol’s equation is derived for the small-amplitude oscillations. SDD differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant-delay differential equation, but explicit analytical construction of solutions are rare. This paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for SDD problems with large delays. PMID:24511255

Singular Hopf bifurcation in a differential equation with large state-dependent delay.

PubMed

Kozyreff, G; Erneux, T

2014-02-08

We study the onset of sustained oscillations in a classical state-dependent delay (SDD) differential equation inspired by control theory. Owing to the large delays considered, the Hopf bifurcation is singular and the oscillations rapidly acquire a sawtooth profile past the instability threshold. Using asymptotic techniques, we explicitly capture the gradual change from nearly sinusoidal to sawtooth oscillations. The dependence of the delay on the solution can be either linear or nonlinear, with at least quadratic dependence. In the former case, an asymptotic connection is made with the Rayleigh oscillator. In the latter, van der Pol's equation is derived for the small-amplitude oscillations. SDD differential equations are currently the subject of intense research in order to establish or amend general theorems valid for constant-delay differential equation, but explicit analytical construction of solutions are rare. This paper illustrates the use of singular perturbation techniques and the unusual way in which solvability conditions can arise for SDD problems with large delays.

Emergence of the bifurcation structure of a Langmuir-Blodgett transfer model

NASA Astrophysics Data System (ADS)

Köpf, Michael H.; Thiele, Uwe

2014-11-01

We explore the bifurcation structure of a modified Cahn-Hilliard equation that describes a system that may undergo a first-order phase transition and is kept permanently out of equilibrium by a lateral driving. This forms a simple model, e.g., for the deposition of stripe patterns of different phases of surfactant molecules through Langmuir-Blodgett transfer. Employing continuation techniques the bifurcation structure is numerically investigated using the non-dimensional transfer velocity as the main control parameter. It is found that the snaking structure of steady front states is intertwined with a large number of branches of time-periodic solutions that emerge from Hopf or period-doubling bifurcations and end in global bifurcations (sniper and homoclinic). Overall the bifurcation diagram has a harp-like appearance. This is complemented by a two-parameter study in non-dimensional transfer velocity and domain size (as a measure of the distance to the phase transition threshold) that elucidates through which local and global codimension 2 bifurcations the entire harp-like structure emerges.

Bifurcation phenomena in cylindrical convection

NASA Astrophysics Data System (ADS)

Tuckerman, Laurette; Boronska, K.; Bordja, L.; Martin-Witkowski, L.; Navarro, M. C.

2008-11-01

We present two bifurcation scenarios occurring in Rayleigh-Benard convection in a small-aspect-ratio cylinder. In water (Pr=6.7) with R/H=2, Hof et al. (1999) observed five convective patterns at Ra=14200. We have computed 14 stable and unstable steady branches, as well as novel time-dependent branches. The resulting complicated bifurcation diagram, can be partitioned according to azimuthal symmetry. For example, three-roll and dipole states arise from an m=1 bifurcation, four-roll and ``pizza'' branches from m=2, and the ``mercedes'' state from an m=3 bifurcation after successive saddle-node bifurcations via ``marigold'', ``mitsubishi'' and ``cloverleaf'' states. The diagram represents a compromise between the physical tendency towards parallel rolls and the mathematical requirement that primary bifurcations be towards trigonometric states. Our second investigation explores the effect of exact counter-rotation of the upper and lower bounding disks on axisymmetric flows with Pr=1 and R/H=1. The convection threshold increases and, for sufficiently high rotation, the instability becomes oscillatory. Limit cycles originating at the Hopf bifurcation are annihilated when their period becomes infinite at saddle-node-on-periodic-orbit (SNOPER) bifurcations.

Positive And Negative Feedback Loops Coupled By Common Transcription Activator And Repressor

NASA Astrophysics Data System (ADS)

Sielewiesiuk, Jan; Łopaciuk, Agata

2015-03-01

Dynamical systems consisting of two interlocked loops with negative and positive feedback have been studied using the linear analysis of stability and numerical solutions. Conditions for saddle-node bifurcation were formulated in a general form. Conditions for Hopf bifurcations were found in a few symmetrical cases. Auto-oscillations, when they exist, are generated by the negative feedback repressive loop. This loop determines the frequency and amplitude of oscillations. The positive feedback loop of activation slightly modifies the oscillations. Oscillations are possible when the difference between Hilll's coefficients of the repression and activation is sufficiently high. The highly cooperative activation loop with a fast turnover slows down or even makes the oscillations impossible. The system under consideration can constitute a component of epigenetic or enzymatic regulation network.

Stability Analysis and Internal Heating Effect on Oscillatory Convection in a Viscoelastic Fluid Saturated Porous Medium Under Gravity Modulation

NASA Astrophysics Data System (ADS)

Bhadauria, B. S.; Singh, M. K.; Singh, A.; Singh, B. K.; Kiran, P.

2016-12-01

In this paper, we investigate the combined effect of internal heating and time periodic gravity modulation in a viscoelastic fluid saturated porous medium by reducing the problem into a complex non-autonomous Ginzgburg-Landau equation. Weak nonlinear stability analysis has been performed by using power series expansion in terms of the amplitude of gravity modulation, which is assumed to be small. The Nusselt number is obtained in terms of the amplitude for oscillatory mode of convection. The influence of viscoelastic parameters on heat transfer has been discussed. Gravity modulation is found to have a destabilizing effect at low frequencies and a stabilizing effect at high frequencies. Finally, it is found that overstability advances the onset of convection, more with internal heating. The conditions for which the complex Ginzgburg-Landau equation undergoes Hopf bifurcation and the amplitude equation undergoes supercritical pitchfork bifurcation are studied.

Dynamics of tuberculosis transmission with exogenous reinfections and endogenous reactivation

NASA Astrophysics Data System (ADS)

Khajanchi, Subhas; Das, Dhiraj Kumar; Kar, Tapan Kumar

2018-05-01

We propose and analyze a mathematical model for tuberculosis (TB) transmission to study the role of exogenous reinfection and endogenous reactivation. The model exhibits two equilibria: a disease free and an endemic equilibria. We observe that the TB model exhibits transcritical bifurcation when basic reproduction number R0 = 1. Our results demonstrate that the disease transmission rate β and exogenous reinfection rate α plays an important role to change the qualitative dynamics of TB. The disease transmission rate β give rises to the possibility of backward bifurcation for R0 < 1, and hence the existence of multiple endemic equilibria one of which is stable and another one is unstable. Our analysis suggests that R0 < 1 may not be sufficient to completely eliminate the disease. We also investigate that our TB transmission model undergoes Hopf-bifurcation with respect to the contact rate β and the exogenous reinfection rate α. We conducted some numerical simulations to support our analytical findings.

Experimental study of complex mixed-mode oscillations generated in a Bonhoeffer-van der Pol oscillator under weak periodic perturbation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Shimizu, Kuniyasu, E-mail: kuniyasu.shimizu@it-chiba.ac.jp; Sekikawa, Munehisa; Inaba, Naohiko

2015-02-15

Bifurcations of complex mixed-mode oscillations denoted as mixed-mode oscillation-incrementing bifurcations (MMOIBs) have frequently been observed in chemical experiments. In a previous study [K. Shimizu et al., Physica D 241, 1518 (2012)], we discovered an extremely simple dynamical circuit that exhibits MMOIBs. Our model was represented by a slow/fast Bonhoeffer-van der Pol circuit under weak periodic perturbation near a subcritical Andronov-Hopf bifurcation point. In this study, we experimentally and numerically verify that our dynamical circuit captures the essence of the underlying mechanism causing MMOIBs, and we observe MMOIBs and chaos with distinctive waveforms in real circuit experiments.

Oscillatory bistability of real-space transfer in semiconductor heterostructures

NASA Astrophysics Data System (ADS)

Do˙ttling, R.; Scho˙ll, E.

1992-01-01

Charge transport parallel to the layers of a modulation-doped GaAs/AlxGa1-xAs heterostructure is studied theoretically. The heating of electrons by the applied electric field leads to real-space transfer of electrons from the GaAs into the adjacent AlxGa1-xAs layer. For sufficiently large dc bias, spontaneous periodic 100-GHz current oscillations, and bistability and hysteretic switching transitions between oscillatory and stationary states are predicted. We present a detailed investigation of complex bifurcation scenarios as a function of the bias voltage U0 and the load resistance RL. For large RL subcritical Hopf bifurcations and global bifurcations of limit cycles are displayed.

Self-localized structures in vertical-cavity surface-emitting lasers with external feedback.

PubMed

Paulau, P V; Gomila, D; Ackemann, T; Loiko, N A; Firth, W J

2008-07-01

In this paper, we analyze a model of broad area vertical-cavity surface-emitting lasers subjected to frequency-selective optical feedback. In particular, we analyze the spatio-temporal regimes arising above threshold and the existence and dynamical properties of cavity solitons. We build the bifurcation diagram of stationary self-localized states, finding that branches of cavity solitons emerge from the degenerate Hopf bifurcations marking the homogeneous solutions with maximal and minimal gain. These branches collide in a saddle-node bifurcation, defining a maximum pump current for soliton existence that lies below the threshold of the laser without feedback. The properties of these cavity solitons are in good agreement with those observed in recent experiments.

Stable cycling in discrete-time genetic models.

PubMed

Hastings, A

1981-11-01

Examples of stable cycling are discussed for two-locus, two-allele, deterministic, discrete-time models with constant fitnesses. The cases that cycle were found by using numerical techniques to search for stable Hopf bifurcations. One consequence of the results is that apparent cases of directional selection may be due to stable cycling.

Phase transitions in tumor growth VI: Epithelial-Mesenchymal transition

NASA Astrophysics Data System (ADS)

Guerra, A.; Rodriguez, D. J.; Montero, S.; Betancourt-Mar, J. A.; Martin, R. R.; Silva, E.; Bizzarri, M.; Cocho, G.; Mansilla, R.; Nieto-Villar, J. M.

2018-06-01

Herewith we discuss a network model of the epithelial-mesenchymal transition (EMT) based on our previous proposed framework. The EMT appears as a "first order" phase transition process, analogous to the transitions observed in the chemical-physical field. Chiefly, EMT should be considered a transition characterized by a supercritical Andronov-Hopf bifurcation, with the emergence of limit cycle and, consequently, a cascade of saddle-foci Shilnikov's bifurcations. We eventually show that the entropy production rate is an EMT-dependent function and, as such, its formalism reminds the van der Waals equation.

A nonlinear wave equation in nonadiabatic flame propagation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Booty, M.R.; Matalon, M.; Matkowsky, B.J.

1988-06-01

The authors derive a nonlinear wave equation from the diffusional thermal model of gaseous combustion to describe the evolution of a flame front. The equation arises as a long wave theory, for values of the volumeric heat loss in a neighborhood of the extinction point (beyond which planar uniformly propagating flames cease to exist), and for Lewis numbers near the critical value beyond which uniformly propagating planar flames lose stability via a degenerate Hopf bifurcation. Analysis of the equation suggests the possibility of a singularity developing in finite time.

Breathing multichimera states in nonlocally coupled phase oscillators

NASA Astrophysics Data System (ADS)

Suda, Yusuke; Okuda, Koji

2018-04-01

Chimera states for the one-dimensional array of nonlocally coupled phase oscillators in the continuum limit are assumed to be stationary states in most studies, but a few studies report the existence of breathing chimera states. We focus on multichimera states with two coherent and incoherent regions and numerically demonstrate that breathing multichimera states, whose global order parameter oscillates temporally, can appear. Moreover, we show that the system exhibits a Hopf bifurcation from a stationary multichimera to a breathing one by the linear stability analysis for the stationary multichimera.

Mode Selection Rules for a Two-Delay System with Positive and Negative Feedback Loops

NASA Astrophysics Data System (ADS)

Takahashi, Kin'ya; Kobayashi, Taizo

2018-04-01

The mode selection rules for a two-delay system, which has negative feedback with a short delay time t1 and positive feedback with a long delay time t2, are studied numerically and theoretically. We find two types of mode selection rules depending on the strength of the negative feedback. When the strength of the negative feedback |α1| (α1 < 0) is sufficiently small compared with that of the positive feedback α2 (> 0), 2m + 1-th harmonic oscillation is well sustained in a neighborhood of t1/t2 = even/odd, i.e., relevant condition. In a neighborhood of the irrelevant condition given by t1/t2 = odd/even or t1/t2 = odd/odd, higher harmonic oscillations are observed. However, if |α1| is slightly less than α2, a different mode selection rule works, where the condition t1/t2 = odd/even is relevant and the conditions t1/t2 = odd/odd and t1/t2 = even/odd are irrelevant. These mode selection rules are different from the mode selection rule of the normal two-delay system with two positive feedback loops, where t1/t2 = odd/odd is relevant and the others are irrelevant. The two types of mode selection rules are induced by individually different mechanisms controlling the Hopf bifurcation, i.e., the Hopf bifurcation controlled by the "boosted bifurcation process" and by the "anomalous bifurcation process", which occur for |α1| below and above the threshold value αth, respectively.

Non-linear dynamic characteristics and optimal control of giant magnetostrictive film subjected to in-plane stochastic excitation

NASA Astrophysics Data System (ADS)

Zhu, Z. W.; Zhang, W. D.; Xu, J.

2014-03-01

The non-linear dynamic characteristics and optimal control of a giant magnetostrictive film (GMF) subjected to in-plane stochastic excitation were studied. Non-linear differential items were introduced to interpret the hysteretic phenomena of the GMF, and the non-linear dynamic model of the GMF subjected to in-plane stochastic excitation was developed. The stochastic stability was analysed, and the probability density function was obtained. The condition of stochastic Hopf bifurcation and noise-induced chaotic response were determined, and the fractal boundary of the system's safe basin was provided. The reliability function was solved from the backward Kolmogorov equation, and an optimal control strategy was proposed in the stochastic dynamic programming method. Numerical simulation shows that the system stability varies with the parameters, and stochastic Hopf bifurcation and chaos appear in the process; the area of the safe basin decreases when the noise intensifies, and the boundary of the safe basin becomes fractal; the system reliability improved through stochastic optimal control. Finally, the theoretical and numerical results were proved by experiments. The results are helpful in the engineering applications of GMF.

Analysis of current-driven oscillatory dynamics of single-layer homoepitaxial islands on crystalline conducting substrates

NASA Astrophysics Data System (ADS)

Dasgupta, Dwaipayan; Kumar, Ashish; Maroudas, Dimitrios

2018-03-01

We report results of a systematic study on the complex oscillatory current-driven dynamics of single-layer homoepitaxial islands on crystalline substrate surfaces and the dependence of this driven dynamical behavior on important physical parameters, including island size, substrate surface orientation, and direction of externally applied electric field. The analysis is based on a nonlinear model of driven island edge morphological evolution that accounts for curvature-driven edge diffusion, edge electromigration, and edge diffusional anisotropy. Using a linear theory of island edge morphological stability, we calculate a critical island size at which the island's equilibrium edge shape becomes unstable, which sets a lower bound for the onset of time-periodic oscillatory dynamical response. Using direct dynamical simulations, we study the edge morphological dynamics of current-driven single-layer islands at larger-than-critical size, and determine the actual island size at which the migrating islands undergo a transition from steady to time-periodic asymptotic states through a subcritical Hopf bifurcation. At the highest symmetry of diffusional anisotropy examined, on {111} surfaces of face-centered cubic crystalline substrates, we find that more complex stable oscillatory states can be reached through period-doubling bifurcation at island sizes larger than those at the Hopf points. We characterize in detail the island morphology and dynamical response at the stable time-periodic asymptotic states, determine the range of stability of these oscillatory states terminated by island breakup, and explain the morphological features of the stable oscillating islands on the basis of linear stability theory.

Bifurcation scenarios for bubbling transition.

PubMed

Zimin, Aleksey V; Hunt, Brian R; Ott, Edward

2003-01-01

Dynamical systems with chaos on an invariant submanifold can exhibit a type of behavior called bubbling, whereby a small random or fixed perturbation to the system induces intermittent bursting. The bifurcation to bubbling occurs when a periodic orbit embedded in the chaotic attractor in the invariant manifold becomes unstable to perturbations transverse to the invariant manifold. Generically the periodic orbit can become transversely unstable through a pitchfork, transcritical, period-doubling, or Hopf bifurcation. In this paper a unified treatment of the four types of bubbling bifurcation is presented. Conditions are obtained determining whether the transition to bubbling is soft or hard; that is, whether the maximum burst amplitude varies continuously or discontinuously with variation of the parameter through its critical value. For soft bubbling transitions, the scaling of the maximum burst amplitude with the parameter is derived. For both hard and soft transitions the scaling of the average interburst time with the bifurcation parameter is deduced. Both random (noise) and fixed (mismatch) perturbations are considered. Results of numerical experiments testing our theoretical predictions are presented.

Dynamics, Analysis and Implementation of a Multiscroll Memristor-Based Chaotic Circuit

NASA Astrophysics Data System (ADS)

Alombah, N. Henry; Fotsin, Hilaire; Ngouonkadi, E. B. Megam; Nguazon, Tekou

This article introduces a novel four-dimensional autonomous multiscroll chaotic circuit which is derived from the actual simplest memristor-based chaotic circuit. A fourth circuit element — another inductor — is introduced to generate the complex behavior observed. A systematic study of the chaotic behavior is performed with the help of some nonlinear tools such as Lyapunov exponents, phase portraits, and bifurcation diagrams. Multiple scroll attractors are observed in Matlab, Pspice environments and also experimentally. We also observe the phenomenon of antimonotonicity, periodic and chaotic bubbles, multiple periodic-doubling bifurcations, Hopf bifurcations, crises and the phenomenon of intermittency. The chaotic dynamics of this circuit is realized by laboratory experiments, Pspice simulations, numerical and analytical investigations. It is observed that the results from the three environments agree to a great extent. This topology is likely convenient to be used to intentionally generate chaos in memristor-based chaotic circuit applications, given the fact that multiscroll chaotic systems have found important applications as broadband signal generators, pseudorandom number generators for communication engineering and also in biometric authentication.

Time-delayed feedback control of breathing localized structures in a three-component reaction-diffusion system.

PubMed

Gurevich, Svetlana V

2014-10-28

The dynamics of a single breathing localized structure in a three-component reaction-diffusion system subjected to time-delayed feedback is investigated. It is shown that variation of the delay time and the feedback strength can lead either to stabilization of the breathing or to delay-induced periodic or quasi-periodic oscillations of the localized structure. A bifurcation analysis of the system in question is provided and an order parameter equation is derived that describes the dynamics of the localized structure in the vicinity of the Andronov-Hopf bifurcation. With the aid of this equation, the boundaries of the stabilization domains as well as the dependence of the oscillation radius on delay parameters can be explicitly derived, providing a robust mechanism to control the behaviour of the breathing localized structure in a straightforward manner. © 2014 The Author(s) Published by the Royal Society. All rights reserved.

Multiple time scale analysis of pressure oscillations in solid rocket motors

NASA Astrophysics Data System (ADS)

Ahmed, Waqas; Maqsood, Adnan; Riaz, Rizwan

2018-03-01

In this study, acoustic pressure oscillations for single and coupled longitudinal acoustic modes in Solid Rocket Motor (SRM) are investigated using Multiple Time Scales (MTS) method. Two independent time scales are introduced. The oscillations occur on fast time scale whereas the amplitude and phase changes on slow time scale. Hopf bifurcation is employed to investigate the properties of the solution. The supercritical bifurcation phenomenon is observed for linearly unstable system. The amplitude of the oscillations result from equal energy gain and loss rates of longitudinal acoustic modes. The effect of linear instability and frequency of longitudinal modes on amplitude and phase of oscillations are determined for both single and coupled modes. For both cases, the maximum amplitude of oscillations decreases with the frequency of acoustic mode and linear instability of SRM. The comparison of analytical MTS results and numerical simulations demonstrate an excellent agreement.

Non-robust dynamic inferences from macroeconometric models: Bifurcation stratification of confidence regions

NASA Astrophysics Data System (ADS)

Barnett, William A.; Duzhak, Evgeniya Aleksandrovna

2008-06-01

Grandmont [J.M. Grandmont, On endogenous competitive business cycles, Econometrica 53 (1985) 995-1045] found that the parameter space of the most classical dynamic models is stratified into an infinite number of subsets supporting an infinite number of different kinds of dynamics, from monotonic stability at one extreme to chaos at the other extreme, and with many forms of multiperiodic dynamics in between. The econometric implications of Grandmont’s findings are particularly important, if bifurcation boundaries cross the confidence regions surrounding parameter estimates in policy-relevant models. Stratification of a confidence region into bifurcated subsets seriously damages robustness of dynamical inferences. Recently, interest in policy in some circles has moved to New-Keynesian models. As a result, in this paper we explore bifurcation within the class of New-Keynesian models. We develop the econometric theory needed to locate bifurcation boundaries in log-linearized New-Keynesian models with Taylor policy rules or inflation-targeting policy rules. Central results needed in this research are our theorems on the existence and location of Hopf bifurcation boundaries in each of the cases that we consider.

A dynamical systems approach to actin-based motility in Listeria monocytogenes

NASA Astrophysics Data System (ADS)

Hotton, S.

2010-11-01

A simple kinematic model for the trajectories of Listeria monocytogenes is generalized to a dynamical system rich enough to exhibit the resonant Hopf bifurcation structure of excitable media and simple enough to be studied geometrically. It is shown how L. monocytogenes trajectories and meandering spiral waves are organized by the same type of attracting set.

Period doubling cascades of prey-predator model with nonlinear harvesting and control of over exploitation through taxation

NASA Astrophysics Data System (ADS)

Gupta, R. P.; Banerjee, Malay; Chandra, Peeyush

2014-07-01

The present study investigates a prey predator type model for conservation of ecological resources through taxation with nonlinear harvesting. The model uses the harvesting function as proposed by Agnew (1979) [1] which accounts for the handling time of the catch and also the competition between standard vessels being utilized for harvesting of resources. In this paper we consider a three dimensional dynamic effort prey-predator model with Holling type-II functional response. The conditions for uniform persistence of the model have been derived. The existence and stability of bifurcating periodic solution through Hopf bifurcation have been examined for a particular set of parameter value. Using numerical examples it is shown that the system admits periodic, quasi-periodic and chaotic solutions. It is observed that the system exhibits periodic doubling route to chaos with respect to tax. Many forms of complexities such as chaotic bands (including periodic windows, period-doubling bifurcations, period-halving bifurcations and attractor crisis) and chaotic attractors have been observed. Sensitivity analysis is carried out and it is observed that the solutions are highly dependent to the initial conditions. Pontryagin's Maximum Principle has been used to obtain optimal tax policy to maximize the monetary social benefit as well as conservation of the ecosystem.

Dynamics of an HIV-1 infection model with cell mediated immunity

NASA Astrophysics Data System (ADS)

Yu, Pei; Huang, Jianing; Jiang, Jiao

2014-10-01

In this paper, we study the dynamics of an improved mathematical model on HIV-1 virus with cell mediated immunity. This new 5-dimensional model is based on the combination of a basic 3-dimensional HIV-1 model and a 4-dimensional immunity response model, which more realistically describes dynamics between the uninfected cells, infected cells, virus, the CTL response cells and CTL effector cells. Our 5-dimensional model may be reduced to the 4-dimensional model by applying a quasi-steady state assumption on the variable of virus. However, it is shown in this paper that virus is necessary to be involved in the modeling, and that a quasi-steady state assumption should be applied carefully, which may miss some important dynamical behavior of the system. Detailed bifurcation analysis is given to show that the system has three equilibrium solutions, namely the infection-free equilibrium, the infectious equilibrium without CTL, and the infectious equilibrium with CTL, and a series of bifurcations including two transcritical bifurcations and one or two possible Hopf bifurcations occur from these three equilibria as the basic reproduction number is varied. The mathematical methods applied in this paper include characteristic equations, Routh-Hurwitz condition, fluctuation lemma, Lyapunov function and computation of normal forms. Numerical simulation is also presented to demonstrate the applicability of the theoretical predictions.

Imitation dynamics of vaccine decision-making behaviours based on the game theory.

PubMed

Yang, Junyuan; Martcheva, Maia; Chen, Yuming

2016-01-01

Based on game theory, we propose an age-structured model to investigate the imitation dynamics of vaccine uptake. We first obtain the existence and local stability of equilibria. We show that Hopf bifurcation can occur. We also establish the global stability of the boundary equilibria and persistence of the disease. The theoretical results are supported by numerical simulations.

Coherence resonance in low-density jets

NASA Astrophysics Data System (ADS)

Zhu, Yuanhang; Gupta, Vikrant; Li, Larry K. B.

2017-11-01

Coherence resonance is a phenomenon in which the response of a stable nonlinear system to noise exhibits a peak in coherence at an intermediate noise amplitude. We report the first experimental evidence of coherence resonance in a purely hydrodynamic system, a low-density jet whose variants can be found in many natural and engineering systems. This evidence comprises four parts: (i) the jet's response amplitude increases as the Reynolds number approaches the instability boundary under a constant noise amplitude; (ii) as the noise amplitude increases, the amplitude distribution of the jet response first becomes unimodal, then bimodal, and finally unimodal again; (iii) a distinct peak emerges in the coherence factor at an intermediate noise amplitude; and (iv) for a subcritical Hopf bifurcation, the decay rate of the autocorrelation function exhibits a maximum at an intermediate noise amplitude, but for a supercritical Hopf bifurcation, the decay rate decreases monotonically with increasing noise amplitude. It is clear that coherence resonance can provide valuable information about a system's nonlinearity even in the unconditionally stable regime, opening up new possibilities for its use in system identification and flow control. This work was supported by the Research Grants Council of Hong Kong (Project No. 16235716 and 26202815).

O(2) Hopf bifurcation of viscous shock waves in a channel

NASA Astrophysics Data System (ADS)

Pogan, Alin; Yao, Jinghua; Zumbrun, Kevin

2015-07-01

Extending work of Texier and Zumbrun in the semilinear non-reflection symmetric case, we study O(2) transverse Hopf bifurcation, or "cellular instability", of viscous shock waves in a channel, for a class of quasilinear hyperbolic-parabolic systems including the equations of thermoviscoelasticity. The main difficulties are to (i) obtain Fréchet differentiability of the time- T solution operator by appropriate hyperbolic-parabolic energy estimates, and (ii) handle O(2) symmetry in the absence of either center manifold reduction (due to lack of spectral gap) or (due to nonstandard quasilinear hyperbolic-parabolic form) the requisite framework for treatment by spatial dynamics on the space of time-periodic functions, the two standard treatments for this problem. The latter issue is resolved by Lyapunov-Schmidt reduction of the time- T map, yielding a four-dimensional problem with O(2) plus approximate S1 symmetry, which we treat "by hand" using direct Implicit Function Theorem arguments. The former is treated by balancing information obtained in Lagrangian coordinates with that from associated constraints. Interestingly, this argument does not apply to gas dynamics or magnetohydrodynamics (MHD), due to the infinite-dimensional family of Lagrangian symmetries corresponding to invariance under arbitrary volume-preserving diffeomorphisms.

Harvesting in delayed food web model with omnivory

NASA Astrophysics Data System (ADS)

Collera, Juancho A.

2016-02-01

We consider a tri-trophic community module called intraguild predation (IGP) that includes a prey and its predator which share a common basal resource for their sustenance. The growth of the basal resource in the absence of predation follows the Hutchinson's equation where the delay parameter arises, while functional responses in our model are of Lotka-Volterra type. Moreover, the basal resource is harvested for its economic value with a constant harvesting rate. This work generalizes the previous works on the same model with no harvesting and no time delay. We show that the harvesting rate has to be small enough in order for the equilibria to exist. Moreover, we show that by increasing the delay parameter the stability of the equilibrium solutions may change, and periodic solutions may emerge through Hopf bifurcations. In the case of the positive equilibrium solution, multiple stability switches are obtained, and numerical continuation shows that a stable branch of periodic solutions emerges once the positive equilibrium loses its stability at the first Hopf bifurcation point. This result is important because it gives an alternative for the coexistence of all three species, avoiding extinction of one or more species when the positive equilibrium becomes unstable.

Non-linear dynamic characteristics and optimal control of giant magnetostrictive film subjected to in-plane stochastic excitation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhu, Z. W., E-mail: zhuzhiwen@tju.edu.cn; Tianjin Key Laboratory of Non-linear Dynamics and Chaos Control, 300072, Tianjin; Zhang, W. D., E-mail: zhangwenditju@126.com

2014-03-15

The non-linear dynamic characteristics and optimal control of a giant magnetostrictive film (GMF) subjected to in-plane stochastic excitation were studied. Non-linear differential items were introduced to interpret the hysteretic phenomena of the GMF, and the non-linear dynamic model of the GMF subjected to in-plane stochastic excitation was developed. The stochastic stability was analysed, and the probability density function was obtained. The condition of stochastic Hopf bifurcation and noise-induced chaotic response were determined, and the fractal boundary of the system's safe basin was provided. The reliability function was solved from the backward Kolmogorov equation, and an optimal control strategy was proposedmore » in the stochastic dynamic programming method. Numerical simulation shows that the system stability varies with the parameters, and stochastic Hopf bifurcation and chaos appear in the process; the area of the safe basin decreases when the noise intensifies, and the boundary of the safe basin becomes fractal; the system reliability improved through stochastic optimal control. Finally, the theoretical and numerical results were proved by experiments. The results are helpful in the engineering applications of GMF.« less

Dynamics in a ratio-dependent predator-prey model with predator harvesting

NASA Astrophysics Data System (ADS)

Xiao, Dongmei; Li, Wenxia; Han, Maoan

2006-12-01

The objective of this paper is to study systematically the dynamical properties of a ratio-dependent predator-prey model with nonzero constant rate predator harvesting. It is shown that the model has at most two equilibria in the first quadrant and can exhibit numerous kinds of bifurcation phenomena, including the bifurcation of cusp type of codimension 2 (i.e., Bogdanov-Takens bifurcation), the subcritical and supercritical Hopf bifurcations. These results reveal far richer dynamics compared to the model with no harvesting and different dynamics compared to the model with nonzero constant rate prey harvesting in [D. Xiao, L. Jennings, Bifurcations of a ratio-dependent predator-prey system with constant rate harvesting, SIAM Appl. Math. 65 (2005) 737-753]. Biologically, it is shown that nonzero constant rate predator harvesting can prevent mutual extinction as a possible outcome of the predator prey interaction, and remove the singularity of the origin, which was regarded as "pathological behavior" for a ratio-dependent predator prey model in [P. Yodzis, Predator-prey theory and management of multispecies fisheries, Ecological Applications 4 (2004) 51-58].

Ecoepidemic predator-prey model with feeding satiation, prey herd behavior and abandoned infected prey.

PubMed

Kooi, Bob W; Venturino, Ezio

2016-04-01

In this paper we analyse a predator-prey model where the prey population shows group defense and the prey individuals are affected by a transmissible disease. The resulting model is of the Rosenzweig-MacArthur predator-prey type with an SI (susceptible-infected) disease in the prey. Modeling prey group defense leads to a square root dependence in the Holling type II functional for the predator-prey interaction term. The system dynamics is investigated using simulations, classical existence and asymptotic stability analysis and numerical bifurcation analysis. A number of bifurcations, such as transcritical and Hopf bifurcations which occur commonly in predator-prey systems will be found. Because of the square root interaction term there is non-uniqueness of the solution and a singularity where the prey population goes extinct in a finite time. This results in a collapse initiated by extinction of the healthy or susceptible prey and thereafter the other population(s). When also a positive attractor exists this leads to bistability similar to what is found in predator-prey models with a strong Allee effect. For the two-dimensional disease-free (i.e. the purely demographic) system the region in the parameter space where bistability occurs is marked by a global bifurcation. At this bifurcation a heteroclinic connection exists between saddle prey-only equilibrium points where a stable limit cycle together with its basin of attraction, are destructed. In a companion paper (Gimmelli et al., 2015) the same model was formulated and analysed in which the disease was not in the prey but in the predator. There we also observed this phenomenon. Here we extend its analysis using a phase portrait analysis. For the three-dimensional ecoepidemic predator-prey system where the prey is affected by the disease, also tangent bifurcations including a cusp bifurcation and a torus bifurcation of limit cycles occur. This leads to new complex dynamics. Continuation by varying one parameter of the emerging quasi-periodic dynamics from a torus bifurcation can lead to its destruction by a collision with a saddle-cycle. Under other conditions the quasi-periodic dynamics changes gradually in a trajectory that lands on a boundary point where the prey go extinct in finite time after which a total collapse of the three-dimensional system occurs. Copyright © 2016 Elsevier Inc. All rights reserved.

Bifurcation and Fractal of the Coupled Logistic Map

NASA Astrophysics Data System (ADS)

Wang, Xingyuan; Luo, Chao

The nature of the fixed points of the coupled Logistic map is researched, and the boundary equation of the first bifurcation of the coupled Logistic map in the parameter space is given out. Using the quantitative criterion and rule of system chaos, i.e., phase graph, bifurcation graph, power spectra, the computation of the fractal dimension, and the Lyapunov exponent, the paper reveals the general characteristics of the coupled Logistic map transforming from regularity to chaos, the following conclusions are shown: (1) chaotic patterns of the coupled Logistic map may emerge out of double-periodic bifurcation and Hopf bifurcation, respectively; (2) during the process of double-period bifurcation, the system exhibits self-similarity and scale transform invariability in both the parameter space and the phase space. From the research of the attraction basin and Mandelbrot-Julia set of the coupled Logistic map, the following conclusions are indicated: (1) the boundary between periodic and quasiperiodic regions is fractal, and that indicates the impossibility to predict the moving result of the points in the phase plane; (2) the structures of the Mandelbrot-Julia sets are determined by the control parameters, and their boundaries have the fractal characteristic.

Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation

NASA Astrophysics Data System (ADS)

Bonciolini, Giacomo; Ebi, Dominik; Boujo, Edouard; Noiray, Nicolas

2018-03-01

Complex systems exhibiting critical transitions when one of their governing parameters varies are ubiquitous in nature and in engineering applications. Despite a vast literature focusing on this topic, there are few studies dealing with the effect of the rate of change of the bifurcation parameter on the tipping points. In this work, we consider a subcritical stochastic Hopf bifurcation under two scenarios: the bifurcation parameter is first changed in a quasi-steady manner and then, with a finite ramping rate. In the latter case, a rate-dependent bifurcation delay is observed and exemplified experimentally using a thermoacoustic instability in a combustion chamber. This delay increases with the rate of change. This leads to a state transition of larger amplitude compared with the one that would be experienced by the system with a quasi-steady change of the parameter. We also bring experimental evidence of a dynamic hysteresis caused by the bifurcation delay when the parameter is ramped back. A surrogate model is derived in order to predict the statistic of these delays and to scrutinize the underlying stochastic dynamics. Our study highlights the dramatic influence of a finite rate of change of bifurcation parameters upon tipping points, and it pinpoints the crucial need of considering this effect when investigating critical transitions.

Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation

PubMed Central

Noiray, Nicolas

2018-01-01

Complex systems exhibiting critical transitions when one of their governing parameters varies are ubiquitous in nature and in engineering applications. Despite a vast literature focusing on this topic, there are few studies dealing with the effect of the rate of change of the bifurcation parameter on the tipping points. In this work, we consider a subcritical stochastic Hopf bifurcation under two scenarios: the bifurcation parameter is first changed in a quasi-steady manner and then, with a finite ramping rate. In the latter case, a rate-dependent bifurcation delay is observed and exemplified experimentally using a thermoacoustic instability in a combustion chamber. This delay increases with the rate of change. This leads to a state transition of larger amplitude compared with the one that would be experienced by the system with a quasi-steady change of the parameter. We also bring experimental evidence of a dynamic hysteresis caused by the bifurcation delay when the parameter is ramped back. A surrogate model is derived in order to predict the statistic of these delays and to scrutinize the underlying stochastic dynamics. Our study highlights the dramatic influence of a finite rate of change of bifurcation parameters upon tipping points, and it pinpoints the crucial need of considering this effect when investigating critical transitions. PMID:29657803

On the linear stability of blood flow through model capillary networks.

PubMed

Davis, Jeffrey M

2014-12-01

Under the approximation that blood behaves as a continuum, a numerical implementation is presented to analyze the linear stability of capillary blood flow through model tree and honeycomb networks that are based on the microvascular structures of biological tissues. The tree network is comprised of a cascade of diverging bifurcations, in which a parent vessel bifurcates into two descendent vessels, while the honeycomb network also contains converging bifurcations, in which two parent vessels merge into one descendent vessel. At diverging bifurcations, a cell partitioning law is required to account for the nonuniform distribution of red blood cells as a function of the flow rate of blood into each descendent vessel. A linearization of the governing equations produces a system of delay differential equations involving the discharge hematocrit entering each network vessel and leads to a nonlinear eigenvalue problem. All eigenvalues in a specified region of the complex plane are captured using a transformation based on contour integrals to construct a linear eigenvalue problem with identical eigenvalues, which are then determined using a standard QR algorithm. The predicted value of the dimensionless exponent in the cell partitioning law at the instability threshold corresponds to a supercritical Hopf bifurcation in numerical simulations of the equations governing unsteady blood flow. Excellent agreement is found between the predictions of the linear stability analysis and nonlinear simulations. The relaxation of the assumption of plug flow made in previous stability analyses typically has a small, quantitative effect on the stability results that depends on the specific network structure. This implementation of the stability analysis can be applied to large networks with arbitrary structure provided only that the connectivity among the network segments is known.

Dynamic Analysis of a Reaction-Diffusion Rumor Propagation Model

NASA Astrophysics Data System (ADS)

Zhao, Hongyong; Zhu, Linhe

2016-06-01

The rapid development of the Internet, especially the emergence of the social networks, leads rumor propagation into a new media era. Rumor propagation in social networks has brought new challenges to network security and social stability. This paper, based on partial differential equations (PDEs), proposes a new SIS rumor propagation model by considering the effect of the communication between the different rumor infected users on rumor propagation. The stabilities of a nonrumor equilibrium point and a rumor-spreading equilibrium point are discussed by linearization technique and the upper and lower solutions method, and the existence of a traveling wave solution is established by the cross-iteration scheme accompanied by the technique of upper and lower solutions and Schauder’s fixed point theorem. Furthermore, we add the time delay to rumor propagation and deduce the conditions of Hopf bifurcation and stability switches for the rumor-spreading equilibrium point by taking the time delay as the bifurcation parameter. Finally, numerical simulations are performed to illustrate the theoretical results.

Wave of chaos in a spatial eco-epidemiological system: Generating realistic patterns of patchiness in rabbit-lynx dynamics.

PubMed

Upadhyay, Ranjit Kumar; Roy, Parimita; Venkataraman, C; Madzvamuse, A

2016-11-01

In the present paper, we propose and analyze an eco-epidemiological model with diffusion to study the dynamics of rabbit populations which are consumed by lynx populations. Existence, boundedness, stability and bifurcation analyses of solutions for the proposed rabbit-lynx model are performed. Results show that in the presence of diffusion the model has the potential of exhibiting Turing instability. Numerical results (finite difference and finite element methods) reveal the existence of the wave of chaos and this appears to be a dominant mode of disease dispersal. We also show the mechanism of spatiotemporal pattern formation resulting from the Hopf bifurcation analysis, which can be a potential candidate for understanding the complex spatiotemporal dynamics of eco-epidemiological systems. Implications of the asymptotic transmission rate on disease eradication among rabbit population which in turn enhances the survival of Iberian lynx are discussed. Crown Copyright © 2016. Published by Elsevier Inc. All rights reserved.

Stability analysis of an HIV/AIDS epidemic model with treatment

NASA Astrophysics Data System (ADS)

Cai, Liming; Li, Xuezhi; Ghosh, Mini; Guo, Baozhu

2009-07-01

An HIV/AIDS epidemic model with treatment is investigated. The model allows for some infected individuals to move from the symptomatic phase to the asymptomatic phase by all sorts of treatment methods. We first establish the ODE treatment model with two infective stages. Mathematical analyses establish that the global dynamics of the spread of the HIV infectious disease are completely determined by the basic reproduction number [real]0. If [real]0<=1, the disease-free equilibrium is globally stable, whereas the unique infected equilibrium is globally asymptotically stable if [real]0>1. Then, we introduce a discrete time delay to the model to describe the time from the start of treatment in the symptomatic stage until treatment effects become visible. The effect of the time delay on the stability of the endemically infected equilibrium is investigated. Moreover, the delay model exhibits Hopf bifurcations by using the delay as a bifurcation parameter. Finally, numerical simulations are presented to illustrate the results.

Ultra-high-frequency chaos in a time-delay electronic device with band-limited feedback.

PubMed

Illing, Lucas; Gauthier, Daniel J

2006-09-01

We report an experimental study of ultra-high-frequency chaotic dynamics generated in a delay-dynamical electronic device. It consists of a transistor-based nonlinearity, commercially-available amplifiers, and a transmission-line for feedback. The feedback is band-limited, allowing tuning of the characteristic time-scales of both the periodic and high-dimensional chaotic oscillations that can be generated with the device. As an example, periodic oscillations ranging from 48 to 913 MHz are demonstrated. We develop a model and use it to compare the experimentally observed Hopf bifurcation of the steady-state to existing theory [Illing and Gauthier, Physica D 210, 180 (2005)]. We find good quantitative agreement of the predicted and the measured bifurcation threshold, bifurcation type and oscillation frequency. Numerical integration of the model yields quasiperiodic and high dimensional chaotic solutions (Lyapunov dimension approximately 13), which match qualitatively the observed device dynamics.

Dynamics in a three species food-web system

NASA Astrophysics Data System (ADS)

Gupta, K.; Gakkhar, S.

2016-04-01

In this paper, the dynamics of a three species food-web system is discussed. The food-web comprises of one predator and two logistically growing competing species. The predator species is taking food from one of the competitors with Holling type II functional response. Another competitor is the amensal species for the predator of first species. The system is shown to be positive and bounded. The stability of various axial points, boundary points and interior point has been investigated. The persistence of the system has been studied. Numerical simulation has been performed to show the occurrence of Hopf bifurcation and stable limit cycle about the interior point. The presence of second competitor and its interaction with predator gives more complex dynamics than the simple prey-predator system. The existence of transcritical bifurcation has been established about two axial points. The existence of periodic attractor having period-2 solution has been shown, when amensal coefficient is chosen as bifurcation parameter.

Streamwise-Localized Solutions with natural 1-fold symmetry

NASA Astrophysics Data System (ADS)

Altmeyer, Sebastian; Willis, Ashley; Hof, Björn

2014-11-01

It has been proposed in recent years that turbulence is organized around unstable invariant solutions, which provide the building blocks of the chaotic dynamics. In direct numerical simulations of pipe flow we show that when imposing a minimal symmetry constraint (reflection in an axial plane only) the formation of turbulence can indeed be explained by dynamical systems concepts. The hypersurface separating laminar from turbulent motion, the edge of turbulence, is spanned by the stable manifolds of an exact invariant solution, a periodic orbit of a spatially localized structure. The turbulent states themselves (turbulent puffs in this case) are shown to arise in a bifurcation sequence from a related localized solution (the upper branch orbit). The rather complex bifurcation sequence involves secondary Hopf bifurcations, frequency locking and a period doubling cascade until eventually turbulent puffs arise. In addition we report preliminary results of the transition sequence for pipe flow without symmetry constraints.

The discrete dynamics of symmetric competition in the plane.

PubMed

Jiang, H; Rogers, T D

1987-01-01

We consider the generalized Lotka-Volterra two-species system xn + 1 = xn exp(r1(1 - xn) - s1yn) yn + 1 = yn exp(r2(1 - yn) - s2xn) originally proposed by R. M. May as a model for competitive interaction. In the symmetric case that r1 = r2 and s1 = s2, a region of ultimate confinement is found and the dynamics therein are described in some detail. The bifurcations of periodic points of low period are studied, and a cascade of period-doubling bifurcations is indicated. Within the confinement region, a parameter region is determined for the stable Hopf bifurcation of a pair of symmetrically placed period-two points, which imposes a second component of oscillation near the stable cycles. It is suggested that the symmetric competitive model contains much of the dynamical complexity to be expected in any discrete two-dimensional competitive model.

Mathematical analysis of a nutrient-plankton system with delay.

PubMed

Rehim, Mehbuba; Zhang, Zhenzhen; Muhammadhaji, Ahmadjan

2016-01-01

A mathematical model describing the interaction of nutrient-plankton is investigated in this paper. In order to account for the time needed by the phytoplankton to mature after which they can release toxins, a discrete time delay is incorporated into the system. Moreover, it is also taken into account discrete time delays which indicates the partially recycled nutrient decomposed by bacteria after the death of biomass. In the first part of our analysis the sufficient conditions ensuring local and global asymptotic stability of the model are obtained. Next, the existence of the Hopf bifurcation as time delay crosses a threshold value is established and, meanwhile, the phenomenon of stability switches is found under certain conditions. Numerical simulations are presented to illustrate the analytical results.

On the structure of cellular solutions in Rayleigh-Benard-Marangoni flows in small-aspect-ratio containers

NASA Technical Reports Server (NTRS)

Dijkstra, Henk A.

1992-01-01

Multiple steady flow patterns occur in surface-tension/buoyancy-driven convection in a liquid layer heated from below (Rayleigh-Benard-Marangoni flows). Techniques of numerical bifurcation theory are used to study the multiplicity and stability of two-dimensional steady flow patterns (rolls) in rectangular small-aspect-ratio containers as the aspect ratio is varied. For pure Marangoni flows at moderate Biot and Prandtl number, the transitions occurring when paths of codimension 1 singularities intersect determine to a large extent the multiplicity of stable patterns. These transitions also lead, for example, to Hopf bifurcations and stable periodic flows for a small range in aspect ratio. The influence of the type of lateral walls on the multiplicity of steady states is considered. 'No-slip' lateral walls lead to hysteresis effects and typically restrict the number of stable flow patterns (with respect to 'slippery' sidewalls) through the occurrence of saddle node bifurcations. In this way 'no-slip' sidewalls induce a selection of certain patterns, which typically have the largest Nusselt number, through secondary bifurcation.

Spatiotemporal Patterns in a Predator-Prey Model with Cross-Diffusion Effect

NASA Astrophysics Data System (ADS)

Sambath, M.; Balachandran, K.; Guin, L. N.

The present research deals with the emergence of spatiotemporal patterns of a two-dimensional (2D) continuous predator-prey system with cross-diffusion effect. First, we work out the critical lines of Hopf and Turing bifurcations of the current model system in a 2D spatial domain by means of bifurcation theory. More specifically, the exact Turing region is specified in a two-parameter space. In effect, by choosing the cross-diffusion coefficient as one of the momentous parameter, we demonstrate that the model system undergoes a sequence of spatiotemporal patterns in a homogeneous environment through diffusion-driven instability. Our results via numerical simulation authenticate that cross-diffusion be able to create stationary patterns which enrich the findings of pattern formation in an ecosystem.

Bursting synchronization dynamics of pancreatic β-cells with electrical and chemical coupling.

PubMed

Meng, Pan; Wang, Qingyun; Lu, Qishao

2013-06-01

Based on bifurcation analysis, the synchronization behaviors of two identical pancreatic β-cells connected by electrical and chemical coupling are investigated, respectively. Various firing patterns are produced in coupled cells when a single cell exhibits tonic spiking or square-wave bursting individually, irrespectively of what the cells are connected by electrical or chemical coupling. On the one hand, cells can burst synchronously for both weak electrical and chemical coupling when an isolated cell exhibits tonic spiking itself. In particular, for electrically coupled cells, under the variation of the coupling strength there exist complex transition processes of synchronous firing patterns such as "fold/limit cycle" type of bursting, then anti-phase continuous spiking, followed by the "fold/torus" type of bursting, and finally in-phase tonic spiking. On the other hand, it is shown that when the individual cell exhibits square-wave bursting, suitable coupling strength can make the electrically coupled system generate "fold/Hopf" bursting via "fold/fold" hysteresis loop; whereas, the chemically coupled cells generate "fold/subHopf" bursting. Especially, chemically coupled bursters can exhibit inverse period-adding bursting sequence. Fast-slow dynamics analysis is applied to explore the generation mechanism of these bursting oscillations. The above analysis of bursting types and the transition may provide us with better insight into understanding the role of coupling in the dynamic behaviors of pancreatic β-cells.

Indo-Pacific ENSO modes in a double-basin Zebiak-Cane model

NASA Astrophysics Data System (ADS)

Wieners, Claudia; de Ruijter, Will; Dijkstra, Henk

2016-04-01

We study Indo-Pacific interactions on ENSO timescales in a double-basin version of the Zebiak-Cane ENSO model, employing both time integrations and bifurcation analysis (continuation methods). The model contains two oceans (the Indian and Pacific Ocean) separated by a meridional wall. Interaction between the basins is possible via the atmosphere overlaying both basins. We focus on the effect of the Indian Ocean (both its mean state and its variability) on ENSO stability. In addition, inspired by analysis of observational data (Wieners et al, Coherent tropical Indo-Pacific interannual climate variability, in review), we investigate the effect of state-dependent atmospheric noise. Preliminary results include the following: 1) The background state of the Indian Ocean stabilises the Pacific ENSO (i.e. the Hopf bifurcation is shifted to higher values of the SST-atmosphere coupling), 2) the West Pacific cooling (warming) co-occurring with El Niño (La Niña) is essential to simulate the phase relations between Pacific and Indian SST anomalies, 3) a non-linear atmosphere is needed to simulate the effect of the Indian Ocean variability onto the Pacific ENSO that is suggested by observations.

Observation of Poincaré-Andronov-Hopf Bifurcation in Cyclotron Maser Emission from a Magnetic Plasma Trap

NASA Astrophysics Data System (ADS)

Shalashov, A. G.; Gospodchikov, E. D.; Izotov, I. V.; Mansfeld, D. A.; Skalyga, V. A.; Tarvainen, O.

2018-04-01

We report the first experimental evidence of a controlled transition from the generation of periodic bursts of electromagnetic radiation into the continuous-wave regime of a cyclotron maser formed in magnetically confined nonequilibrium plasma. The kinetic cyclotron instability of the extraordinary wave of weakly inhomogeneous magnetized plasma is driven by the anisotropic electron population resulting from electron cyclotron plasma heating in a MHD-stable minimum-B open magnetic trap.

On the stability and compressive nonlinearity of a physiologically based model of the cochlea

DOE Office of Scientific and Technical Information (OSTI.GOV)

Nankali, Amir; Grosh, Karl; Department of Biomedical Engineering, University of Michigan, Ann Arbor, Michigan

Hearing relies on a series of coupled electrical, acoustical (fluidic) and mechanical interactions inside the cochlea that enable sound processing. A positive feedback mechanism within the cochlea, called the cochlear amplifier, provides amplitude and frequency selectivity in the mammalian auditory system. The cochlear amplifier and stability are studied using a nonlinear, micromechanical model of the Organ of Corti (OoC) coupled to the electrical potentials in the cochlear ducts. It is observed that the mechano-electrical transduction (MET) sensitivity and somatic motility of the outer hair cell (OHC), control the cochlear stability. Increasing MET sensitivity beyond a critical value, while electromechanical couplingmore » coefficient is within a specific range, causes instability. We show that instability in this model is generated through a supercritical Hopf bifurcation. A reduced order model of the system is approximated and it is shown that the tectorial membrane (TM) transverse mode effect on the dynamics is significant while the radial mode can be simplified from the equations. The cochlear amplifier in this model exhibits good agreement with the experimental data. A comprehensive 3-dimensional model based on the cross sectional model is simulated and the results are compared. It is indicated that the global model qualitatively inherits some characteristics of the local model, but the longitudinal coupling along the cochlea shifts the stability boundary (i.e., Hopf bifurcation point) and enhances stability.« less

Non-equilibrium thermodynamical description of rhythmic motion patterns of active systems: a canonical-dissipative approach.

PubMed

Dotov, D G; Kim, S; Frank, T D

2015-02-01

We derive explicit expressions for the non-equilibrium thermodynamical variables of a canonical-dissipative limit cycle oscillator describing rhythmic motion patterns of active systems. These variables are statistical entropy, non-equilibrium internal energy, and non-equilibrium free energy. In particular, the expression for the non-equilibrium free energy is derived as a function of a suitable control parameter. The control parameter determines the Hopf bifurcation point of the deterministic active system and describes the effective pumping of the oscillator. In analogy to the equilibrium free energy of the Landau theory, it is shown that the non-equilibrium free energy decays as a function of the control parameter. In doing so, a similarity between certain equilibrium and non-equilibrium phase transitions is pointed out. Data from an experiment on human rhythmic movements is presented. Estimates for pumping intensity as well as the thermodynamical variables are reported. It is shown that in the experiment the non-equilibrium free energy decayed when pumping intensity was increased, which is consistent with the theory. Moreover, pumping intensities close to zero could be observed at relatively slow intended rhythmic movements. In view of the Hopf bifurcation underlying the limit cycle oscillator model, this observation suggests that the intended limit cycle movements were actually more similar to trajectories of a randomly perturbed stable focus. Copyright © 2015 Elsevier Ireland Ltd. All rights reserved.

Frequency locking in auditory hair cells: Distinguishing between additive and parametric forcing

NASA Astrophysics Data System (ADS)

Edri, Yuval; Bozovic, Dolores; Yochelis, Arik

2016-10-01

The auditory system displays remarkable sensitivity and frequency discrimination, attributes shown to rely on an amplification process that involves a mechanical as well as a biochemical response. Models that display proximity to an oscillatory onset (also known as Hopf bifurcation) exhibit a resonant response to distinct frequencies of incoming sound, and can explain many features of the amplification phenomenology. To understand the dynamics of this resonance, frequency locking is examined in a system near the Hopf bifurcation and subject to two types of driving forces: additive and parametric. Derivation of a universal amplitude equation that contains both forcing terms enables a study of their relative impact on the hair cell response. In the parametric case, although the resonant solutions are 1 : 1 frequency locked, they show the coexistence of solutions obeying a phase shift of π, a feature typical of the 2 : 1 resonance. Different characteristics are predicted for the transition from unlocked to locked solutions, leading to smooth or abrupt dynamics in response to different types of forcing. The theoretical framework provides a more realistic model of the auditory system, which incorporates a direct modulation of the internal control parameter by an applied drive. The results presented here can be generalized to many other media, including Faraday waves, chemical reactions, and elastically driven cardiomyocytes, which are known to exhibit resonant behavior.

Dynamics and Physiological Roles of Stochastic Firing Patterns Near Bifurcation Points

NASA Astrophysics Data System (ADS)

Jia, Bing; Gu, Huaguang

2017-06-01

Different stochastic neural firing patterns or rhythms that appeared near polarization or depolarization resting states were observed in biological experiments on three nervous systems, and closely matched those simulated near bifurcation points between stable equilibrium point and limit cycle in a theoretical model with noise. The distinct dynamics of spike trains and interspike interval histogram (ISIH) of these stochastic rhythms were identified and found to build a relationship to the coexisting behaviors or fixed firing frequency of four different types of bifurcations. Furthermore, noise evokes coherence resonances near bifurcation points and plays important roles in enhancing information. The stochastic rhythms corresponding to Hopf bifurcation points with fixed firing frequency exhibited stronger coherence degree and a sharper peak in the power spectrum of the spike trains than those corresponding to saddle-node bifurcation points without fixed firing frequency. Moreover, the stochastic firing patterns changed to a depolarization resting state as the extracellular potassium concentration increased for the injured nerve fiber related to pathological pain or static blood pressure level increased for aortic depressor nerve fiber, and firing frequency decreased, which were different from the physiological viewpoint that firing frequency increased with increasing pressure level or potassium concentration. This shows that rhythms or firing patterns can reflect pressure or ion concentration information related to pathological pain information. Our results present the dynamics of stochastic firing patterns near bifurcation points, which are helpful for the identification of both dynamics and physiological roles of complex neural firing patterns or rhythms, and the roles of noise.

Analysis of Prey-Predator Three Species Fishery Model with Harvesting Including Prey Refuge and Migration

NASA Astrophysics Data System (ADS)

Roy, Sankar Kumar; Roy, Banani

In this article, a prey-predator system with Holling type II functional response for the predator population including prey refuge region has been analyzed. Also a harvesting effort has been considered for the predator population. The density-dependent mortality rate for the prey, predator and super predator has been considered. The equilibria of the proposed system have been determined. Local and global stabilities for the system have been discussed. We have used the analytic approach to derive the global asymptotic stabilities of the system. The maximal predator per capita consumption rate has been considered as a bifurcation parameter to evaluate Hopf bifurcation in the neighborhood of interior equilibrium point. Also, we have used fishing effort to harvest predator population of the system as a control to develop a dynamic framework to investigate the optimal utilization of the resource, sustainability properties of the stock and the resource rent is earned from the resource. Finally, we have presented some numerical simulations to verify the analytic results and the system has been analyzed through graphical illustrations.

Constructing Hopf bifurcation lines for the stability of nonlinear systems with two time delays

NASA Astrophysics Data System (ADS)

Nguimdo, Romain Modeste

2018-03-01

Although the plethora real-life systems modeled by nonlinear systems with two independent time delays, the algebraic expressions for determining the stability of their fixed points remain the Achilles' heel. Typically, the approach for studying the stability of delay systems consists in finding the bifurcation lines separating the stable and unstable parameter regions. This work deals with the parametric construction of algebraic expressions and their use for the determination of the stability boundaries of fixed points in nonlinear systems with two independent time delays. In particular, we concentrate on the cases for which the stability of the fixed points can be ascertained from a characteristic equation corresponding to that of scalar two-delay differential equations, one-component dual-delay feedback, or nonscalar differential equations with two delays for which the characteristic equation for the stability analysis can be reduced to that of a scalar case. Then, we apply our obtained algebraic expressions to identify either the parameter regions of stable microwaves generated by dual-delay optoelectronic oscillators or the regions of amplitude death in identical coupled oscillators.

A theoretical approach on controlling agricultural pest by biological controls.

PubMed

Mondal, Prasanta Kumar; Jana, Soovoojeet; Kar, T K

2014-03-01

In this paper we propose and analyze a prey-predator type dynamical system for pest control where prey population is treated as the pest. We consider two classes for the pest namely susceptible pest and infected pest and the predator population is the natural enemy of the pest. We also consider average delay for both the predation rate i.e. predation to the susceptible pest and infected pest. Considering a subsystem of original system in the absence of infection, we analyze the existence of all possible non-negative equilibria and their stability criteria for both the subsystem as well as the original system. We present the conditions for transcritical bifurcation and Hopf bifurcation in the disease free system. The theoretical evaluations are demonstrated through numerical simulations.

Chimera states in multi-strain epidemic models with temporary immunity

NASA Astrophysics Data System (ADS)

Bauer, Larissa; Bassett, Jason; Hövel, Philipp; Kyrychko, Yuliya N.; Blyuss, Konstantin B.

2017-11-01

We investigate a time-delayed epidemic model for multi-strain diseases with temporary immunity. In the absence of cross-immunity between strains, dynamics of each individual strain exhibit emergence and annihilation of limit cycles due to a Hopf bifurcation of the endemic equilibrium, and a saddle-node bifurcation of limit cycles depending on the time delay associated with duration of temporary immunity. Effects of all-to-all and non-local coupling topologies are systematically investigated by means of numerical simulations, and they suggest that cross-immunity is able to induce a diverse range of complex dynamical behaviors and synchronization patterns, including discrete traveling waves, solitary states, and amplitude chimeras. Interestingly, chimera states are observed for narrower cross-immunity kernels, which can have profound implications for understanding the dynamics of multi-strain diseases.

Interacting Turing-Hopf Instabilities Drive Symmetry-Breaking Transitions in a Mean-Field Model of the Cortex: A Mechanism for the Slow Oscillation

NASA Astrophysics Data System (ADS)

Steyn-Ross, Moira L.; Steyn-Ross, D. A.; Sleigh, J. W.

2013-04-01

Electrical recordings of brain activity during the transition from wake to anesthetic coma show temporal and spectral alterations that are correlated with gross changes in the underlying brain state. Entry into anesthetic unconsciousness is signposted by the emergence of large, slow oscillations of electrical activity (≲1Hz) similar to the slow waves observed in natural sleep. Here we present a two-dimensional mean-field model of the cortex in which slow spatiotemporal oscillations arise spontaneously through a Turing (spatial) symmetry-breaking bifurcation that is modulated by a Hopf (temporal) instability. In our model, populations of neurons are densely interlinked by chemical synapses, and by interneuronal gap junctions represented as an inhibitory diffusive coupling. To demonstrate cortical behavior over a wide range of distinct brain states, we explore model dynamics in the vicinity of a general-anesthetic-induced transition from “wake” to “coma.” In this region, the system is poised at a codimension-2 point where competing Turing and Hopf instabilities coexist. We model anesthesia as a moderate reduction in inhibitory diffusion, paired with an increase in inhibitory postsynaptic response, producing a coma state that is characterized by emergent low-frequency oscillations whose dynamics is chaotic in time and space. The effect of long-range axonal white-matter connectivity is probed with the inclusion of a single idealized point-to-point connection. We find that the additional excitation from the long-range connection can provoke seizurelike bursts of cortical activity when inhibitory diffusion is weak, but has little impact on an active cortex. Our proposed dynamic mechanism for the origin of anesthetic slow waves complements—and contrasts with—conventional explanations that require cyclic modulation of ion-channel conductances. We postulate that a similar bifurcation mechanism might underpin the slow waves of natural sleep and comment on the possible consequences of chaotic dynamics for memory processing and learning.

Dynamics of absence seizures

NASA Astrophysics Data System (ADS)

Deeba, Farah; Sanz-Leon, Paula; Robinson, Peter

A neural field model of the corticothalamic system is used to investigate the dynamics of absence seizures in the presence of temporally varying connection strength between the cerebral cortex and thalamus. Variation of connection strength from cortex to thalamus drives the system into seizure once a threshold is passed and a supercritical Hopf bifurcation occurs. The dynamics and spectral characteristics of the resulting seizures are explored as functions of maximum connection strength, time above threshold, and ramp rate. The results enable spectral and temporal characteristics of seizures to be related to underlying physiological variations via nonlinear dynamics and neural field theory. Notably, this analysis adds to neural field modeling of a wide variety of brain activity phenomena and measurements in recent years. Australian Research Council Grants FL1401000225 and CE140100007.

Explicit solutions of normal form of driven oscillatory systems in entrainment bands

NASA Astrophysics Data System (ADS)

Tsarouhas, George E.; Ross, John

1988-11-01

As in a prior article (Ref. 1), we consider an oscillatory dissipative system driven by external sinusoidal perturbations of given amplitude Q and frequency ω. The kinetic equations are transformed to normal form and solved for small Q near a Hopf bifurcation to oscillations in the autonomous system. Whereas before we chose irrational ratios of the frequency of the autonomous system ωn to ω, with quasiperiodic response of the system to the perturbation, we now choose rational coprime ratios, with periodic response (entrainment). The dissipative system has either two variables or is adequately described by two variables near the bifurcation. We obtain explicit solutions and develop these in detail for ωn/ω=1; 1:2; 2:1; 1:3; 3:1. We choose a specific dissipative model (Brusselator) and test the theory by comparison with full numerical solutions. The analytic solutions of the theory give an excellent approximation for the autonomous system near the bifurcation. The theoretically predicted and calculated entrainment bands agree very well for small Q in the vicinity of the bifurcation (small μ); deviations increase with increasing Q and μ. The theory is applicable to one or two external periodic perturbations.

A model of economic growth with physical and human capital: The role of time delays.

PubMed

Gori, Luca; Guerrini, Luca; Sodini, Mauro

2016-09-01

This article aims at analysing a two-sector economic growth model with discrete delays. The focus is on the dynamic properties of the emerging system. In particular, this study concentrates on the stability properties of the stationary solution, characterised by analytical results and geometrical techniques (stability crossing curves), and the conditions under which oscillatory dynamics emerge (through Hopf bifurcations). In addition, this article proposes some numerical simulations to illustrate the behaviour of the system when the stationary equilibrium is unstable.

Stability and oscillations in a CML model

NASA Astrophysics Data System (ADS)

Badralexi, Irina; Halanay, Andrei

2017-01-01

We capture the evolution in competition of healthy and leukemic cells in Chronic Myelogenous Leukemia (CML) taking into consideration the response of the immune system. Delay-differential equations in a Mackey-Glass approach are used. We start with the study of stability of the equilibrium points of the system. Conditions on parameters for the local stability are given. Oscillatory behaviors occur naturally in biological phenomena. Thus, we investigate the periodic behavior of solutions and we obtain conditions for periodic solutions to appear through a Hopf bifurcation.

Femtojoule-scale all-optical latching and modulation via cavity nonlinear optics.

PubMed

Kwon, Yeong-Dae; Armen, Michael A; Mabuchi, Hideo

2013-11-15

We experimentally characterize Hopf bifurcation phenomena at femtojoule energy scales in a multiatom cavity quantum electrodynamical (cavity QED) system and demonstrate how such behaviors can be exploited in the design of all-optical memory and modulation devices. The data are analyzed by using a semiclassical model that explicitly treats heterogeneous coupling of atoms to the cavity mode. Our results highlight the interest of cavity QED systems for ultralow power photonic signal processing as well as for fundamental studies of mesoscopic nonlinear dynamics.

Explodator: A new skeleton mechanism for the halate driven chemical oscillators

NASA Astrophysics Data System (ADS)

Noszticzius, Z.; Farkas, H.; Schelly, Z. A.

1984-06-01

In the first part of this work, some shortcomings in the present theories of the Belousov-Zhabotinskii oscillating reaction are discussed. In the second part, a new oscillatory scheme, the limited Explodator, is proposed as an alternative skeleton mechanism. This model contains an always unstable three-variable Lotka-Volterra core (the ``Explodator'') and a stabilizing limiting reaction. The new scheme exhibits Hopf bifurcation and limit cycle oscillations. Finally, some possibilities and problems of a generalization are mentioned.

A Model for Amplification of Hair-Bundle Motion by Cyclical Binding of Ca2+ to Mechanoelectrical-Transduction Channels

NASA Astrophysics Data System (ADS)

Choe, Yong; Magnasco, Marcelo O.; Hudspeth, A. J.

1998-12-01

Amplification of auditory stimuli by hair cells augments the sensitivity of the vertebrate inner ear. Cell-body contractions of outer hair cells are thought to mediate amplification in the mammalian cochlea. In vertebrates that lack these cells, and perhaps in mammals as well, active movements of hair bundles may underlie amplification. We have evaluated a mathematical model in which amplification stems from the activity of mechanoelectrical-transduction channels. The intracellular binding of Ca2+ to channels is posited to promote their closure, which increases the tension in gating springs and exerts a negative force on the hair bundle. By enhancing bundle motion, this force partially compensates for viscous damping by cochlear fluids. Linear stability analysis of a six-state kinetic model reveals Hopf bifurcations for parameter values in the physiological range. These bifurcations signal conditions under which the system's behavior changes from a damped oscillatory response to spontaneous limit-cycle oscillation. By varying the number of stereocilia in a bundle and the rate constant for Ca2+ binding, we calculate bifurcation frequencies spanning the observed range of auditory sensitivity for a representative receptor organ, the chicken's cochlea. Simulations using prebifurcation parameter values demonstrate frequency-selective amplification with a striking compressive nonlinearity. Because transduction channels occur universally in hair cells, this active-channel model describes a mechanism of auditory amplification potentially applicable across species and hair-cell types.

Sustained and transient oscillations and chaos induced by delayed antiviral immune response in an immunosuppressive infection model.

PubMed

Shu, Hongying; Wang, Lin; Watmough, James

2014-01-01

Sustained and transient oscillations are frequently observed in clinical data for immune responses in viral infections such as human immunodeficiency virus, hepatitis B virus, and hepatitis C virus. To account for these oscillations, we incorporate the time lag needed for the expansion of immune cells into an immunosuppressive infection model. It is shown that the delayed antiviral immune response can induce sustained periodic oscillations, transient oscillations and even sustained aperiodic oscillations (chaos). Both local and global Hopf bifurcation theorems are applied to show the existence of periodic solutions, which are illustrated by bifurcation diagrams and numerical simulations. Two types of bistability are shown to be possible: (i) a stable equilibrium can coexist with another stable equilibrium, and (ii) a stable equilibrium can coexist with a stable periodic solution.

Semi-analytical solutions of the Schnakenberg model of a reaction-diffusion cell with feedback

NASA Astrophysics Data System (ADS)

Al Noufaey, K. S.

2018-06-01

This paper considers the application of a semi-analytical method to the Schnakenberg model of a reaction-diffusion cell. The semi-analytical method is based on the Galerkin method which approximates the original governing partial differential equations as a system of ordinary differential equations. Steady-state curves, bifurcation diagrams and the region of parameter space in which Hopf bifurcations occur are presented for semi-analytical solutions and the numerical solution. The effect of feedback control, via altering various concentrations in the boundary reservoirs in response to concentrations in the cell centre, is examined. It is shown that increasing the magnitude of feedback leads to destabilization of the system, whereas decreasing this parameter to negative values of large magnitude stabilizes the system. The semi-analytical solutions agree well with numerical solutions of the governing equations.

Photon-phonon parametric oscillation induced by quadratic coupling in an optomechanical resonator

NASA Astrophysics Data System (ADS)

Zhang, Lin; Ji, Fengzhou; Zhang, Xu; Zhang, Weiping

2017-07-01

A direct photon-phonon parametric effect of quadratic coupling on the mean-field dynamics of an optomechanical resonator in the large-scale-movement regime is found and investigated. Under a weak pumping power, the mechanical resonator damps to a steady state with a nonlinear static response sensitively modified by the quadratic coupling. When the driving power increases beyond the static energy balance, the steady states lose their stabilities via Hopf bifurcations, and the resonator produces stable self-sustained oscillation (limit-circle behavior) of discrete energies with step-like amplitudes due to the parametric effect of quadratic coupling, which can be understood roughly by the power balance between gain and loss on the resonator. A further increase in the pumping power can induce a chaotic dynamic of the resonator via a typical routine of period-doubling bifurcation, but which can be stabilized by the parametric effect through an inversion-bifurcation process back to the limit-circle states. The bifurcation-to-inverse-bifurcation transitions are numerically verified by the maximal Lyapunov exponents of the dynamics, which indicate an efficient way of suppressing the chaotic behavior of the optomechanical resonator by quadratic coupling. Furthermore, the parametric effect of quadratic coupling on the dynamic transitions of an optomechanical resonator can be conveniently detected or traced by the output power spectrum of the cavity field.

Passive broadband targeted energy transfers and control of self-excited vibrations

NASA Astrophysics Data System (ADS)

Lee, Young S.

This work consists of the three main parts---Nonlinear energy pumping (that is, passive broadband targeted energy transfers---TETs), and its applications to theoretical and experimental suppression of aeroelastic instabilities. In the first part, nonlinear energy pumping (or TETs) in coupled oscillators is studied. The system is composed of a primary linear subsystem coupled through an essentially nonlinear stiffness and a linear viscous damper to an additional mass (which is called, as a whole, a nonlinear energy sink---NES). By considering the linear damping as a perturbation to the system, periodic solutions of the underlying Hamiltonian system are formulated by means of the non-smooth temporal transformation and solved numerically by a shooting method. The special periodic orbits, which are corresponding to the impulsive initial conditions for the primary subsystem, bear their importance as baits for initiating localized transfers of a significant portion of energy to the NES. The second part theoretically deals with suppression of limit cycle oscillations (LCOs) in self-excited systems by means of passive energy localizations. As a pilot scheme, suppression or even complete elimination of the LCO in a van der Pol (VDP) oscillator coupled with two types of NESS---grounded and ungrounded---is studied. Computational parametric study proves the efficacy of LCO elimination by means of passive nonlinear energy pumping from the VDP oscillator to appropriately designed NESs. The numerical study of the transient dynamics of the system showed that the dynamical mechanism for LCO suppression is a series of 1:1 and 1:3 transient resonance captures, with the damped transient dynamics following closely corresponding resonant manifolds of the underlying Hamiltonian system. It is through the TRCs that energy gets transferred from the VDP oscillator to the NES, thus causing LCO suppression. By performing an additional bifurcation analysis of the steady state responses through a numerical continuation of equilibria and periodic solutions, the parameter dependence and bifurcations of the steady-state solutions are examined. It is also proved that a Hopf bifurcation is the global dynamical mechanism for generation and elimination of the LCOs in the configurations considered. The bifurcation analysis revealed that it is possible to design grounded or ungrounded NESs that robustly and completely eliminate the LCO instability of the system. This should be possible when the system parameters are chosen such that a subcritical Hopf bifurcation occurs, thus assuring the existence of a unique global trivial attractor of the dynamics in the parameter ranges of interest. Then, triggering mechanisms of aeroelastic instability is investigated for a two-DOF rigid wing model in subsonic flow with cubic nonlinear stiffnesses at the support. Based on the observation of the instability triggering, a single-degree-of-freedom (SDOF) NES is applied to the wing model. The NES is attached at an offset from the elastic axis for its additional interaction with the pitch mode, as well as being parallel with the heave mode, primarily to hinder initial triggering of the heave mode by the flow. It is shown that it is feasible to partially or even completely suppress aeroelastic instabilities of the wing by passively transferring vibration energy from the wing to the NES in a one-way irreversible fashion. Moreover, this aeroelastic instability suppression is performed by partially or completely eliminating the triggering mechanists for aeroelastic suppression. Through numerical parametric studies three main mechanisms for suppressing aeroelastic instability are identified: (i) Recurring burst-out and suppression; (ii) intermediate suppression; (iii) complete elimination of instability. In general, the relative occurrence of one of the two limit point cycle (LPC) bifurcations with respect to the Hopf bifurcation decides whether or not the suppression mechanisms are robust. In order to improve robustness of instability suppression, several types of multi-DOF NES configurations are introduced. In the last part, experimental suppression of aeroelastic instability by means of targeted energy transfers is investigated. In order to gain insights into the experiments, theoretical triggering mechanism of the aeroelastic instability in the nonlinear aeroelastic test apparatus (NATA) in a low-speed wind tunnel at Texas A&M University is studied. Finally, experimental results are presented in connection to the theoretical investigation, and all the predictions on the instability suppression mechanisms are demonstrated experimentally. It is also revealed that the dry friction affects only the robustness of an instability suppression by changing the unstable trivial equilibrium into an equilibrium set. (Abstract shortened by UMI.)

Mean-field models for heterogeneous networks of two-dimensional integrate and fire neurons.

PubMed

Nicola, Wilten; Campbell, Sue Ann

2013-01-01

We analytically derive mean-field models for all-to-all coupled networks of heterogeneous, adapting, two-dimensional integrate and fire neurons. The class of models we consider includes the Izhikevich, adaptive exponential and quartic integrate and fire models. The heterogeneity in the parameters leads to different moment closure assumptions that can be made in the derivation of the mean-field model from the population density equation for the large network. Three different moment closure assumptions lead to three different mean-field systems. These systems can be used for distinct purposes such as bifurcation analysis of the large networks, prediction of steady state firing rate distributions, parameter estimation for actual neurons and faster exploration of the parameter space. We use the mean-field systems to analyze adaptation induced bursting under realistic sources of heterogeneity in multiple parameters. Our analysis demonstrates that the presence of heterogeneity causes the Hopf bifurcation associated with the emergence of bursting to change from sub-critical to super-critical. This is confirmed with numerical simulations of the full network for biologically reasonable parameter values. This change decreases the plausibility of adaptation being the cause of bursting in hippocampal area CA3, an area with a sizable population of heavily coupled, strongly adapting neurons.

Quantitative and qualitative characterization of zigzag spatiotemporal chaos in a system of amplitude equations for nematic electroconvection.

PubMed

Oprea, Iuliana; Triandaf, Ioana; Dangelmayr, Gerhard; Schwartz, Ira B

2007-06-01

It has been suggested by experimentalists that a weakly nonlinear analysis of the recently introduced equations of motion for the nematic electroconvection by M. Treiber and L. Kramer [Phys. Rev. E 58, 1973 (1998)] has the potential to reproduce the dynamics of the zigzag-type extended spatiotemporal chaos and localized solutions observed near onset in experiments [M. Dennin, D. S. Cannell, and G. Ahlers, Phys. Rev. E 57, 638 (1998); J. T. Gleeson (private communication)]. In this paper, we study a complex spatiotemporal pattern, identified as spatiotemporal chaos, that bifurcates at the onset from a spatially uniform solution of a system of globally coupled complex Ginzburg-Landau equations governing the weakly nonlinear evolution of four traveling wave envelopes. The Ginzburg-Landau system can be derived directly from the weak electrolyte model for electroconvection in nematic liquid crystals when the primary instability is a Hopf bifurcation to oblique traveling rolls. The chaotic nature of the pattern and the resemblance to the observed experimental spatiotemporal chaos in the electroconvection of nematic liquid crystals are confirmed through a combination of techniques including the Karhunen-Loeve decomposition, time-series analysis of the amplitudes of the dominant modes, statistical descriptions, and normal form theory, showing good agreement between theory and experiments.

Mean-field models for heterogeneous networks of two-dimensional integrate and fire neurons

PubMed Central

Nicola, Wilten; Campbell, Sue Ann

2013-01-01

We analytically derive mean-field models for all-to-all coupled networks of heterogeneous, adapting, two-dimensional integrate and fire neurons. The class of models we consider includes the Izhikevich, adaptive exponential and quartic integrate and fire models. The heterogeneity in the parameters leads to different moment closure assumptions that can be made in the derivation of the mean-field model from the population density equation for the large network. Three different moment closure assumptions lead to three different mean-field systems. These systems can be used for distinct purposes such as bifurcation analysis of the large networks, prediction of steady state firing rate distributions, parameter estimation for actual neurons and faster exploration of the parameter space. We use the mean-field systems to analyze adaptation induced bursting under realistic sources of heterogeneity in multiple parameters. Our analysis demonstrates that the presence of heterogeneity causes the Hopf bifurcation associated with the emergence of bursting to change from sub-critical to super-critical. This is confirmed with numerical simulations of the full network for biologically reasonable parameter values. This change decreases the plausibility of adaptation being the cause of bursting in hippocampal area CA3, an area with a sizable population of heavily coupled, strongly adapting neurons. PMID:24416013

Localized states in an unbounded neural field equation with smooth firing rate function: a multi-parameter analysis.

PubMed

Faye, Grégory; Rankin, James; Chossat, Pascal

2013-05-01

The existence of spatially localized solutions in neural networks is an important topic in neuroscience as these solutions are considered to characterize working (short-term) memory. We work with an unbounded neural network represented by the neural field equation with smooth firing rate function and a wizard hat spatial connectivity. Noting that stationary solutions of our neural field equation are equivalent to homoclinic orbits in a related fourth order ordinary differential equation, we apply normal form theory for a reversible Hopf bifurcation to prove the existence of localized solutions; further, we present results concerning their stability. Numerical continuation is used to compute branches of localized solution that exhibit snaking-type behaviour. We describe in terms of three parameters the exact regions for which localized solutions persist.

Memristive Model of the Barnacle Giant Muscle Fibers

NASA Astrophysics Data System (ADS)

Sah, Maheshwar Pd.; Kim, Hyongsuk; Eroglu, Abdullah; Chua, Leon

The generation of action potentials (oscillations) in biological systems is a complex, yet poorly understood nonlinear dynamical phenomenon involving ions. This paper reveals that the time-varying calcium ion and the time-varying potassium ion, which are essential for generating action potentials in Barnacle giant muscle fibers are in fact generic memristors in the perspective of electrical circuit theory. We will show that these two ions exhibit all the fingerprints of memristors from the equations of the Morris-Lecar model of the Barnacle giant muscle fibers. This paper also gives a textbook reference to understand the difference between memristor and nonlinear resistor via analysis of the potassium ion-channel memristor and calcium ion-channel nonlinear resistor. We will also present a comprehensive in-depth analysis of the generation of action potentials (oscillations) in memristive Morris-Lecar model using small-signal circuit model and the Hopf bifurcation theorem.

Effects of planar shear on the three-dimensional instability in flow past a circular cylinder

NASA Astrophysics Data System (ADS)

Park, Doohyun; Yang, Kyung-Soo

2018-03-01

A Floquet stability analysis has been carried out in order to investigate how a planar shear in wake flow affects the three-dimensional (3D) instability in the near-wake region. We consider a circular cylinder immersed in a freestream with planar shear. The cylinder was implemented in a Cartesian grid system by means of an immersed boundary method. Planar shear tends to promote the primary instability, known as Hopf bifurcation where steady flow bifurcates into time-periodic flow, in the sense that its critical Reynolds number decreases with increasing planar shear. The effects of planar shear on the 3D instability are different depending on the type of 3D instability. The flow asymmetry caused by the planar shear suppresses a QP-type mode but generates a C-type mode. The conventional A and B modes are stabilized by the planar shear, whereas mode C is intensified with increasing shear. The criticality of each 3D mode is discussed, and the neutral stability curves for each 3D mode are presented. The current Floquet results have been validated by using direct numerical simulation for some selected cases of flow parameters.

The temporal patterns of disease severity and prevalence in schistosomiasis

DOE Office of Scientific and Technical Information (OSTI.GOV)

Ciddio, Manuela; Gatto, Marino, E-mail: marino.gatto@polimi.it; Casagrandi, Renato, E-mail: renato.casagrandi@polimi.it

2015-03-15

Schistosomiasis is one of the most widespread public health problems in the world. In this work, we introduce an eco-epidemiological model for its transmission and dynamics with the purpose of explaining both intra- and inter-annual fluctuations of disease severity and prevalence. The model takes the form of a system of nonlinear differential equations that incorporate biological complexity associated with schistosome's life cycle, including a prepatent period in snails (i.e., the time between initial infection and onset of infectiousness). Nonlinear analysis is used to explore the parametric conditions that produce different temporal patterns (stationary, endemic, periodic, and chaotic). For the time-invariantmore » model, we identify a transcritical and a Hopf bifurcation in the space of the human and snail infection parameters. The first corresponds to the occurrence of an endemic equilibrium, while the latter marks the transition to interannual periodic oscillations. We then investigate a more realistic time-varying model in which fertility of the intermediate host population is assumed to seasonally vary. We show that seasonality can give rise to a cascade of period-doubling bifurcations leading to chaos for larger, though realistic, values of the amplitude of the seasonal variation of fertility.« less

The temporal patterns of disease severity and prevalence in schistosomiasis

NASA Astrophysics Data System (ADS)

Ciddio, Manuela; Mari, Lorenzo; Gatto, Marino; Rinaldo, Andrea; Casagrandi, Renato

2015-03-01

Schistosomiasis is one of the most widespread public health problems in the world. In this work, we introduce an eco-epidemiological model for its transmission and dynamics with the purpose of explaining both intra- and inter-annual fluctuations of disease severity and prevalence. The model takes the form of a system of nonlinear differential equations that incorporate biological complexity associated with schistosome's life cycle, including a prepatent period in snails (i.e., the time between initial infection and onset of infectiousness). Nonlinear analysis is used to explore the parametric conditions that produce different temporal patterns (stationary, endemic, periodic, and chaotic). For the time-invariant model, we identify a transcritical and a Hopf bifurcation in the space of the human and snail infection parameters. The first corresponds to the occurrence of an endemic equilibrium, while the latter marks the transition to interannual periodic oscillations. We then investigate a more realistic time-varying model in which fertility of the intermediate host population is assumed to seasonally vary. We show that seasonality can give rise to a cascade of period-doubling bifurcations leading to chaos for larger, though realistic, values of the amplitude of the seasonal variation of fertility.

A Mathematical Model for the Bee Hive of Apis Mellifera

NASA Astrophysics Data System (ADS)

Antonioni, Alberto; Bellom, Fabio Enrici; Montabone, Andrea; Venturino, Ezio

2010-09-01

In this work we introduce and discuss a model for the bee hive, in which only adult bees and drones are modeled. The role that the latter have in the system is interesting, their population can retrieve even if they are totally absent from the bee hive. The feasibility and stability of the equilibria is studied numerically. A simplified version of the model shows the importance of the drones' role, in spite of the fact that it allows only a trivial equilibrium. For this simplified system, no Hopf bifurcations are shown to arise.

A Cross-Validation Approach to Approximate Basis Function Selection of the Stall Flutter Response of a Rectangular Wing in a Wind Tunnel

NASA Technical Reports Server (NTRS)

Kukreja, Sunil L.; Vio, Gareth A.; Andrianne, Thomas; azak, Norizham Abudl; Dimitriadis, Grigorios

2012-01-01

The stall flutter response of a rectangular wing in a low speed wind tunnel is modelled using a nonlinear difference equation description. Static and dynamic tests are used to select a suitable model structure and basis function. Bifurcation criteria such as the Hopf condition and vibration amplitude variation with airspeed were used to ensure the model was representative of experimentally measured stall flutter phenomena. Dynamic test data were used to estimate model parameters and estimate an approximate basis function.

Numerical simulation of nonstationary dissipative structures in 3D double-diffusive convection at large Rayleigh numbers

NASA Astrophysics Data System (ADS)

Kozitskiy, Sergey

2018-06-01

Numerical simulation of nonstationary dissipative structures in 3D double-diffusive convection has been performed by using the previously derived system of complex Ginzburg-Landau type amplitude equations, valid in a neighborhood of Hopf bifurcation points. Simulation has shown that the state of spatiotemporal chaos develops in the system. It has the form of nonstationary structures that depend on the parameters of the system. The shape of structures does not depend on the initial conditions, and a limited number of spectral components participate in their formation.

Numerical simulation of nonstationary dissipative structures in 3D double-diffusive convection at large Rayleigh numbers

NASA Astrophysics Data System (ADS)

Kozitskiy, Sergey

2018-05-01

Numerical simulation of nonstationary dissipative structures in 3D double-diffusive convection has been performed by using the previously derived system of complex Ginzburg-Landau type amplitude equations, valid in a neighborhood of Hopf bifurcation points. Simulation has shown that the state of spatiotemporal chaos develops in the system. It has the form of nonstationary structures that depend on the parameters of the system. The shape of structures does not depend on the initial conditions, and a limited number of spectral components participate in their formation.

Defect chaos of oscillating hexagons in rotating convection

PubMed

Echebarria; Riecke

2000-05-22

Using coupled Ginzburg-Landau equations, the dynamics of hexagonal patterns with broken chiral symmetry are investigated, as they appear in rotating non-Boussinesq or surface-tension-driven convection. We find that close to the secondary Hopf bifurcation to oscillating hexagons the dynamics are well described by a single complex Ginzburg-Landau equation (CGLE) coupled to the phases of the hexagonal pattern. At the band center these equations reduce to the usual CGLE and the system exhibits defect chaos. Away from the band center a transition to a frozen vortex state is found.

Dynamic interaction of monowheel inclined vehicle-vibration platform coupled system with quadratic and cubic nonlinearities

NASA Astrophysics Data System (ADS)

Zhou, Shihua; Song, Guiqiu; Sun, Maojun; Ren, Zhaohui; Wen, Bangchun

2018-01-01

In order to analyze the nonlinear dynamics and stability of a novel design for the monowheel inclined vehicle-vibration platform coupled system (MIV-VPCS) with intermediate nonlinearity support subjected to a harmonic excitation, a multi-degree of freedom lumped parameter dynamic model taking into account the dynamic interaction of the MIV-VPCS with quadratic and cubic nonlinearities is presented. The dynamical equations of the coupled system are derived by applying the displacement relationship, interaction force relationship at the contact position and Lagrange's equation, which are further discretized into a set of nonlinear ordinary differential equations with coupled terms by Galerkin's truncation. Based on the mathematical model, the coupled multi-body nonlinear dynamics of the vibration system is investigated by numerical method, and the parameters influences of excitation amplitude, mass ratio and inclined angle on the dynamic characteristics are precisely analyzed and discussed by bifurcation diagram, Largest Lyapunov exponent and 3-D frequency spectrum. Depending on different ranges of system parameters, the results show that the different motions and jump discontinuity appear, and the coupled system enters into chaotic behavior through different routes (period-doubling bifurcation, inverse period-doubling bifurcation, saddle-node bifurcation and Hopf bifurcation), which are strongly attributed to the dynamic interaction of the MIV-VPCS. The decreasing excitation amplitude and inclined angle could reduce the higher order bifurcations, and effectively control the complicated nonlinear dynamic behaviors under the perturbation of low rotational speed. The first bifurcation and chaotic motion occur at lower value of inclined angle, and the chaotic behavior lasts for larger intervals with higher rotational speed. The investigation results could provide a better understanding of the nonlinear dynamic behaviors for the dynamic interaction of the MIV-VPCS.

Limit cycles and the benefits of a short memory in rock-paper-scissors games.

PubMed

Burridge, James

2015-10-01

When playing games in groups, it is an advantage for individuals to have accurate statistical information on the strategies of their opponents. Such information may be obtained by remembering previous interactions. We consider a rock-paper-scissors game in which agents are able to recall their last m interactions, used to estimate the behavior of their opponents. At critical memory length, a Hopf bifurcation leads to the formation of stable limit cycles. In a mixed population, agents with longer memories have an advantage, provided the system has a stable fixed point, and there is some asymmetry in the payoffs of the pure strategies. However, at a critical concentration of long memory agents, the appearance of limit cycles destroys their advantage. By introducing population dynamics that favors successful agents, we show that the system evolves toward the bifurcation point.

Limit cycles and the benefits of a short memory in rock-paper-scissors games

NASA Astrophysics Data System (ADS)

Burridge, James

2015-10-01

When playing games in groups, it is an advantage for individuals to have accurate statistical information on the strategies of their opponents. Such information may be obtained by remembering previous interactions. We consider a rock-paper-scissors game in which agents are able to recall their last m interactions, used to estimate the behavior of their opponents. At critical memory length, a Hopf bifurcation leads to the formation of stable limit cycles. In a mixed population, agents with longer memories have an advantage, provided the system has a stable fixed point, and there is some asymmetry in the payoffs of the pure strategies. However, at a critical concentration of long memory agents, the appearance of limit cycles destroys their advantage. By introducing population dynamics that favors successful agents, we show that the system evolves toward the bifurcation point.

Mathematical Model of Three Species Food Chain Interaction with Mixed Functional Response

NASA Astrophysics Data System (ADS)

Ws, Mada Sanjaya; Mohd, Ismail Bin; Mamat, Mustafa; Salleh, Zabidin

In this paper, we study mathematical model of ecology with a tritrophic food chain composed of a classical Lotka-Volterra functional response for prey and predator, and a Holling type-III functional response for predator and super predator. There are two equilibrium points of the system. In the parameter space, there are passages from instability to stability, which are called Hopf bifurcation points. For the first equilibrium point, it is possible to find bifurcation points analytically and to prove that the system has periodic solutions around these points. Furthermore the dynamical behaviors of this model are investigated. Models for biologically reasonable parameter values, exhibits stable, unstable periodic and limit cycles. The dynamical behavior is found to be very sensitive to parameter values as well as the parameters of the practical life. Computer simulations are carried out to explain the analytical findings.

Self-organized pattern dynamics of somitogenesis model in embryos

NASA Astrophysics Data System (ADS)

Guan, Linan; Shen, Jianwei

2018-09-01

Somitogenesis, the sequential formation of a periodic pattern along the anteroposterior axis of vertebrate embryos, is one of the most obvious examples of the segmental patterning processes that take place during embryogenesis and also one of the major unresolved events in developmental biology. In this paper, we investigate the effect of diffusion on pattern formation use a modified two dimensional model which can be used to explain somitogenesis during embryonic development. This model is suitable for exploring a design space of somitogenesis and can explain many aspects of somitogenesis that previous models cannot. In the present paper, by analyzing the local linear stability of the equation, we acquired the conditions of Hopf bifurcation and Turing bifurcation. In addition, the amplitude equation near the Turing bifurcation point is obtained by using the methods of multi-scale expansion and symmetry analysis. By analyzing the stability of the amplitude equation, we know that there are various complex phenomena, including Spot pattern, mixture of spot-stripe patterns and labyrinthine. Finally, numerical simulation are given to verify the correctness of our theoretical results. Somitogenesis occupies an important position in the process of biological development, and as a pattern process can be used to investigate many aspects of embryogenesis. Therefore, our study helps greatly to cell differentiation, gene expression and embryonic development. What is more, it is of great significance for the diagnosis and treatment of human diseases to study the related knowledge of model biology.

Beam Flutter and Energy Harvesting in Internal Flow

NASA Astrophysics Data System (ADS)

Tosi, Luis Phillipe; Colonius, Tim; Sherrit, Stewart; Lee, Hyeong Jae

2017-11-01

Aeroelastic flutter, largely studied for causing engineering failures, has more recently been used as a means of extracting energy from the flow. Particularly, flutter of a cantilever or an elastically mounted plate in a converging-diverging flow passage has shown promise as an energy harvesting concept for internal flow applications. The instability onset is observed as a function of throat velocity, internal wall geometry, fluid and structure material properties. To enable these devices, our work explores features of the fluid-structure coupled dynamics as a function of relevant nondimensional parameters. The flutter boundary is examined through stability analysis of a reduced order model, and corroborated with numerical simulations at low Reynolds number. Experiments for an energy harvester design are qualitatively compared to results from analytical and numerical work, suggesting a robust limit cycle ensues due to a subcritical Hopf bifurcation. Bosch Corporation.

Restoration and recovery of damaged eco-epidemiological systems: application to the Salton Sea, California, USA.

PubMed

Upadhyay, Ranjit Kumar; Raw, S N; Roy, P; Rai, Vikas

2013-04-01

In this paper, we have proposed and analysed a mathematical model to figure out possible ways to rescue a damaged eco-epidemiological system. Our strategy of rescue is based on the realization of the fact that chaotic dynamics often associated with excursions of system dynamics to extinction-sized densities. Chaotic dynamics of the model is depicted by 2D scans, bifurcation analysis, largest Lyapunov exponent and basin boundary calculations. 2D scan results show that μ, the total death rate of infected prey should be brought down in order to avoid chaotic dynamics. We have carried out linear and nonlinear stability analysis and obtained Hopf-bifurcation and persistence criteria of the proposed model system. The other outcome of this study is a suggestion which involves removal of infected fishes at regular interval of time. The estimation of timing and periodicity of the removal exercises would be decided by the nature of infection more than anything else. If this suggestion is carefully worked out and implemented, it would be most effective in restoring the health of the ecosystem which has immense ecological, economic and aesthetic potential. We discuss the implications of this result to Salton Sea, California, USA. The restoration of the Salton Sea provides a perspective for conservation and management strategy. Copyright © 2013 Elsevier Inc. All rights reserved.

Bifurcations and chaos in convection taking non-Fourier heat-flux

NASA Astrophysics Data System (ADS)

Layek, G. C.; Pati, N. C.

2017-11-01

In this Letter, we report the influences of thermal time-lag on the onset of convection, its bifurcations and chaos of a horizontal layer of Boussinesq fluid heated underneath taking non-Fourier Cattaneo-Christov hyperbolic model for heat propagation. A five-dimensional nonlinear system is obtained for a low-order Galerkin expansion, and it reduces to Lorenz system for Cattaneo number tending to zero. The linear stability agreed with existing results that depend on Cattaneo number C. It also gives a threshold Cattaneo number, CT, above which only oscillatory solutions can persist. The oscillatory solutions branch terminates at the subcritical steady branch with a heteroclinic loop connecting a pair of saddle points for subcritical steady-state solutions. For subcritical onset of convection two stable solutions coexist, that is, hysteresis phenomenon occurs at this stage. The steady solution undergoes a Hopf bifurcation and is of subcritical type for small value of C, while it becomes supercritical for moderate Cattaneo number. The system goes through period-doubling/noisy period-doubling transition to chaos depending on the control parameters. There after the system exhibits Shil'nikov chaos via homoclinic explosion. The complexity of spiral strange attractor is analyzed using fractal dimension and return map.

Behavior of an aeroelastic system beyond critical point of instability

NASA Astrophysics Data System (ADS)

Sekar, T. Chandra; Agarwal, Ravindra; Mandal, Alakesh Chandra; Kushari, Abhijit

2017-11-01

Understanding the behavior of an aeroelastic system beyond the critical point is essential for effective implementation of any active control scheme since the control system design depends on the type of instability (bifurcation) the system encounters. Previous studies had found the aeroelastic system to enter into chaos beyond the point of instability. In the present work, an attempt has been made to carry out an experimental study on an aeroelastic model placed in a wind tunnel, to understand the behavior of aerodynamics around a wing section undergoing classical flutter. Wind speed was increased from zero until the model encountered flutter. Pressure at various locations along the surface of wing and acceleration at multiple points on the wing were measured in real time for the entire duration of experiment. A Leading Edge Separation Bubble (LSB) was observed beyond the critical point. The growing strength of the LSB with increasing wind speed was found to alter the aerodynamic moment acting on the system, which forced the system to enter into a second bifurcation. Based on the nature of the response, the system appears to undergo periodic doubling bifurcation rather than Hopf-bifurcation, resulting in chaotic motion. Eliminating the LSB can help in preventing the system from entering chaos. Any active flow control scheme that can avoid or counter the formation of leading edge separation bubble can be a potential solution to control the classical flutter.

Noise-induced shifts in the population model with a weak Allee effect

NASA Astrophysics Data System (ADS)

Bashkirtseva, Irina; Ryashko, Lev

2018-02-01

We consider the Truscott-Brindley system of interacting phyto- and zooplankton populations with a weak Allee effect. We add a random noise to the parameter of the prey carrying capacity, and study how the noise affects the dynamic behavior of this nonlinear prey-predator model. Phenomena of the stochastic excitement and noise-induced shifts in zones of the Andronov-Hopf bifurcation and Canard explosion are analyzed on the base of the direct numerical simulation and stochastic sensitivity functions technique. A relationship of these phenomena with transitions between order and chaos is discussed.

Frequency combs with weakly lasing exciton-polariton condensates.

PubMed

Rayanov, K; Altshuler, B L; Rubo, Y G; Flach, S

2015-05-15

We predict the spontaneous modulated emission from a pair of exciton-polariton condensates due to coherent (Josephson) and dissipative coupling. We show that strong polariton-polariton interaction generates complex dynamics in the weak-lasing domain way beyond Hopf bifurcations. As a result, the exciton-polariton condensates exhibit self-induced oscillations and emit an equidistant frequency comb light spectrum. A plethora of possible emission spectra with asymmetric peak distributions appears due to spontaneously broken time-reversal symmetry. The lasing dynamics is affected by the shot noise arising from the influx of polaritons. That results in a complex inhomogeneous line broadening.

Twirling and Whirling: Viscous Dynamics of Rotating Elastica

NASA Astrophysics Data System (ADS)

Powers, Thomas R.; Wolgemuth, Charles W.; Goldstein, Raymond E.

1999-11-01

Motivated by diverse phenomena in cellular biophysics, including bacterial flagellar motion and DNA transcription and replication, we study the overdamped nonlinear dynamics of a rotationally forced filament with twist and bend elasticity. The competition between twist diffusion and writhing instabilities is described by a novel pair of coupled PDEs for twist and bend evolution. Analytical and numerical methods elucidate the twist-bend coupling and reveal two dynamical regimes separated by a Hopf bifurcation: (i) diffusion-dominated axial rotation, or twirling, and (ii) steady-state crankshafting motion, or whirling. The consequences of these phenomena for self-propulsion are investigated, and experimental tests proposed.

Control of Oscillation Patterns in a Symmetric Coupled Biological Oscillator System

NASA Astrophysics Data System (ADS)

Takamatsu, Atsuko; Tanaka, Reiko; Yamamoto, Takatoki; Fujii, Teruo

2003-08-01

A chain of three-oscillator system was constructed with living biological oscillators of phasmodial slime mold, Physarum polycehalum and the oscillation patterns were analyzed by the symmetric Hopf bifurcation theory using group theory. Multi-stability of oscillation patterns was observed, even when the coupling strength was fixed. This suggests that the coupling strength is not an effective parameter to obtain a desired oscillation pattern among the multiple patterns. Here we propose a method to control oscillation patterns using resonance to external stimulus and demonstrate pattern switching induced by frequency resonance given to only one of oscillators in the system.

Finding Limit Cycles in self-excited oscillators with infinite-series damping functions

NASA Astrophysics Data System (ADS)

Das, Debapriya; Banerjee, Dhruba; Bhattacharjee, Jayanta K.

2015-03-01

In this paper we present a simple method for finding the location of limit cycles of self excited oscillators whose damping functions can be represented by some infinite convergent series. We have used standard results of first-order perturbation theory to arrive at amplitude equations. The approach has been kept pedagogic by first working out the cases of finite polynomials using elementary algebra. Then the method has been extended to various infinite polynomials, where the fixed points of the corresponding amplitude equations cannot be found out. Hopf bifurcations for systems with nonlinear powers in velocities have also been discussed.

Dynamic states of a unidirectional ring of chen oscillators

DOE Office of Scientific and Technical Information (OSTI.GOV)

Carvalho, Ana; Pinto, Carla M.A.

2015-03-10

We study curious dynamical patterns appearing in a network of a unidirectional ring of Chen oscillators coupled to a ‘buffer’ cell. The network has Z{sub 3} exact symmetry group. We simulate the coupled cell systems associated to the two networks and obtain steady-states, rotating waves, quasiperiodic behavior, and chaos. The different patterns appear to arise through a sequence of Hopf, period-doubling and period-halving bifurcations. The network architecture appears to explain some patterns, whereas the properties of the chaotic oscillator may explain others. We use XPPAUT and MATLAB to compute numerically the relevant states.

Mixed-mode oscillations in a three-store calcium dynamics model

NASA Astrophysics Data System (ADS)

Liu, Peng; Liu, Xijun; Yu, Pei

2017-11-01

Calcium ions are important in cell process, which control cell functions. Many models on calcium oscillation have been proposed. Most of existing literature analyzed calcium oscillations using numerical methods, and found rich dynamical behaviours. In this paper, we explore a further study on an established three-store model, which contains endoplasmic reticulum (ER), mitochondria and calcium binding proteins. We conduct bifurcation analysis to identify two Hopf bifurcations, and apply normal form theory to study their stability and show that one of them is supercritical while the other is subcritical. Further, we transform the model into a slow-fast system, and then apply the geometrical singular perturbation theory to investigate the mechanism of generating slow-fast motions. The study reveals that the mechanism of generating the slow-fast oscillating behaviour in the three-store calcium model for certain parameter values is due to the relative fast change in the free calcium in cytosol, and relative slow changes in the free calcium in mitochondria and in the bounded Ca2+ binding sites on the cytosolic proteins. A further parametric study may provide some useful information for controlling harmful effect, by adjusting the amount of calcium in a human body. Numerical simulations are present to demonstrate the correct analytical predictions.

Recent advances in symmetric and network dynamics

NASA Astrophysics Data System (ADS)

Golubitsky, Martin; Stewart, Ian

2015-09-01

We summarize some of the main results discovered over the past three decades concerning symmetric dynamical systems and networks of dynamical systems, with a focus on pattern formation. In both of these contexts, extra constraints on the dynamical system are imposed, and the generic phenomena can change. The main areas discussed are time-periodic states, mode interactions, and non-compact symmetry groups such as the Euclidean group. We consider both dynamics and bifurcations. We summarize applications of these ideas to pattern formation in a variety of physical and biological systems, and explain how the methods were motivated by transferring to new contexts René Thom's general viewpoint, one version of which became known as "catastrophe theory." We emphasize the role of symmetry-breaking in the creation of patterns. Topics include equivariant Hopf bifurcation, which gives conditions for a periodic state to bifurcate from an equilibrium, and the H/K theorem, which classifies the pairs of setwise and pointwise symmetries of periodic states in equivariant dynamics. We discuss mode interactions, which organize multiple bifurcations into a single degenerate bifurcation, and systems with non-compact symmetry groups, where new technical issues arise. We transfer many of the ideas to the context of networks of coupled dynamical systems, and interpret synchrony and phase relations in network dynamics as a type of pattern, in which space is discretized into finitely many nodes, while time remains continuous. We also describe a variety of applications including animal locomotion, Couette-Taylor flow, flames, the Belousov-Zhabotinskii reaction, binocular rivalry, and a nonlinear filter based on anomalous growth rates for the amplitude of periodic oscillations in a feed-forward network.

On the validity of Zeeman's classification for three dimensional competitive differential equations with linearly determined nullclines

NASA Astrophysics Data System (ADS)

Jiang, Jifa; Niu, Lei

2017-12-01

We study three dimensional competitive differential equations with linearly determined nullclines and prove that they always have 33 stable nullcline classes in total. Each class is given in terms of inequalities on the intrinsic growth rates and competitive coefficients and is independent of generating functions. The common characteristics are that every trajectory converges to an equilibrium in classes 1-25, that Hopf bifurcations do not occur within class 32, and that there is always a heteroclinic cycle in class 27. Nontrivial dynamical behaviors, such as the existence and multiplicity of limit cycles, only may occur in classes 26-33, but these nontrivial dynamical behaviors depend on generating functions. We show that Hopf bifurcation can occur within each of classes 26-31 for continuous-time Leslie/Gower system and Ricker system, the same as Lotka-Volterra system; but it only occurs in classes 26 and 27 for continuous-time Atkinson/Allen system and Gompertz system. There is an apparent distinction between Lotka-Volterra system and Leslie/Gower system, Ricker system, Atkinson/Allen system, and Gompertz system with the identical growth rate. Lotka-Volterra system with the identical growth rate has no limit cycle, but admits a center on the carrying simplex in classes 26 and 27. But Leslie/Gower system, Ricker system, Atkinson/Allen system, and Gompertz system with the identical growth rate do possess limit cycles. At last, we provide examples to show that Leslie/Gower system and Ricker system can also admit two limit cycles. This general classification greatly widens applications of Zeeman's method and makes it possible to investigate the existence and multiplicity of limit cycles, centers and stability of heteroclinic cycles for three dimensional competitive systems with linearly determined nullclines, as done in planar systems.

Effect of Carreau-Yasuda rheological parameters on subcritical Lapwood convection in horizontal porous cavity saturated by shear-thinning fluid

NASA Astrophysics Data System (ADS)

Khechiba, Khaled; Mamou, Mahmoud; Hachemi, Madjid; Delenda, Nassim; Rebhi, Redha

2017-06-01

The present study is focused on Lapwood convection in isotropic porous media saturated with non-Newtonian shear thinning fluid. The non-Newtonian rheological behavior of the fluid is modeled using the general viscosity model of Carreau-Yasuda. The convection configuration consists of a shallow porous cavity with a finite aspect ratio and subject to a vertical constant heat flux, whereas the vertical walls are maintained impermeable and adiabatic. An approximate analytical solution is developed on the basis of the parallel flow assumption, and numerical solutions are obtained by solving the full governing equations. The Darcy model with the Boussinesq approximation and energy transport equations are solved numerically using a finite difference method. The results are obtained in terms of the Nusselt number and the flow fields as functions of the governing parameters. A good agreement is obtained between the analytical approximation and the numerical solution of the full governing equations. The effects of the rheological parameters of the Carreau-Yasuda fluid and Rayleigh number on the onset of subcritical convection thresholds are demonstrated. Regardless of the aspect ratio of the enclosure and thermal boundary condition type, the subcritical convective flows are seen to occur below the onset of stationary convection. Correlations are proposed to estimate the subcritical Rayleigh number for the onset of finite amplitude convection as a function of the fluid rheological parameters. Linear stability of the convective motion, predicted by the parallel flow approximation, is studied, and the onset of Hopf bifurcation, from steady convective flow to oscillatory behavior, is found to depend strongly on the rheological parameters. In general, Hopf bifurcation is triggered earlier as the fluid becomes more and more shear-thinning.

Coupled Leidenfrost states as a monodisperse granular clock

NASA Astrophysics Data System (ADS)

Liu, Rui; Yang, Mingcheng; Chen, Ke; Hou, Meiying; To, Kiwing

2016-08-01

Using an event-driven molecular dynamics simulation, we show that simple monodisperse granular beads confined in coupled columns may oscillate as a different type of granular clock. To trigger this oscillation, the system needs to be driven against gravity into a density-inverted state, with a high-density clustering phase supported from below by a gaslike low-density phase (Leidenfrost effect) in each column. Our analysis reveals that the density-inverted structure and the relaxation dynamics between the phases can amplify any small asymmetry between the columns, and lead to a giant oscillation. The oscillation occurs only for an intermediate range of the coupling strength, and the corresponding phase diagram can be universally described with a characteristic height of the density-inverted structure. A minimal two-phase model is proposed and a linear stability analysis shows that the triggering mechanism of the oscillation can be explained as a switchable two-parameter Andronov-Hopf bifurcation. Numerical solutions of the model also reproduce similar oscillatory dynamics to the simulation results.

Nonlinear dynamics of attractive magnetic bearings

NASA Technical Reports Server (NTRS)

Hebbale, K. V.; Taylor, D. L.

1987-01-01

The nonlinear dynamics of a ferromagnetic shaft suspended by the force of attraction of 1, 2, or 4 independent electromagnets is presented. Each model includes a state variable feedback controller which has been designed using the pole placement method. The constitutive relationships for the magnets are derived analytically from magnetic circuit theory, and the effects of induced eddy currents due to the rotation of the journal are included using Maxwell's field relations. A rotor suspended by four electro-magnets with closed loop feedback is shown to have nine equilibrium points within the bearing clearance space. As the rotor spin speed increases, the system is shown to pass through a Hopf bifurcation (a flutter instability). Using center manifold theory, this bifurcation can be shown to be of the subcritical type, indicating an unstable limit cycle below the critical speed. The bearing is very sensitive to initial conditions, and the equilibrium position is easily upset by transient excitation. The results are confirmed by numerical simulation.

Time delay in the Kuramoto model of coupled-phase oscillators

NASA Astrophysics Data System (ADS)

Yeung, Man Kit Stephen

1999-10-01

The Kuramoto model is a mean-field model of coupled phase oscillators with distributed natural frequencies. It was proposed to study collective synchronization in large systems of nonlinear oscillators. Here we generalize this model to allow time-delayed interactions. Despite the delay, synchronization is still possible. We derive exact stability conditions for the incoherent state, and for synchronized states and clustering states in the special case of noiseless identical oscillators. We also study the bifurcations of these states. We find that the incoherent state loses stability in a Hopf bifurcation. In the absence of noise, this leads to partial synchrony, where some oscillators are entrained to a common frequency. New phenomena caused by the delay include multistability among synchronization, incoherence, and clustering; and unsteady solutions with time-dependent order parameters. The experimental implications of the model are discussed for populations of chirping crickets, where the finite speed of sound causes communication delays, and for physical systems such as coupled phase- locked loops, lasers, and communication satellites.

Periodicity and Chaos Amidst Twisting and Folding in Two-Dimensional Maps

NASA Astrophysics Data System (ADS)

Garst, Swier; Sterk, Alef E.

We study the dynamics of three planar, noninvertible maps which rotate and fold the plane. Two maps are inspired by real-world applications whereas the third map is constructed to serve as a toy model for the other two maps. The dynamics of the three maps are remarkably similar. A stable fixed point bifurcates through a Hopf-Neĭmark-Sacker which leads to a countably infinite set of resonance tongues in the parameter plane of the map. Within a resonance tongue a periodic point can bifurcate through a period-doubling cascade. At the end of the cascade we detect Hénon-like attractors which are conjectured to be the closure of the unstable manifold of a saddle periodic point. These attractors have a folded structure which can be explained by means of the concept of critical lines. We also detect snap-back repellers which can either coexist with Hénon-like attractors or which can be formed when the saddle-point of a Hénon-like attractor becomes a source.

Symmetry breaking motion of a vortex pair in a driven cavity

NASA Astrophysics Data System (ADS)

McHugh, John; Osman, Kahar; Farias, Jason

2002-11-01

The two-dimensional driven cavity problem with an anti-symmetric sinusoidal forcing has been found to exhibit a subcritical symmetry breaking bifurcation (Farias and McHugh, Phys. Fluids, 2002). Equilibrium solutions are either a symmetric vortex pair or an asymmetric motion. The asymmetric motion is an asymmetric vortex pair at low Reynolds numbers, but merges into a three vortex motion at higher Reynolds numbers. The asymmetric solution is obtained by initiating the flow with a single vortex centered in the domain. Symmetric motion is obtained with no initial vortex, or weak initial vortex. The steady three-vortex motion occurs at a Reynolds number of approximately 3000, where the symmetric vortex pair has already gone through a Hopf bifurcation. Further two-dimensional results show that forcing with two full oscillations across the top of the cavity results in two steady vortex motions, depending on initial conditions. Three-dimensional results have even more steady solutions. The results are computational and theoretical.

Boundary effects and the onset of Taylor vortices

NASA Astrophysics Data System (ADS)

Rucklidge, A. M.; Champneys, A. R.

2004-05-01

It is well established that the onset of spatially periodic vortex states in the Taylor-Couette flow between rotating cylinders occurs at the value of Reynolds number predicted by local bifurcation theory. However, the symmetry breaking induced by the top and bottom plates means that the true situation should be a disconnected pitchfork. Indeed, experiments have shown that the fold on the disconnected branch can occur at more than double the Reynolds number of onset. This leads to an apparent contradiction: why should Taylor vortices set in so sharply at the Reynolds number predicted by the symmetric theory, given such large symmetry-breaking effects caused by the boundary conditions? This paper offers a generic explanation. The details are worked out using a Swift-Hohenberg pattern formation model that shares the same qualitative features as the Taylor-Couette flow. Onset occurs via a wall mode whose exponential tail penetrates further into the bulk of the domain as the driving parameter increases. In a large domain of length L, we show that the wall mode creates significant amplitude in the centre at parameter values that are O( L-2) away from the value of onset in the problem with ideal boundary conditions. We explain this as being due to a Hamiltonian Hopf bifurcation in space, which occurs at the same parameter value as the pitchfork bifurcation of the temporal dynamics. The disconnected anomalous branch remains O(1) away from the onset parameter since it does not arise as a bifurcation from the wall mode.

Period locking due to delayed feedback in a laser with saturable absorber.

PubMed

Carr, T W

2003-08-01

We consider laser with saturable absorber operating in the pulsating regime that is subject to delayed feedback. Alone, both the saturable absorber and delayed feedback cause the clockwise output to become unstable to periodic output via Hopf bifurcations. The delay feedback causes the laser pulse period to lock to an integer fraction of the feedback time. We derive a map from the original model to describe the periodic pulsations of the laser. Equations for the period of the laser predict the occurrence of the different locking states as well as the value of the pump when there is a switch between the locked states.

Dynamics of a linear magnetic “microswimmer molecule”

NASA Astrophysics Data System (ADS)

Babel, S.; Löwen, H.; Menzel, A. M.

2016-03-01

In analogy to nanoscopic molecules that are composed of individual atoms, we consider an active “microswimmer molecule”. It is made of three individual magnetic colloidal microswimmers that are connected by harmonic springs and interact hydrodynamically. In the ground state, they form a linear straight molecule. We analyze the relaxation dynamics for perturbations of this straight configuration. As a central result, with increasing self-propulsion, we observe an oscillatory instability in accord with a subcritical Hopf bifurcation scenario. It is accompanied by a corkscrew-like swimming trajectory of increasing radius. Our results can be tested experimentally, using, for instance, magnetic self-propelled Janus particles, supposably linked by DNA molecules.

Trajectories of Listeria-type motility in two dimensions

NASA Astrophysics Data System (ADS)

Wen, Fu-Lai; Leung, Kwan-tai; Chen, Hsuan-Yi

2012-12-01

Force generated by actin polymerization is essential in cell motility and the locomotion of organelles or bacteria such as Listeria monocytogenes. Both in vivo and in vitro experiments on actin-based motility have observed geometrical trajectories including straight lines, circles, S-shaped curves, and translating figure eights. This paper reports a phenomenological model of an actin-propelled disk in two dimensions that generates geometrical trajectories. Our model shows that when the evolutions of actin density and force per filament on the disk are strongly coupled to the disk self-rotation, it is possible for a straight trajectory to lose its stability. When the instability is due to a pitchfork bifurcation, the resulting trajectory is a circle; a straight trajectory can also lose stability through a Hopf bifurcation, and the resulting trajectory is an S-shaped curve. We also show that a half-coated disk, which mimics the distribution of functionalized proteins in Listeria, also undergoes similar symmetry-breaking bifurcations when the straight trajectory loses stability. For both a fully coated disk and a half-coated disk, when the trajectory is an S-shaped curve, the angular frequency of the disk self-rotation is different from that of the disk trajectory. However, for circular trajectories, these angular frequencies are different for a fully coated disk but the same for a half-coated disk.

On the stability analysis of a pair of van der Pol oscillators with delayed self-connection, position and velocity couplings

DOE Office of Scientific and Technical Information (OSTI.GOV)

Hu, Kun; Department of Mathematics, City University of Hong Kong, 83 Tat Chee Avenue, Kowloon; Chung, Kwok-wai, E-mail: makchung@cityu.edu.hk

2013-11-15

In this paper, we perform a stability analysis of a pair of van der Pol oscillators with delayed self-connection, position and velocity couplings. Bifurcation diagram of the damping, position and velocity coupling strengths is constructed, which gives insight into how stability boundary curves come into existence and how these curves evolve from small closed loops into open-ended curves. The van der Pol oscillator has been considered by many researchers as the nodes for various networks. It is inherently unstable at the zero equilibrium. Stability control of a network is always an important problem. Currently, the stabilization of the zero equilibriummore » of a pair of van der Pol oscillators can be achieved only for small damping strength by using delayed velocity coupling. An interesting question arises naturally: can the zero equilibrium be stabilized for an arbitrarily large value of the damping strength? We prove that it can be. In addition, a simple condition is given on how to choose the feedback parameters to achieve such goal. We further investigate how the in-phase mode or the out-of-phase mode of a periodic solution is related to the stability boundary curve that it emerges from a Hopf bifurcation. Analytical expression of a periodic solution is derived using an integration method. Some illustrative examples show that the theoretical prediction and numerical simulation are in good agreement.« less

Heterogeneity induces spatiotemporal oscillations in reaction-diffusion systems

NASA Astrophysics Data System (ADS)

Krause, Andrew L.; Klika, Václav; Woolley, Thomas E.; Gaffney, Eamonn A.

2018-05-01

We report on an instability arising in activator-inhibitor reaction-diffusion (RD) systems with a simple spatial heterogeneity. This instability gives rise to periodic creation, translation, and destruction of spike solutions that are commonly formed due to Turing instabilities. While this behavior is oscillatory in nature, it occurs purely within the Turing space such that no region of the domain would give rise to a Hopf bifurcation for the homogeneous equilibrium. We use the shadow limit of the Gierer-Meinhardt system to show that the speed of spike movement can be predicted from well-known asymptotic theory, but that this theory is unable to explain the emergence of these spatiotemporal oscillations. Instead, we numerically explore this system and show that the oscillatory behavior is caused by the destabilization of a steady spike pattern due to the creation of a new spike arising from endogeneous activator production. We demonstrate that on the edge of this instability, the period of the oscillations goes to infinity, although it does not fit the profile of any well-known bifurcation of a limit cycle. We show that nearby stationary states are either Turing unstable or undergo saddle-node bifurcations near the onset of the oscillatory instability, suggesting that the periodic motion does not emerge from a local equilibrium. We demonstrate the robustness of this spatiotemporal oscillation by exploring small localized heterogeneity and showing that this behavior also occurs in the Schnakenberg RD model. Our results suggest that this phenomenon is ubiquitous in spatially heterogeneous RD systems, but that current tools, such as stability of spike solutions and shadow-limit asymptotics, do not elucidate understanding. This opens several avenues for further mathematical analysis and highlights difficulties in explaining how robust patterning emerges from Turing's mechanism in the presence of even small spatial heterogeneity.

Stability analysis of BWR nuclear-coupled thermal-hyraulics using a simple model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Karve, A.A.; Rizwan-uddin; Dorning, J.J.

1995-09-01

A simple mathematical model is developed to describe the dynamics of the nuclear-coupled thermal-hydraulics in a boiling water reactor (BWR) core. The model, which incorporates the essential features of neutron kinetics, and single-phase and two-phase thermal-hydraulics, leads to simple dynamical system comprised of a set of nonlinear ordinary differential equations (ODEs). The stability boundary is determined and plotted in the inlet-subcooling-number (enthalpy)/external-reactivity operating parameter plane. The eigenvalues of the Jacobian matrix of the dynamical system also are calculated at various steady-states (fixed points); the results are consistent with those of the direct stability analysis and indicate that a Hopf bifurcationmore » occurs as the stability boundary in the operating parameter plane is crossed. Numerical simulations of the time-dependent, nonlinear ODEs are carried out for selected points in the operating parameter plane to obtain the actual damped and growing oscillations in the neutron number density, the channel inlet flow velocity, and the other phase variables. These indicate that the Hopf bifurcation is subcritical, hence, density wave oscillations with growing amplitude could result from a finite perturbation of the system even where the steady-state is stable. The power-flow map, frequently used by reactor operators during start-up and shut-down operation of a BWR, is mapped to the inlet-subcooling-number/neutron-density (operating-parameter/phase-variable) plane, and then related to the stability boundaries for different fixed inlet velocities corresponding to selected points on the flow-control line. The stability boundaries for different fixed inlet subcooling numbers corresponding to those selected points, are plotted in the neutron-density/inlet-velocity phase variable plane and then the points on the flow-control line are related to their respective stability boundaries in this plane.« less

Epidemic spreading on adaptively weighted scale-free networks.

PubMed

Sun, Mengfeng; Zhang, Haifeng; Kang, Huiyan; Zhu, Guanghu; Fu, Xinchu

2017-04-01

We introduce three modified SIS models on scale-free networks that take into account variable population size, nonlinear infectivity, adaptive weights, behavior inertia and time delay, so as to better characterize the actual spread of epidemics. We develop new mathematical methods and techniques to study the dynamics of the models, including the basic reproduction number, and the global asymptotic stability of the disease-free and endemic equilibria. We show the disease-free equilibrium cannot undergo a Hopf bifurcation. We further analyze the effects of local information of diseases and various immunization schemes on epidemic dynamics. We also perform some stochastic network simulations which yield quantitative agreement with the deterministic mean-field approach.

Discrete Self-Similarity in Interfacial Hydrodynamics and the Formation of Iterated Structures.

PubMed

Dallaston, Michael C; Fontelos, Marco A; Tseluiko, Dmitri; Kalliadasis, Serafim

2018-01-19

The formation of iterated structures, such as satellite and subsatellite drops, filaments, and bubbles, is a common feature in interfacial hydrodynamics. Here we undertake a computational and theoretical study of their origin in the case of thin films of viscous fluids that are destabilized by long-range molecular or other forces. We demonstrate that iterated structures appear as a consequence of discrete self-similarity, where certain patterns repeat themselves, subject to rescaling, periodically in a logarithmic time scale. The result is an infinite sequence of ridges and filaments with similarity properties. The character of these discretely self-similar solutions as the result of a Hopf bifurcation from ordinarily self-similar solutions is also described.

External noise-induced transitions in a current-biased Josephson junction

DOE Office of Scientific and Technical Information (OSTI.GOV)

Huang, Qiongwei; Xue, Changfeng, E-mail: cfxue@163.com; Tang, Jiashi

We investigate noise-induced transitions in a current-biased and weakly damped Josephson junction in the presence of multiplicative noise. By using the stochastic averaging procedure, the averaged amplitude equation describing dynamic evolution near a constant phase difference is derived. Numerical results show that a stochastic Hopf bifurcation between an absorbing and an oscillatory state occurs. This means the external controllable noise triggers a transition into the non-zero junction voltage state. With the increase of noise intensity, the stationary probability distribution peak shifts and is characterised by increased width and reduced height. And the different transition rates are shown for large andmore » small bias currents.« less

Nonlinear dynamics of the rock-paper-scissors game with mutations.

PubMed

Toupo, Danielle F P; Strogatz, Steven H

2015-05-01

We analyze the replicator-mutator equations for the rock-paper-scissors game. Various graph-theoretic patterns of mutation are considered, ranging from a single unidirectional mutation pathway between two of the species, to global bidirectional mutation among all the species. Our main result is that the coexistence state, in which all three species exist in equilibrium, can be destabilized by arbitrarily small mutation rates. After it loses stability, the coexistence state gives birth to a stable limit cycle solution created in a supercritical Hopf bifurcation. This attracting periodic solution exists for all the mutation patterns considered, and persists arbitrarily close to the limit of zero mutation rate and a zero-sum game.

Efferent feedback can explain many hearing phenomena

NASA Astrophysics Data System (ADS)

Holmes, W. Harvey; Flax, Matthew R.

2015-12-01

The mixed mode cochlear amplifier (MMCA) model was presented at the last Mechanics of Hearing workshop [4]. The MMCA consists principally of a nonlinear feedback loop formed when an efferent-controlled outer hair cell (OHC) is combined with the cochlear mechanics and the rest of the relevant neurobiology. Essential elements of this model are efferent control of the OHC motility and a delay in the feedback to the OHC. The input to the MMCA is the passive travelling wave. In the MMCA amplification is localized where both the neural and tuned mechanical systems meet in the Organ of Corti (OoC). The simplest model based on this idea is a nonlinear delay line resonator (DLR), which is mathematically described by a nonlinear delay-differential equation (DDE). This model predicts possible Hopf bifurcations and exhibits its most interesting behaviour when operating near a bifurcation. This contribution presents some simulation results using the DLR model. These show that various observed hearing phenomena can be accounted for by this model, at least qualitatively, including compression effects, two-tone suppression and some forms of otoacoustic emissions (OAEs).

Elastic wake instabilities in a creeping flow between two obstacles

NASA Astrophysics Data System (ADS)

Varshney, Atul; Steinberg, Victor

2017-05-01

It is shown that a channel flow of a dilute polymer solution between two widely spaced cylinders hindering the flow is an important paradigm of an unbounded flow in the case in which the channel wall is located sufficiently far from the cylinders. The quantitative characterization of instabilities in a creeping viscoelastic channel flow between two widely spaced cylinders reveals two elastically driven transitions, which are associated with the breaking of time-reversal and mirror symmetries: Hopf and forward bifurcations described by two order parameters vrms and ω ¯, respectively. We suggest that a decrease of the normalized distance between the obstacles leads to a collapse of the two bifurcations into a codimension-2 point, a situation general for many nonequilibrium systems. However, the striking and unexpected result is the discovery of a mechanism of the vorticity growth via an increase of a vortex length at the preserved streamline curvature in a viscoelastic flow, which is in sharp contrast to the well-known suppression of the vorticity in a Newtonian flow by polymer additives.

Dynamical Behaviors in Complex-Valued Love Model With or Without Time Delays

NASA Astrophysics Data System (ADS)

Deng, Wei; Liao, Xiaofeng; Dong, Tao

2017-12-01

In this paper, a novel version of nonlinear model, i.e. a complex-valued love model with two time delays between two individuals in a love affair, has been proposed. A notable feature in this model is that we separate the emotion of one individual into real and imaginary parts to represent the variation and complexity of psychophysiological emotion in romantic relationship instead of just real domain, and make our model much closer to reality. This is because love is a complicated cognitive and social phenomenon, full of complexity, diversity and unpredictability, which refers to the coexistence of different aspects of feelings, states and attitudes ranging from joy and trust to sadness and disgust. By analyzing associated characteristic equation of linearized equations for our model, it is found that the Hopf bifurcation occurs when the sum of time delays passes through a sequence of critical value. Stability of bifurcating cyclic love dynamics is also derived by applying the normal form theory and the center manifold theorem. In addition, it is also shown that, for some appropriate chosen parameters, chaotic behaviors can appear even without time delay.

Mathematical Analysis of an SIQR Influenza Model with Imperfect Quarantine.

PubMed

Erdem, Mustafa; Safan, Muntaser; Castillo-Chavez, Carlos

2017-07-01

The identification of mechanisms responsible for recurrent epidemic outbreaks, such as age structure, cross-immunity and variable delays in the infective classes, has challenged and fascinated epidemiologists and mathematicians alike. This paper addresses, motivated by mathematical work on influenza models, the impact of imperfect quarantine on the dynamics of SIR-type models. A susceptible-infectious-quarantine-recovered (SIQR) model is formulated with quarantined individuals altering the transmission dynamics process through their possibly reduced ability to generate secondary cases of infection. Mathematical and numerical analyses of the model of the equilibria and their stability have been carried out. Uniform persistence of the model has been established. Numerical simulations show that the model supports Hopf bifurcation as a function of the values of the quarantine effectiveness and other parameters. The upshot of this work is somewhat surprising since it is shown that SIQR model oscillatory behavior, as shown by multiple researchers, is in fact not robust to perturbations in the quarantine regime.

Structure and Dynamics of Replication-Mutation Systems

NASA Astrophysics Data System (ADS)

Schuster, Peter

1987-03-01

The kinetic equations of polynucleotide replication can be brought into fairly simple form provided certain environmental conditions are fulfilled. Two flow reactors, the continuously stirred tank reactor (CSTR) and a special dialysis reactor are particularly suitable for the analysis of replication kinetics. An experimental setup to study the chemical reaction network of RNA synthesis was derived from the bacteriophage Qβ. It consists of a virus specific RNA polymerase, Qβ replicase, the activated ribonucleosides GTP, ATP, CTP and UTP as well as a template suitable for replication. The ordinary differential equations for replication and mutation under the conditions of the flow reactors were analysed by the qualitative methods of bifurcation theory as well as by numerical integration. The various kinetic equations are classified according to their dynamical properties: we distinguish "quasilinear systems" which have uniquely stable point attractors and "nonlinear systems" with inherent nonlinearities which lead to multiple steady states, Hopf bifuractions, Feigenbaum-like sequences and chaotic dynamics for certain parameter ranges. Some examples which are relevant in molecular evolution and population genetics are discussed in detail.

The effect of topography on the evolution of unstable disturbances in a baroclinic atmosphere

NASA Technical Reports Server (NTRS)

Clark, J. H. E.

1985-01-01

A two layer spectral quasi-geostrophic model is used to simulate the effects of topography on the equilibria, their stability, and the long term evolution of incipient unstable waves. The flow is forced by latitudinally dependent radiative heating. Dissipation is in the form of Rayleigh friction. An analytical solution is found for the propagating finite amplitude waves which result from baroclinic instability of the zonal winds when topography is absent. The appearance of this solution for wavelengths just longer than the Rossby radius of deformation and disappearance of ultra-long wavelengths is interpreted in terms of the Hopf bifurcation theory. Simple dynamic and thermodynamic criteria for the existence of periodic Rossby solutions are presented. A Floquet stability analysis shows that the waves are neutral. The nature of the form drag instability of high index equilibria is investigated. The proximity of the equilibrium shear to a resonant value is essential for the instability, provided the equilibrium occurs at a slightly stronger shear than resonance.

Mathematical analysis of a multiple strain, multi-locus-allele system for antigenically variable infectious diseases revisited.

PubMed

Cherif, Alhaji

2015-09-01

Many important pathogens such as HIV/AIDS, influenza, malaria, dengue and meningitis generally exist in phenotypically distinct serotypes that compete for hosts. Models used to study these diseases appear as meta-population systems. Herein, we revisit one of the multiple strain models that have been used to investigate the dynamics of infectious diseases with co-circulating serotypes or strains, and provide analytical results underlying the numerical investigations. In particular, we establish the necessary conditions for the local asymptotic stability of the steady states and for the existence of oscillatory behaviors via Hopf bifurcation. In addition, we show that the existence of discrete antigenic forms among pathogens can either fully or partially self-organize, where (i) strains exhibit no strain structures and coexist or (ii) antigenic variants sort into non-overlapping or minimally overlapping clusters that either undergo the principle of competitive exclusion exhibiting discrete strain structures, or co-exist cyclically. Copyright © 2015. Published by Elsevier Inc.

A model for oscillations and pattern formation in protoplasmic droplets of Physarum polycephalum

NASA Astrophysics Data System (ADS)

Radszuweit, M.; Engel, H.; Bär, M.

2010-12-01

A mechano-chemical model for the spatiotemporal dynamics of free calcium and the thickness in protoplasmic droplets of the true slime mold Physarum polycephalum is derived starting from a physiologically detailed description of intracellular calcium oscillations proposed by Smith and Saldana (Biopys. J. 61, 368 (1992)). First, we have modified the Smith-Saldana model for the temporal calcium dynamics in order to reproduce the experimentally observed phase relation between calcium and mechanical tension oscillations. Then, we formulate a model for spatiotemporal dynamics by adding spatial coupling in the form of calcium diffusion and advection due to calcium-dependent mechanical contraction. In another step, the resulting reaction-diffusion model with mechanical coupling is simplified to a reaction-diffusion model with global coupling that approximates the mechanical part. We perform a bifurcation analysis of the local dynamics and observe a Hopf bifurcation upon increase of a biochemical activity parameter. The corresponding reaction-diffusion model with global coupling shows regular and chaotic spatiotemporal behaviour for parameters with oscillatory dynamics. In addition, we show that the global coupling leads to a long-wavelength instability even for parameters where the local dynamics possesses a stable spatially homogeneous steady state. This instability causes standing waves with a wavelength of twice the system size in one dimension. Simulations of the model in two dimensions are found to exhibit defect-mediated turbulence as well as various types of spiral wave patterns in qualitative agreement with earlier experimental observation by Takagi and Ueda (Physica D, 237, 420 (2008)).

Input-output relation and energy efficiency in the neuron with different spike threshold dynamics.

PubMed

Yi, Guo-Sheng; Wang, Jiang; Tsang, Kai-Ming; Wei, Xi-Le; Deng, Bin

2015-01-01

Neuron encodes and transmits information through generating sequences of output spikes, which is a high energy-consuming process. The spike is initiated when membrane depolarization reaches a threshold voltage. In many neurons, threshold is dynamic and depends on the rate of membrane depolarization (dV/dt) preceding a spike. Identifying the metabolic energy involved in neural coding and their relationship to threshold dynamic is critical to understanding neuronal function and evolution. Here, we use a modified Morris-Lecar model to investigate neuronal input-output property and energy efficiency associated with different spike threshold dynamics. We find that the neurons with dynamic threshold sensitive to dV/dt generate discontinuous frequency-current curve and type II phase response curve (PRC) through Hopf bifurcation, and weak noise could prohibit spiking when bifurcation just occurs. The threshold that is insensitive to dV/dt, instead, results in a continuous frequency-current curve, a type I PRC and a saddle-node on invariant circle bifurcation, and simultaneously weak noise cannot inhibit spiking. It is also shown that the bifurcation, frequency-current curve and PRC type associated with different threshold dynamics arise from the distinct subthreshold interactions of membrane currents. Further, we observe that the energy consumption of the neuron is related to its firing characteristics. The depolarization of spike threshold improves neuronal energy efficiency by reducing the overlap of Na(+) and K(+) currents during an action potential. The high energy efficiency is achieved at more depolarized spike threshold and high stimulus current. These results provide a fundamental biophysical connection that links spike threshold dynamics, input-output relation, energetics and spike initiation, which could contribute to uncover neural encoding mechanism.

Input-output relation and energy efficiency in the neuron with different spike threshold dynamics

PubMed Central

Yi, Guo-Sheng; Wang, Jiang; Tsang, Kai-Ming; Wei, Xi-Le; Deng, Bin

2015-01-01

Neuron encodes and transmits information through generating sequences of output spikes, which is a high energy-consuming process. The spike is initiated when membrane depolarization reaches a threshold voltage. In many neurons, threshold is dynamic and depends on the rate of membrane depolarization (dV/dt) preceding a spike. Identifying the metabolic energy involved in neural coding and their relationship to threshold dynamic is critical to understanding neuronal function and evolution. Here, we use a modified Morris-Lecar model to investigate neuronal input-output property and energy efficiency associated with different spike threshold dynamics. We find that the neurons with dynamic threshold sensitive to dV/dt generate discontinuous frequency-current curve and type II phase response curve (PRC) through Hopf bifurcation, and weak noise could prohibit spiking when bifurcation just occurs. The threshold that is insensitive to dV/dt, instead, results in a continuous frequency-current curve, a type I PRC and a saddle-node on invariant circle bifurcation, and simultaneously weak noise cannot inhibit spiking. It is also shown that the bifurcation, frequency-current curve and PRC type associated with different threshold dynamics arise from the distinct subthreshold interactions of membrane currents. Further, we observe that the energy consumption of the neuron is related to its firing characteristics. The depolarization of spike threshold improves neuronal energy efficiency by reducing the overlap of Na+ and K+ currents during an action potential. The high energy efficiency is achieved at more depolarized spike threshold and high stimulus current. These results provide a fundamental biophysical connection that links spike threshold dynamics, input-output relation, energetics and spike initiation, which could contribute to uncover neural encoding mechanism. PMID:26074810

Dynamical properties of a prey-predator-scavenger model with quadratic harvesting

NASA Astrophysics Data System (ADS)

Gupta, R. P.; Chandra, Peeyush

2017-08-01

In this paper, we propose and analyze an extended model for the prey-predator-scavenger in presence of harvesting to study the effects of harvesting of predator as well as scavenger. The positivity, boundedness and persistence conditions are derived for the proposed model. The model undergoes a Hopf-bifurcation around the co-existing equilibrium point. It is also observed that the model is capable of exhibiting period doubling route to chaos. It is pointed out that a suitable amount of harvesting of predator can control the chaotic dynamics and make the system stable. An extensive numerical simulation is performed to validate the analytic findings. The associated control problem for the proposed model has been analyzed for optimal harvesting.

Noisy Oscillations in the Actin Cytoskeleton of Chemotactic Amoeba.

PubMed

Negrete, Jose; Pumir, Alain; Hsu, Hsin-Fang; Westendorf, Christian; Tarantola, Marco; Beta, Carsten; Bodenschatz, Eberhard

2016-09-30

Biological systems with their complex biochemical networks are known to be intrinsically noisy. Here we investigate the dynamics of actin polymerization of amoeboid cells, which are close to the onset of oscillations. We show that the large phenotypic variability in the polymerization dynamics can be accurately captured by a generic nonlinear oscillator model in the presence of noise. We determine the relative role of the noise with a single dimensionless, experimentally accessible parameter, thus providing a quantitative description of the variability in a population of cells. Our approach, which rests on a generic description of a system close to a Hopf bifurcation and includes the effect of noise, can characterize the dynamics of a large class of noisy systems close to an oscillatory instability.

Twirling and Whirling: Viscous Dynamics of Rotating Elastica

NASA Astrophysics Data System (ADS)

Wolgemuth, Charles; Powers, Thomas; Goldstein, Raymond

1999-10-01

The stability of forced elastic filaments arise in several important biological settings involving bend and twist elasticity at low Reynolds number. Examples include DNA transcription and replication and bacterial flagellar motion. In order to elucidate fundamental processes common to these systems, we consider the model problem of a rotationally forced filament with twist and bend elasticity. Competition between twist injection, twist diffusion, and writhing instabilities is described by a novel pair of PDEs for twist and bend evolution. Analytical and numerical methods elucidate the twist/bend coupling and reveal two dynamical regimes seperated by a Hopf bifurcation: (i) diffusion-dominated axial rotation, or twirling, and (ii) steady-state crankshafting motion, or whirling. Experiments are proposed to examine these phenomena and the consequences for swimming investigated.

Noisy Oscillations in the Actin Cytoskeleton of Chemotactic Amoeba

NASA Astrophysics Data System (ADS)

Negrete, Jose; Pumir, Alain; Hsu, Hsin-Fang; Westendorf, Christian; Tarantola, Marco; Beta, Carsten; Bodenschatz, Eberhard

2016-09-01

Biological systems with their complex biochemical networks are known to be intrinsically noisy. Here we investigate the dynamics of actin polymerization of amoeboid cells, which are close to the onset of oscillations. We show that the large phenotypic variability in the polymerization dynamics can be accurately captured by a generic nonlinear oscillator model in the presence of noise. We determine the relative role of the noise with a single dimensionless, experimentally accessible parameter, thus providing a quantitative description of the variability in a population of cells. Our approach, which rests on a generic description of a system close to a Hopf bifurcation and includes the effect of noise, can characterize the dynamics of a large class of noisy systems close to an oscillatory instability.

Anomalous neuronal responses to fluctuated inputs

NASA Astrophysics Data System (ADS)

Hosaka, Ryosuke; Sakai, Yutaka

2015-10-01

The irregular firing of a cortical neuron is thought to result from a highly fluctuating drive that is generated by the balance of excitatory and inhibitory synaptic inputs. A previous study reported anomalous responses of the Hodgkin-Huxley neuron to the fluctuated inputs where an irregularity of spike trains is inversely proportional to an input irregularity. In the current study, we investigated the origin of these anomalous responses with the Hindmarsh-Rose neuron model, map-based models, and a simple mixture of interspike interval distributions. First, we specified the parameter regions for the bifurcations in the Hindmarsh-Rose model, and we confirmed that the model reproduced the anomalous responses in the dynamics of the saddle-node and subcritical Hopf bifurcations. For both bifurcations, the Hindmarsh-Rose model shows bistability in the resting state and the repetitive firing state, which indicated that the bistability was the origin of the anomalous input-output relationship. Similarly, the map-based model that contained bistability reproduced the anomalous responses, while the model without bistability did not. These results were supported by additional findings that the anomalous responses were reproduced by mimicking the bistable firing with a mixture of two different interspike interval distributions. Decorrelation of spike trains is important for neural information processing. For such spike train decorrelation, irregular firing is key. Our results indicated that irregular firing can emerge from fluctuating drives, even weak ones, under conditions involving bistability. The anomalous responses, therefore, contribute to efficient processing in the brain.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Zhu, Z. W., E-mail: zhuzhiwen@tju.edu.cn; Li, X. M., E-mail: lixinmiaotju@163.com; Xu, J., E-mail: xujia-ld@163.com

A kind of magnetic shape memory alloy (MSMA) microgripper is proposed in this paper, and its nonlinear dynamic characteristics are studied when the stochastic perturbation is considered. Nonlinear differential items are introduced to explain the hysteretic phenomena of MSMA, and the constructive relationships among strain, stress, and magnetic field intensity are obtained by the partial least-square regression method. The nonlinear dynamic model of a MSMA microgripper subjected to in-plane stochastic excitation is developed. The stationary probability density function of the system’s response is obtained, the transition sets of the system are determined, and the conditions of stochastic bifurcation are obtained.more » The homoclinic and heteroclinic orbits of the system are given, and the boundary of the system’s safe basin is obtained by stochastic Melnikov integral method. The numerical and experimental results show that the system’s motion depends on its parameters, and stochastic Hopf bifurcation appears in the variation of the parameters; the area of the safe basin decreases with the increase of the stochastic excitation, and the boundary of the safe basin becomes fractal. The results of this paper are helpful for the application of MSMA microgripper in engineering fields.« less

Complex dynamics in the Leslie-Gower type of the food chain system with multiple delays

NASA Astrophysics Data System (ADS)

Guo, Lei; Song, Zi-Gen; Xu, Jian

2014-08-01

In this paper, we present a Leslie-Gower type of food chain system composed of three species, which are resource, consumer, and predator, respectively. The digestion time delays corresponding to consumer-eat-resource and predator-eat-consumer are introduced for more realistic consideration. It is called the resource digestion delay (RDD) and consumer digestion delay (CDD) for simplicity. Analyzing the corresponding characteristic equation, the stabilities of the boundary and interior equilibrium points are studied. The food chain system exhibits the species coexistence for the small values of digestion delays. Large RDD/CDD may destabilize the species coexistence and induce the system dynamic into recurrent bloom or system collapse. Further, the present of multiple delays can control species population into the stable coexistence. To investigate the effect of time delays on the recurrent bloom of species population, the Hopf bifurcation and periodic solution are investigated in detail in terms of the central manifold reduction and normal form method. Finally, numerical simulations are performed to display some complex dynamics, which include multiple periodic solution and chaos motion for the different values of system parameters. The system dynamic behavior evolves into the chaos motion by employing the period-doubling bifurcation.

The effect of sexual transmission on Zika virus dynamics.

PubMed

Saad-Roy, C M; Ma, Junling; van den Driessche, P

2018-04-25

Zika virus is a human disease that may lead to neurological disorders in affected individuals, and may be transmitted vectorially (by mosquitoes) or sexually. A mathematical model of Zika virus transmission is formulated, taking into account mosquitoes, sexually active males and females, inactive individuals, and considering both vector transmission and sexual transmission from infectious males to susceptible females. Basic reproduction numbers are computed, and disease control strategies are evaluated. The effect of the incidence function used to model sexual transmission from infectious males to susceptible females is investigated. It is proved that for such functions that are sublinear, if the basic reproduction [Formula: see text], then the disease dies out and [Formula: see text] is a sharp threshold. Moreover, under certain conditions on model parameters and assuming mass action incidence for sexual transmission, it is proved that if [Formula: see text], there exists a unique endemic equilibrium that is globally asymptotically stable. However, under nonlinear incidence, it is shown that for certain functions backward bifurcation and Hopf bifurcation may occur, giving rise to subthreshold equilibria and periodic solutions, respectively. Numerical simulations for various parameter values are displayed to illustrate these behaviours.

Amphibian sacculus and the forced Kuramoto model with intrinsic noise and frequency dispersion

NASA Astrophysics Data System (ADS)

Ji, Seung; Bozovic, Dolores; Bruinsma, Robijn

2018-04-01

The amphibian sacculus (AS) is an end organ that specializes in the detection of low-frequency auditory and vestibular signals. In this paper, we propose a model for the AS in the form of an array of phase oscillators with long-range coupling, subject to a steady load that suppresses spontaneous oscillations. The array is exposed to significant levels of frequency dispersion and intrinsic noise. We show that such an array can be a sensitive and robust subthreshold detector of low-frequency stimuli, though without significant frequency selectivity. The effects of intrinsic noise and frequency dispersion are contrasted. Intermediate levels of intrinsic noise greatly enhance the sensitivity through stochastic resonance. Frequency dispersion, on the other hand, only degrades detection sensitivity. However, frequency dispersion can play a useful role in terms of the suppression of spontaneous activity. As a model for the AS, the array parameters are such that the system is poised near a saddle-node bifurcation on an invariant circle. However, by a change of array parameters, the same system also can be poised near an emergent Andronov-Hopf bifurcation and thereby function as a frequency-selective detector.

An epidemic model to evaluate the homogeneous mixing assumption

NASA Astrophysics Data System (ADS)

Turnes, P. P.; Monteiro, L. H. A.

2014-11-01

Many epidemic models are written in terms of ordinary differential equations (ODE). This approach relies on the homogeneous mixing assumption; that is, the topological structure of the contact network established by the individuals of the host population is not relevant to predict the spread of a pathogen in this population. Here, we propose an epidemic model based on ODE to study the propagation of contagious diseases conferring no immunity. The state variables of this model are the percentages of susceptible individuals, infectious individuals and empty space. We show that this dynamical system can experience transcritical and Hopf bifurcations. Then, we employ this model to evaluate the validity of the homogeneous mixing assumption by using real data related to the transmission of gonorrhea, hepatitis C virus, human immunodeficiency virus, and obesity.

Nonreciprocal wave scattering on nonlinear string-coupled oscillators

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lepri, Stefano, E-mail: stefano.lepri@isc.cnr.it; Istituto Nazionale di Fisica Nucleare, Sezione di Firenze, via G. Sansone 1, I-50019 Sesto Fiorentino; Pikovsky, Arkady

2014-12-01

We study scattering of a periodic wave in a string on two lumped oscillators attached to it. The equations can be represented as a driven (by the incident wave) dissipative (due to radiation losses) system of delay differential equations of neutral type. Nonlinearity of oscillators makes the scattering non-reciprocal: The same wave is transmitted differently in two directions. Periodic regimes of scattering are analyzed approximately, using amplitude equation approach. We show that this setup can act as a nonreciprocal modulator via Hopf bifurcations of the steady solutions. Numerical simulations of the full system reveal nontrivial regimes of quasiperiodic and chaoticmore » scattering. Moreover, a regime of a “chaotic diode,” where transmission is periodic in one direction and chaotic in the opposite one, is reported.« less

Effect of Multiple Delays in an Eco-Epidemiological Model with Strong Allee Effect

NASA Astrophysics Data System (ADS)

Ghosh, Kakali; Biswas, Santanu; Samanta, Sudip; Tiwari, Pankaj Kumar; Alshomrani, Ali Saleh; Chattopadhyay, Joydev

In the present article, we make an attempt to investigate the effect of two time delays, logistic delay and gestation delay, on an eco-epidemiological model. In the proposed model, strong Allee effect is considered in the growth term of the prey population. We incorporate two time lags and inspect elementary mathematical characteristic of the proposed model such as boundedness, uniform persistence, stability and Hopf-bifurcation for all possible combinations of both delays at the interior equilibrium point of the system. We observe that increase in gestation delay leads to chaotic solutions through the limit cycle. We also observe that the Allee effect play a major role in controlling the chaos. We execute several numerical simulations to illustrate the proposed mathematical model and our analytical findings.

Synchronization of glycolytic oscillations in a yeast cell population.

PubMed

Danø, S; Hynne, F; De Monte, S; d'Ovidio, F; Sørensen, P G; Westerhoff, H

2001-01-01

The mechanism of active phase synchronization in a suspension of oscillatory yeast cells has remained a puzzle for almost half a century. The difficulty of the problem stems from the fact that the synchronization phenomenon involves the entire metabolic network of glycolysis and fermentation, and consequently it cannot be addressed at the level of a single enzyme or a single chemical species. In this paper it is shown how this system in a CSTR (continuous flow stirred tank reactor) can be modelled quantitatively as a population of Stuart-Landau oscillators interacting by exchange of metabolites through the extracellular medium, thus reducing the complexity of the problem without sacrificing the biochemical realism. The parameters of the model can be derived by a systematic expansion from any full-scale model of the yeast cell kinetics with a supercritical Hopf bifurcation. Some parameter values can also be obtained directly from analysis of perturbation experiments. In the mean-field limit, equations for the study of populations having a distribution of frequencies are used to simulate the effect of the inherent variations between cells.

A Compact Synchronous Cellular Model of Nonlinear Calcium Dynamics: Simulation and FPGA Synthesis Results.

PubMed

Soleimani, Hamid; Drakakis, Emmanuel M

2017-06-01

Recent studies have demonstrated that calcium is a widespread intracellular ion that controls a wide range of temporal dynamics in the mammalian body. The simulation and validation of such studies using experimental data would benefit from a fast large scale simulation and modelling tool. This paper presents a compact and fully reconfigurable cellular calcium model capable of mimicking Hopf bifurcation phenomenon and various nonlinear responses of the biological calcium dynamics. The proposed cellular model is synthesized on a digital platform for a single unit and a network model. Hardware synthesis, physical implementation on FPGA, and theoretical analysis confirm that the proposed cellular model can mimic the biological calcium behaviors with considerably low hardware overhead. The approach has the potential to speed up large-scale simulations of slow intracellular dynamics by sharing more cellular units in real-time. To this end, various networks constructed by pipelining 10 k to 40 k cellular calcium units are compared with an equivalent simulation run on a standard PC workstation. Results show that the cellular hardware model is, on average, 83 times faster than the CPU version.

A stoichiometric organic matter decomposition model in a chemostat culture.

PubMed

Kong, Jude D; Salceanu, Paul; Wang, Hao

2018-02-01

Biodegradation, the disintegration of organic matter by microorganism, is essential for the cycling of environmental organic matter. Understanding and predicting the dynamics of this biodegradation have increasingly gained attention from the industries and government regulators. Since changes in environmental organic matter are strenuous to measure, mathematical models are essential in understanding and predicting the dynamics of organic matters. Empirical evidence suggests that grazers' preying activity on microorganism helps to facilitate biodegradation. In this paper, we formulate and investigate a stoichiometry-based organic matter decomposition model in a chemostat culture that incorporates the dynamics of grazers. We determine the criteria for the uniform persistence and extinction of the species and chemicals. Our results show that (1) if at the unique internal steady state, the per capita growth rate of bacteria is greater than the sum of the bacteria's death and dilution rates, then the bacteria will persist uniformly; (2) if in addition to this, (a) the grazers' per capita growth rate is greater than the sum of the dilution rate and grazers' death rate, and (b) the death rate of bacteria is less than some threshold, then the grazers will persist uniformly. These conditions can be achieved simultaneously if there are sufficient resources in the feed bottle. As opposed to the microcosm decomposition models' results, in a chemostat culture, chemicals always persist. Besides the transcritical bifurcation observed in microcosm models, our chemostat model exhibits Hopf bifurcation and Rosenzweig's paradox of enrichment phenomenon. Our sensitivity analysis suggests that the most effective way to facilitate degradation is to decrease the dilution rate.

Circuit bounds on stochastic transport in the Lorenz equations

NASA Astrophysics Data System (ADS)

Weady, Scott; Agarwal, Sahil; Wilen, Larry; Wettlaufer, J. S.

2018-07-01

In turbulent Rayleigh-Bénard convection one seeks the relationship between the heat transport, captured by the Nusselt number, and the temperature drop across the convecting layer, captured by the Rayleigh number. In experiments, one measures the Nusselt number for a given Rayleigh number, and the question of how close that value is to the maximal transport is a key prediction of variational fluid mechanics in the form of an upper bound. The Lorenz equations have traditionally been studied as a simplified model of turbulent Rayleigh-Bénard convection, and hence it is natural to investigate their upper bounds, which has previously been done numerically and analytically, but they are not as easily accessible in an experimental context. Here we describe a specially built circuit that is the experimental analogue of the Lorenz equations and compare its output to the recently determined upper bounds of the stochastic Lorenz equations [1]. The circuit is substantially more efficient than computational solutions, and hence we can more easily examine the system. Because of offsets that appear naturally in the circuit, we are motivated to study unique bifurcation phenomena that arise as a result. Namely, for a given Rayleigh number, we find a reentrant behavior of the transport on noise amplitude and this varies with Rayleigh number passing from the homoclinic to the Hopf bifurcation.

Topological semimetals carrying arbitrary Hopf numbers: Fermi surface topologies of a Hopf link, Solomon's knot, trefoil knot, and other linked nodal varieties

NASA Astrophysics Data System (ADS)

Ezawa, Motohiko

2017-07-01

We propose a type of Hopf semimetal indexed by a pair of numbers (p ,q ) , where the Hopf number is given by p q . The Fermi surface is given by a preimage of the Hopf map, which consists of loops nontrivially linked for a nonzero Hopf number. The Fermi surface forms a torus link, whose examples are a Hopf link indexed by (1 ,1 ) , Solomon's knot (2 ,1 ) , a double Hopf link (2 ,2 ) , and a double trefoil knot (3 ,2 ) . We may choose p or q to be a half integer, where the Fermi surface is a torus knot, such as a trefoil knot (3 /2 ,1 ) . It is even possible to make the Hopf number an arbitrary rational number, where a semimetal whose Fermi surface forms open strings is generated.

Critical Slowing Down Governs the Transition to Neuron Spiking

PubMed Central

Meisel, Christian; Klaus, Andreas; Kuehn, Christian; Plenz, Dietmar

2015-01-01

Many complex systems have been found to exhibit critical transitions, or so-called tipping points, which are sudden changes to a qualitatively different system state. These changes can profoundly impact the functioning of a system ranging from controlled state switching to a catastrophic break-down; signals that predict critical transitions are therefore highly desirable. To this end, research efforts have focused on utilizing qualitative changes in markers related to a system’s tendency to recover more slowly from a perturbation the closer it gets to the transition—a phenomenon called critical slowing down. The recently studied scaling of critical slowing down offers a refined path to understand critical transitions: to identify the transition mechanism and improve transition prediction using scaling laws. Here, we outline and apply this strategy for the first time in a real-world system by studying the transition to spiking in neurons of the mammalian cortex. The dynamical system approach has identified two robust mechanisms for the transition from subthreshold activity to spiking, saddle-node and Hopf bifurcation. Although theory provides precise predictions on signatures of critical slowing down near the bifurcation to spiking, quantitative experimental evidence has been lacking. Using whole-cell patch-clamp recordings from pyramidal neurons and fast-spiking interneurons, we show that 1) the transition to spiking dynamically corresponds to a critical transition exhibiting slowing down, 2) the scaling laws suggest a saddle-node bifurcation governing slowing down, and 3) these precise scaling laws can be used to predict the bifurcation point from a limited window of observation. To our knowledge this is the first report of scaling laws of critical slowing down in an experiment. They present a missing link for a broad class of neuroscience modeling and suggest improved estimation of tipping points by incorporating scaling laws of critical slowing down as a strategy applicable to other complex systems. PMID:25706912

Dynamics of an eco-epidemiological model with saturated incidence rate

NASA Astrophysics Data System (ADS)

Suryanto, Agus

2017-03-01

In this paper we study the effect of prey infection on the modified Leslie-Gower predator-prey model with saturated incidence rate. The model will be analyzed dynamically to find the equilibria and their existence conditions as well as their local stability conditions. It is found that there are six type of equilibria, namely the extinction of both prey and predator point, the extinction of infective prey and predator point, the extinction of predator point, the extinction of prey point, the extinction of infective prey point and the interior point. The first four equilibrium points are always unstable, while the last two equilibria are conditionally stable. We also find that the system undergoes Hopf bifurcation around the interior point which is controlled by the rate of infection. To illustrate our analytical results, we show some numerical results.

Early warning signals for critical transitions in a thermoacoustic system

PubMed Central

Gopalakrishnan, E. A.; Sharma, Yogita; John, Tony; Dutta, Partha Sharathi; Sujith, R. I.

2016-01-01

Dynamical systems can undergo critical transitions where the system suddenly shifts from one stable state to another at a critical threshold called the tipping point. The decrease in recovery rate to equilibrium (critical slowing down) as the system approaches the tipping point can be used to identify the proximity to a critical transition. Several measures have been adopted to provide early indications of critical transitions that happen in a variety of complex systems. In this study, we use early warning indicators to predict subcritical Hopf bifurcation occurring in a thermoacoustic system by analyzing the observables from experiments and from a theoretical model. We find that the early warning measures perform as robust indicators in the presence and absence of external noise. Thus, we illustrate the applicability of these indicators in an engineering system depicting critical transitions. PMID:27767065

Vibrational resonance in an inhomogeneous medium with periodic dissipation

NASA Astrophysics Data System (ADS)

Roy-Layinde, T. O.; Laoye, J. A.; Popoola, O. O.; Vincent, U. E.; McClintock, P. V. E.

2017-09-01

The role of nonlinear dissipation in vibrational resonance (VR) is investigated in an inhomogeneous system characterized by a symmetric and spatially periodic potential and subjected to nonuniform state-dependent damping and a biharmonic driving force. The contributions of the parameters of the high-frequency signal to the system's effective dissipation are examined theoretically in comparison to linearly damped systems, for which the parameter of interest is the effective stiffness in the equation of slow vibration. We show that the VR effect can be enhanced by varying the nonlinear dissipation parameters and that it can be induced by a parameter that is shared by the damping inhomogeneity and the system potential. Furthermore, we have apparently identified the origin of the nonlinear-dissipation-enhanced response: We provide evidence of its connection to a Hopf bifurcation, accompanied by monotonic attractor enlargement in the VR regime.

Disease-induced mortality in density-dependent discrete-time S-I-S epidemic models.

PubMed

Franke, John E; Yakubu, Abdul-Aziz

2008-12-01

The dynamics of simple discrete-time epidemic models without disease-induced mortality are typically characterized by global transcritical bifurcation. We prove that in corresponding models with disease-induced mortality a tiny number of infectious individuals can drive an otherwise persistent population to extinction. Our model with disease-induced mortality supports multiple attractors. In addition, we use a Ricker recruitment function in an SIS model and obtained a three component discrete Hopf (Neimark-Sacker) cycle attractor coexisting with a fixed point attractor. The basin boundaries of the coexisting attractors are fractal in nature, and the example exhibits sensitive dependence of the long-term disease dynamics on initial conditions. Furthermore, we show that in contrast to corresponding models without disease-induced mortality, the disease-free state dynamics do not drive the disease dynamics.

Rotating non-Boussinesq convection: oscillating hexagons

NASA Astrophysics Data System (ADS)

Moroz, Vadim; Riecke, Hermann; Pesch, Werner

2000-11-01

Within weakly nonlinear theory hexagon patterns are expected to undergo a Hopf bifurcation to oscillating hexagons when the chiral symmetry of the system is broken. Quite generally, the oscillating hexagons are expected to exhibit bistability of spatio-temporal defect chaos and periodic dynamics. This regime is described by the complex Ginzburg-Landau equation, which has been investigated theoretically in great detail. Its complex dynamics have, however, not been observed in experiments. Starting from the Navier-Stokes equations with realistic boundary conditions, we derive the three coupled real Ginzburg-Landau equations describing hexagons in rotating non-Boussinesq convection. We use them to provide quantitative results for the wavenumber range of stability of the stationary hexagons as well as the range of existence and stability of the oscillating hexagons. Our investigation is complemented by direct numerical simulations of the Navier-Stokes equations.

Analysis of recruitment and industrial human resources management for optimal productivity in the presence of the HIV/AIDS epidemic.

PubMed

Okosun, Kazeem O; Makinde, Oluwole D; Takaidza, Isaac

2013-01-01

The aim of this paper is to analyze the recruitment effects of susceptible and infected individuals in order to assess the productivity of an organizational labor force in the presence of HIV/AIDS with preventive and HAART treatment measures in enhancing the workforce output. We consider constant controls as well as time-dependent controls. In the constant control case, we calculate the basic reproduction number and investigate the existence and stability of equilibria. The model is found to exhibit backward and Hopf bifurcations, implying that for the disease to be eradicated, the basic reproductive number must be below a critical value of less than one. We also investigate, by calculating sensitivity indices, the sensitivity of the basic reproductive number to the model's parameters. In the time-dependent control case, we use Pontryagin's maximum principle to derive necessary conditions for the optimal control of the disease. Finally, numerical simulations are performed to illustrate the analytical results. The cost-effectiveness analysis results show that optimal efforts on recruitment (HIV screening of applicants, etc.) is not the most cost-effective strategy to enhance productivity in the organizational labor force. Hence, to enhance employees' productivity, effective education programs and strict adherence to preventive measures should be promoted.

The Hunt Opinion Model-An Agent Based Approach to Recurring Fashion Cycles.

PubMed

Apriasz, Rafał; Krueger, Tyll; Marcjasz, Grzegorz; Sznajd-Weron, Katarzyna

2016-01-01

We study a simple agent-based model of the recurring fashion cycles in the society that consists of two interacting communities: "snobs" and "followers" (or "opinion hunters", hence the name of the model). Followers conform to all other individuals, whereas snobs conform only to their own group and anticonform to the other. The model allows to examine the role of the social structure, i.e. the influence of the number of inter-links between the two communities, as well as the role of the stability of links. The latter is accomplished by considering two versions of the same model-quenched (parameterized by fraction L of fixed inter-links) and annealed (parameterized by probability p that a given inter-link exists). Using Monte Carlo simulations and analytical treatment (the latter only for the annealed model), we show that there is a critical fraction of inter-links, above which recurring cycles occur. For p ≤ 0.5 we derive a relation between parameters L and p that allows to compare both models and show that the critical value of inter-connections, p*, is the same for both versions of the model (annealed and quenched) but the period of a fashion cycle is shorter for the quenched model. Near the critical point, the cycles are irregular and a change of fashion is difficult to predict. For the annealed model we also provide a deeper theoretical analysis. We conjecture on topological grounds that the so-called saddle node heteroclinic bifurcation appears at p*. For p ≥ 0.5 we show analytically the existence of the second critical value of p, for which the system undergoes Hopf's bifurcation.

Zero Prandtl-number rotating magnetoconvection

NASA Astrophysics Data System (ADS)

Ghosh, Manojit; Pal, Pinaki

2017-12-01

We investigate instabilities and chaos near the onset of Rayleigh-Bénard convection of electrically conducting fluids with free-slip, perfectly electrically and thermally conducting boundary conditions in the presence of uniform rotation about the vertical axis and horizontal external magnetic field by considering zero Prandtl-number limit (Pr → 0). Direct numerical simulations (DNSs) and low-dimensional modeling of the system are done for the investigation. Values of the Chandrasekhar number (Q) and the Taylor number (Ta) are varied in the range 0 < Q, Ta ≤ 50. Depending on the values of the parameters in the chosen range and the choice of initial conditions, the onset of convection is found be either periodic or chaotic. Interestingly, it is found that chaos at the onset can occur through four different routes, namely, homoclinic, intermittent, period doubling, and quasiperiodic routes. Homoclinic and intermittent routes to chaos at the onset occur in the presence of weak magnetic field (Q < 2), while the period doubling route is observed for relatively stronger magnetic field (Q ≥ 2) for one set of initial conditions. On the other hand, the quasiperiodic route to chaos at the onset is observed for another set of initial conditions. However, the rotation rate (value of Ta) also plays an important role in determining the nature of convection at the onset. Analysis of the system simultaneously with DNSs and low-dimensional modeling helps us to clearly identify different flow regimes concentrated near the onset of convection and understand their origins. The periodic or chaotic convection at the onset is found to be connected with rich bifurcation structures involving subcritical pitchfork, imperfect pitchfork, supercritical Hopf, imperfect homoclinic gluing, and Neimark-Sacker bifurcations.

Delay-induced depinning of localized structures in a spatially inhomogeneous Swift-Hohenberg model

NASA Astrophysics Data System (ADS)

Tabbert, Felix; Schelte, Christian; Tlidi, Mustapha; Gurevich, Svetlana V.

2017-03-01

We report on the dynamics of localized structures in an inhomogeneous Swift-Hohenberg model describing pattern formation in the transverse plane of an optical cavity. This real order parameter equation is valid close to the second-order critical point associated with bistability. The optical cavity is illuminated by an inhomogeneous spatial Gaussian pumping beam and subjected to time-delayed feedback. The Gaussian injection beam breaks the translational symmetry of the system by exerting an attracting force on the localized structure. We show that the localized structure can be pinned to the center of the inhomogeneity, suppressing the delay-induced drift bifurcation that has been reported in the particular case where the injection is homogeneous, assuming a continuous wave operation. Under an inhomogeneous spatial pumping beam, we perform the stability analysis of localized solutions to identify different instability regimes induced by time-delayed feedback. In particular, we predict the formation of two-arm spirals, as well as oscillating and depinning dynamics caused by the interplay of an attracting inhomogeneity and destabilizing time-delayed feedback. The transition from oscillating to depinning solutions is investigated by means of numerical continuation techniques. Analytically, we use an order parameter approach to derive a normal form of the delay-induced Hopf bifurcation leading to an oscillating solution. Additionally we model the interplay of an attracting inhomogeneity and destabilizing time delay by describing the localized solution as an overdamped particle in a potential well generated by the inhomogeneity. In this case, the time-delayed feedback acts as a driving force. Comparing results from the later approach with the full Swift-Hohenberg model, we show that the approach not only provides an instructive description of the depinning dynamics, but also is numerically accurate throughout most of the parameter regime.

Spontaneous Symmetry-Breaking in a Network Model for Quadruped Locomotion

NASA Astrophysics Data System (ADS)

Stewart, Ian

2017-12-01

Spontaneous symmetry-breaking proves a mechanism for pattern generation in legged locomotion of animals. The basic timing patterns of animal gaits are produced by a network of spinal neurons known as a Central Pattern Generator (CPG). Animal gaits are primarily characterized by phase differences between leg movements in a periodic gait cycle. Many different gaits occur, often having spatial or spatiotemporal symmetries. A natural way to explain gait patterns is to assume that the CPG is symmetric, and to classify the possible symmetry-breaking periodic motions. Pinto and Golubitsky have discussed a four-node model CPG network for biped gaits with ℤ2 × ℤ2 symmetry, classifying the possible periodic states that can arise. A more specific rate model with this structure has been analyzed in detail by Stewart. Here we extend these methods to quadruped gaits, using an eight-node network with ℤ4 × ℤ2 symmetry proposed by Golubitsky and coworkers. We formulate a rate model and calculate how the first steady or Hopf bifurcation depends on its parameters, which represent four connection strengths. The calculations involve a distinction between “real” gaits with one or two phase shifts (pronk, bound, pace, trot) and “complex” gaits with four phase shifts (forward and reverse walk, forward and reverse buck). The former correspond to real eigenvalues of the connection matrix, the latter to complex conjugate pairs. The partition of parameter space according to the first bifurcation, ignoring complex gaits, is described explicitly. The complex gaits introduce further complications, not yet fully understood. All eight gaits can occur as the first bifurcation from a fully synchronous equilibrium, for suitable parameters, and numerical simulations indicate that they can be asymptotically stable.

Heat and Momentum Transfer Studies in High Reynolds Number Wavy Films at Normal and Reduced Gravity Conditions

NASA Technical Reports Server (NTRS)

Balakotaiah, V.

1996-01-01

We examined the effect of the gas flow on the liquid film when the gas flows in the countercurrent direction in a vertical pipe at normal gravity conditions. The most dramatic effect of the simultaneous flow of gas and liquid in pipes is the greatly increased transport rates of heat, mass, and momentum. In practical situations this enhancement can be a benefit or it can result in serious operational problems. For example, gas-liquid flow always results in substantially higher pressure drop and this is usually undesirable. However, much higher heat transfer coefficients can be expected and this can obviously be of benefit for purposes of design. Unfortunately, designers know so little of the behavior of such two phase systems and as a result these advantages are not utilized. Due to the complexity of the second order boundary model as well as the fact that the pressure variation across the film is small compared to the imposed gas phase pressure, the countercurrent gas flow affect was studied for the standard boundary layer model. A different stream function that can compensate the shear stress affect was developed and this stream function also can predict periodic solutions. The discretized model equations were transformed to a traveling wave coordinate system. A stability analysis of these sets of equations showed the presence of a Hopf bifurcation for certain values of the traveling wave velocity and the shear stress. The Hopf celerity was increased due to the countercurrent shear. For low flow rate the increases of celerity are more than for the high flow rate, which was also observed in experiments. Numerical integration of a traveling wave simplification of the model also predicts the existence of chaotic large amplitude, nonperiodic waves as observed in the experiments. The film thickness was increased by the shear.

Triggered dynamics in a model of different fault creep regimes

PubMed Central

Kostić, Srđan; Franović, Igor; Perc, Matjaž; Vasović, Nebojša; Todorović, Kristina

2014-01-01

The study is focused on the effect of transient external force induced by a passing seismic wave on fault motion in different creep regimes. Displacement along the fault is represented by the movement of a spring-block model, whereby the uniform and oscillatory motion correspond to the fault dynamics in post-seismic and inter-seismic creep regime, respectively. The effect of the external force is introduced as a change of block acceleration in the form of a sine wave scaled by an exponential pulse. Model dynamics is examined for variable parameters of the induced acceleration changes in reference to periodic oscillations of the unperturbed system above the supercritical Hopf bifurcation curve. The analysis indicates the occurrence of weak irregular oscillations if external force acts in the post-seismic creep regime. When fault motion is exposed to external force in the inter-seismic creep regime, one finds the transition to quasiperiodic- or chaos-like motion, which we attribute to the precursory creep regime and seismic motion, respectively. If the triggered acceleration changes are of longer duration, a reverse transition from inter-seismic to post-seismic creep regime is detected on a larger time scale. PMID:24954397

Nonlinear dynamics near the stability margin in rotating pipe flow

NASA Technical Reports Server (NTRS)

Yang, Z.; Leibovich, S.

1991-01-01

The nonlinear evolution of marginally unstable wave packets in rotating pipe flow is studied. These flows depend on two control parameters, which may be taken to be the axial Reynolds number R and a Rossby number, q. Marginal stability is realized on a curve in the (R, q)-plane, and the entire marginal stability boundary is explored. As the flow passes through any point on the marginal stability curve, it undergoes a supercritical Hopf bifurcation and the steady base flow is replaced by a traveling wave. The envelope of the wave system is governed by a complex Ginzburg-Landau equation. The Ginzburg-Landau equation admits Stokes waves, which correspond to standing modulations of the linear traveling wavetrain, as well as traveling wave modulations of the linear wavetrain. Bands of wavenumbers are identified in which the nonlinear modulated waves are subject to a sideband instability.

Evolution of secondary whirls in thermoconvective vortices: Strengthening, weakening, and disappearance in the route to chaos

NASA Astrophysics Data System (ADS)

Castaño, D.; Navarro, M. C.; Herrero, H.

2016-01-01

The appearance, evolution, and disappearance of periodic and quasiperiodic dynamics of fluid flows in a cylindrical annulus locally heated from below are analyzed using nonlinear simulations. The results reveal a route of the transition from a steady axisymmetric vertical vortex to a chaotic flow. The chaotic flow regime is reached after a sequence of successive supercritical Hopf bifurcations to periodic, quasiperiodic, and chaotic flow regimes. A scenario similar to the Ruelle-Takens-Newhouse scenario is verified in this convective flow. In the transition to chaos we find the appearance of subvortices embedded in the primary axisymmetric vortex, flows where the subvortical structure strengthens and weakens, that almost disappears before reforming again, leading to a more disorganized flow to a final chaotic regime. Results are remarkable as they connect to observations describing formation, weakening, and virtual disappearance before revival of subvortices in some atmospheric swirls such as dust devils.

Activation and synchronization of the oscillatory morphodynamics in multicellular monolayer

PubMed Central

Lin, Shao-Zhen; Li, Bo; Lan, Ganhui; Feng, Xi-Qiao

2017-01-01

Oscillatory morphodynamics provides necessary mechanical cues for many multicellular processes. Owing to their collective nature, these processes require robustly coordinated dynamics of individual cells, which are often separated too distantly to communicate with each other through biomaterial transportation. Although it is known that the mechanical balance generally plays a significant role in the systems’ morphologies, it remains elusive whether and how the mechanical components may contribute to the systems’ collective morphodynamics. Here, we study the collective oscillations in the Drosophila amnioserosa tissue to elucidate the regulatory roles of the mechanical components. We identify that the tensile stress is the key activator that switches the collective oscillations on and off. This regulatory role is shown analytically using the Hopf bifurcation theory. We find that the physical properties of the tissue boundary are directly responsible for synchronizing the oscillatory intensity and polarity of all inner cells and for orchestrating the spatial oscillation patterns inthe tissue. PMID:28716911

Voice tracking and spoken word recognition in the presence of other voices

NASA Astrophysics Data System (ADS)

Litong-Palima, Marisciel; Violanda, Renante; Saloma, Caesar

2004-12-01

We study the human hearing process by modeling the hair cell as a thresholded Hopf bifurcator and compare our calculations with experimental results involving human subjects in two different multi-source listening tasks of voice tracking and spoken-word recognition. In the model, we observed noise suppression by destructive interference between noise sources which weakens the effective noise strength acting on the hair cell. Different success rate characteristics were observed for the two tasks. Hair cell performance at low threshold levels agree well with results from voice-tracking experiments while those of word-recognition experiments are consistent with a linear model of the hearing process. The ability of humans to track a target voice is robust against cross-talk interference unlike word-recognition performance which deteriorates quickly with the number of uncorrelated noise sources in the environment which is a response behavior that is associated with linear systems.

Characterization of active hair-bundle motility by a mechanical-load clamp

NASA Astrophysics Data System (ADS)

Salvi, Joshua D.; Maoiléidigh, Dáibhid Ó.; Fabella, Brian A.; Tobin, Mélanie; Hudspeth, A. J.

2015-12-01

Active hair-bundle motility endows hair cells with several traits that augment auditory stimuli. The activity of a hair bundle might be controlled by adjusting its mechanical properties. Indeed, the mechanical properties of bundles vary between different organisms and along the tonotopic axis of a single auditory organ. Motivated by these biological differences and a dynamical model of hair-bundle motility, we explore how adjusting the mass, drag, stiffness, and offset force applied to a bundle control its dynamics and response to external perturbations. Utilizing a mechanical-load clamp, we systematically mapped the two-dimensional state diagram of a hair bundle. The clamp system used a real-time processor to tightly control each of the virtual mechanical elements. Increasing the stiffness of a hair bundle advances its operating point from a spontaneously oscillating regime into a quiescent regime. As predicted by a dynamical model of hair-bundle mechanics, this boundary constitutes a Hopf bifurcation.

Non-linear vibrating systems excited by a nonideal energy source with a large slope characteristic

NASA Astrophysics Data System (ADS)

González-Carbajal, Javier; Domínguez, Jaime

2017-11-01

This paper revisits the problem of an unbalanced motor attached to a fixed frame by means of a nonlinear spring and a linear damper. The excitation provided by the motor is, in general, nonideal, which means it is affected by the vibratory response. Since the system behaviour is highly dependent on the order of magnitude of the motor characteristic slope, the case of large slope is considered herein. Some Perturbation Methods are applied to the system of equations, which allows transforming the original 4D system into a much simpler 2D system. The fixed points of this reduced system and their stability are carefully studied. We find the existence of a Hopf bifurcation which, to the authors' knowledge, has not been addressed before in the literature. These analytical results are supported by numerical simulations. We also compare our approach and results with those published by other authors.

Study of a tri-trophic prey-dependent food chain model of interacting populations.

PubMed

Haque, Mainul; Ali, Nijamuddin; Chakravarty, Santabrata

2013-11-01

The current paper accounts for the influence of intra-specific competition among predators in a prey dependent tri-trophic food chain model of interacting populations. We offer a detailed mathematical analysis of the proposed food chain model to illustrate some of the significant results that has arisen from the interplay of deterministic ecological phenomena and processes. Biologically feasible equilibria of the system are observed and the behaviours of the system around each of them are described. In particular, persistence, stability (local and global) and bifurcation (saddle-node, transcritical, Hopf-Andronov) analysis of this model are obtained. Relevant results from previous well known food chain models are compared with the current findings. Global stability analysis is also carried out by constructing appropriate Lyapunov functions. Numerical simulations show that the present system is capable enough to produce chaotic dynamics when the rate of self-interaction is very low. On the other hand such chaotic behaviour disappears for a certain value of the rate of self interaction. In addition, numerical simulations with experimented parameters values confirm the analytical results and shows that intra-specific competitions bears a potential role in controlling the chaotic dynamics of the system; and thus the role of self interactions in food chain model is illustrated first time. Finally, a discussion of the ecological applications of the analytical and numerical findings concludes the paper. Copyright © 2013 Elsevier Inc. All rights reserved.

Observation of Topological Links Associated with Hopf Insulators in a Solid-State Quantum Simulator

NASA Astrophysics Data System (ADS)

Yuan, X.-X.; He, L.; Wang, S.-T.; Deng, D.-L.; Wang, F.; Lian, W.-Q.; Wang, X.; Zhang, C.-H.; Zhang, H.-L.; Chang, X.-Y.; Duan, L.-M.

2017-06-01

Hopf insulators are intriguing three-dimensional topological insulators characterized by an integer topological invariant. They originate from the mathematical theory of Hopf fibration and epitomize the deep connection between knot theory and topological phases of matter, which distinguishes them from other classes of topological insulators. Here, we implement a model Hamiltonian for Hopf insulators in a solid-state quantum simulator and report the first experimental observation of their topological properties, including fascinating topological links associated with the Hopf fibration and the integer-valued topological invariant obtained from a direct tomographic measurement. Our observation of topological links and Hopf fibration in a quantum simulator opens the door to probe rich topological properties of Hopf insulators in experiments. The quantum simulation and probing methods are also applicable to the study of other intricate three-dimensional topological model Hamiltonians.

The Poincaré-Hopf Theorem for line fields revisited

NASA Astrophysics Data System (ADS)

Crowley, Diarmuid; Grant, Mark

2017-07-01

A Poincaré-Hopf Theorem for line fields with point singularities on orientable surfaces can be found in Hopf's 1956 Lecture Notes on Differential Geometry. In 1955 Markus presented such a theorem in all dimensions, but Markus' statement only holds in even dimensions 2 k ≥ 4. In 1984 Jänich presented a Poincaré-Hopf theorem for line fields with more complicated singularities and focussed on the complexities arising in the generalized setting. In this expository note we review the Poincaré-Hopf Theorem for line fields with point singularities, presenting a careful proof which is valid in all dimensions.

A minimal model of predator–swarm interactions

PubMed Central

Chen, Yuxin; Kolokolnikov, Theodore

2014-01-01

We propose a minimal model of predator–swarm interactions which captures many of the essential dynamics observed in nature. Different outcomes are observed depending on the predator strength. For a ‘weak’ predator, the swarm is able to escape the predator completely. As the strength is increased, the predator is able to catch up with the swarm as a whole, but the individual prey is able to escape by ‘confusing’ the predator: the prey forms a ring with the predator at the centre. For higher predator strength, complex chasing dynamics are observed which can become chaotic. For even higher strength, the predator is able to successfully capture the prey. Our model is simple enough to be amenable to a full mathematical analysis, which is used to predict the shape of the swarm as well as the resulting predator–prey dynamics as a function of model parameters. We show that, as the predator strength is increased, there is a transition (owing to a Hopf bifurcation) from confusion state to chasing dynamics, and we compute the threshold analytically. Our analysis indicates that the swarming behaviour is not helpful in avoiding the predator, suggesting that there are other reasons why the species may swarm. The complex shape of the swarm in our model during the chasing dynamics is similar to the shape of a flock of sheep avoiding a shepherd. PMID:24598204

Flow past a rotating cylinder

NASA Astrophysics Data System (ADS)

Mittal, Sanjay; Kumar, Bhaskar

2003-02-01

Flow past a spinning circular cylinder placed in a uniform stream is investigated via two-dimensional computations. A stabilized finite element method is utilized to solve the incompressible Navier Stokes equations in the primitive variables formulation. The Reynolds number based on the cylinder diameter and free-stream speed of the flow is 200. The non-dimensional rotation rate, [alpha] (ratio of the surface speed and freestream speed), is varied between 0 and 5. The time integration of the flow equations is carried out for very large dimensionless time. Vortex shedding is observed for [alpha] < 1.91. For higher rotation rates the flow achieves a steady state except for 4.34 < [alpha] < 4:70 where the flow is unstable again. In the second region of instability, only one-sided vortex shedding takes place. To ascertain the instability of flow as a function of [alpha] a stabilized finite element formulation is proposed to carry out a global, non-parallel stability analysis of the two-dimensional steady-state flow for small disturbances. The formulation and its implementation are validated by predicting the Hopf bifurcation for flow past a non-rotating cylinder. The results from the stability analysis for the rotating cylinder are in very good agreement with those from direct numerical simulations. For large rotation rates, very large lift coefficients can be obtained via the Magnus effect. However, the power requirement for rotating the cylinder increases rapidly with rotation rate.

A minimal model of predator-swarm interactions.

PubMed

Chen, Yuxin; Kolokolnikov, Theodore

2014-05-06

We propose a minimal model of predator-swarm interactions which captures many of the essential dynamics observed in nature. Different outcomes are observed depending on the predator strength. For a 'weak' predator, the swarm is able to escape the predator completely. As the strength is increased, the predator is able to catch up with the swarm as a whole, but the individual prey is able to escape by 'confusing' the predator: the prey forms a ring with the predator at the centre. For higher predator strength, complex chasing dynamics are observed which can become chaotic. For even higher strength, the predator is able to successfully capture the prey. Our model is simple enough to be amenable to a full mathematical analysis, which is used to predict the shape of the swarm as well as the resulting predator-prey dynamics as a function of model parameters. We show that, as the predator strength is increased, there is a transition (owing to a Hopf bifurcation) from confusion state to chasing dynamics, and we compute the threshold analytically. Our analysis indicates that the swarming behaviour is not helpful in avoiding the predator, suggesting that there are other reasons why the species may swarm. The complex shape of the swarm in our model during the chasing dynamics is similar to the shape of a flock of sheep avoiding a shepherd.

Big Bang Bifurcation Analysis and Allee Effect in Generic Growth Functions

NASA Astrophysics Data System (ADS)

Leonel Rocha, J.; Taha, Abdel-Kaddous; Fournier-Prunaret, D.

2016-06-01

The main purpose of this work is to study the dynamics and bifurcation properties of generic growth functions, which are defined by the population size functions of the generic growth equation. This family of unimodal maps naturally incorporates a principal focus of ecological and biological research: the Allee effect. The analysis of this kind of extinction phenomenon allows to identify a class of Allee’s functions and characterize the corresponding Allee’s effect region and Allee’s bifurcation curve. The bifurcation analysis is founded on the performance of fold and flip bifurcations. The dynamical behavior is rich with abundant complex bifurcation structures, the big bang bifurcations of the so-called “box-within-a-box” fractal type being the most outstanding. Moreover, these bifurcation cascades converge to different big bang bifurcation curves with distinct kinds of boxes, where for the corresponding parameter values several attractors are associated. To the best of our knowledge, these results represent an original contribution to clarify the big bang bifurcation analysis of continuous 1D maps.

Plane wave diffraction by a finite plate with impedance boundary conditions.

PubMed

Nawaz, Rab; Ayub, Muhammad; Javaid, Akmal

2014-01-01

In this study we have examined a plane wave diffraction problem by a finite plate having different impedance boundaries. The Fourier transforms were used to reduce the governing problem into simultaneous Wiener-Hopf equations which are then solved using the standard Wiener-Hopf procedure. Afterwards the separated and interacted fields were developed asymptotically by using inverse Fourier transform and the modified stationary phase method. Detailed graphical analysis was also made for various physical parameters we were interested in.

Scaling up complexity in host-pathogens interaction models. Comment on "Coupled disease-behavior dynamics on complex networks: A review" by Z. Wang et al.

NASA Astrophysics Data System (ADS)

Aguiar, Maíra

2015-12-01

Caused by micro-organisms that are pathogenic to the host, infectious diseases have caused debilitation and premature death to large portions of the human population, leading to serious social-economic concerns. The persistence and increase in the occurrence of infectious diseases as well the emergence or resurgence of vector-borne diseases are closely related with demographic factors such as the uncontrolled urbanization and remarkable population growth, political, social and economical changes, deforestation, development of resistance to insecticides and drugs and increased human travel. In recent years, mathematical modeling became an important tool for the understanding of infectious disease epidemiology and dynamics, addressing ideas about the components of host-pathogen interactions. Acting as a possible tool to understand, predict the spread of infectious diseases these models are also used to evaluate the introduction of intervention strategies like vector control and vaccination. Many scientific papers have been published recently on these topics, and most of the models developed try to incorporate factors focusing on several different aspects of the disease (and eventually biological aspects of the vector), which can imply rich dynamic behavior even in the most basic dynamical models. As one example to be cited, there is a minimalistic dengue model that has shown rich dynamic structures, with bifurcations (Hopf, pitchfork, torus and tangent bifurcations) up to chaotic attractors in unexpected parameter regions [1,2], which was able to describe the large fluctuations observed in empirical outbreak data [3,4].

Length Scales of Chaotic patterns near the onset of of Electroconvection in the Nematic Liquid Crystal I52

NASA Astrophysics Data System (ADS)

Xu, Xiaochao

2005-03-01

We report experimental results for Electroconvection of the nematic Liquid Crystal I52 with planar alignment and a conductivity of 1.0x10-8,φ,)-1. The cell spacing was 19.4,m and the driving frequency was 25.0 Hz. Spatio-temporal chaos consisting of a superposition of zig and zag oblique rolls evolved by means of a supercritical Hopf bifurcation from the uniform conduction state.ootnotetextM. Dennin, G. Ahlers and D. S. Cannell, Science, 272, 388 (1996). For small ɛ≡V^2/ Vc^2 -1, we measured the correlation lengths of the envelopes of both zig and zag patterns. These lengths could be fit to a power law in ɛ with an exponent smaller than that predicted from amplitude equations. The disagreement with theory is similar to that found previously for domain chaos in rotating Rayleigh-Benard convection.ootnotetextY. Hu, R. E. Ecke and G. Ahlers, Phys. Rev. Lett. 74, 5040 (1995).

Dynamics of one- and two-dimensional fronts in a bistable equation with time-delayed global feedback: Propagation failure and control mechanisms

DOE Office of Scientific and Technical Information (OSTI.GOV)

Boubendir, Yassine; Mendez, Vicenc; Rotstein, Horacio G.

2010-09-15

We study the evolution of fronts in a bistable equation with time-delayed global feedback in the fast reaction and slow diffusion regime. This equation generalizes the Hodgkin-Grafstein and Allen-Cahn equations. We derive a nonlinear equation governing the motion of fronts, which includes a term with delay. In the one-dimensional case this equation is linear. We study the motion of one- and two-dimensional fronts, finding a much richer dynamics than for the previously studied cases (without time-delayed global feedback). We explain the mechanism by which localized fronts created by inhibitory global coupling loose stability in a Hopf bifurcation as the delaymore » time increases. We show that for certain delay times, the prevailing phase is different from that corresponding to the system in the absence of global coupling. Numerical simulations of the partial differential equation are in agreement with the analytical predictions.« less

Hopf algebras of rooted forests, cocyles, and free Rota-Baxter algebras

NASA Astrophysics Data System (ADS)

Zhang, Tianjie; Gao, Xing; Guo, Li

2016-10-01

The Hopf algebra and the Rota-Baxter algebra are the two algebraic structures underlying the algebraic approach of Connes and Kreimer to renormalization of perturbative quantum field theory. In particular, the Hopf algebra of rooted trees serves as the "baby model" of Feynman graphs in their approach and can be characterized by certain universal properties involving a Hochschild 1-cocycle. Decorated rooted trees have also been applied to study Feynman graphs. We will continue the study of universal properties of various spaces of decorated rooted trees with such a 1-cocycle, leading to the concept of a cocycle Hopf algebra. We further apply the universal properties to equip a free Rota-Baxter algebra with the structure of a cocycle Hopf algebra.

Mathematical assessment of the role of temperature and rainfall on mosquito population dynamics.

PubMed

Abdelrazec, Ahmed; Gumel, Abba B

2017-05-01

A new stage-structured model for the population dynamics of the mosquito (a major vector for numerous vector-borne diseases), which takes the form of a deterministic system of non-autonomous nonlinear differential equations, is designed and used to study the effect of variability in temperature and rainfall on mosquito abundance in a community. Two functional forms of eggs oviposition rate, namely the Verhulst-Pearl logistic and Maynard-Smith-Slatkin functions, are used. Rigorous analysis of the autonomous version of the model shows that, for any of the oviposition functions considered, the trivial equilibrium of the model is locally- and globally-asymptotically stable if a certain vectorial threshold quantity is less than unity. Conditions for the existence and global asymptotic stability of the non-trivial equilibrium solutions of the model are also derived. The model is shown to undergo a Hopf bifurcation under certain conditions (and that increased density-dependent competition in larval mortality reduces the likelihood of such bifurcation). The analyses reveal that the Maynard-Smith-Slatkin oviposition function sustains more oscillations than the Verhulst-Pearl logistic function (hence, it is more suited, from ecological viewpoint, for modeling the egg oviposition process). The non-autonomous model is shown to have a globally-asymptotically stable trivial periodic solution, for each of the oviposition functions, when the associated reproduction threshold is less than unity. Furthermore, this model, in the absence of density-dependent mortality rate for larvae, has a unique and globally-asymptotically stable periodic solution under certain conditions. Numerical simulations of the non-autonomous model, using mosquito surveillance and weather data from the Peel region of Ontario, Canada, show a peak mosquito abundance for temperature and rainfall values in the range [Formula: see text]C and [15-35] mm, respectively. These ranges are recorded in the Peel region between July and August (hence, this study suggests that anti-mosquito control effects should be intensified during this period).

Nonlinear Dynamic Characteristics of the Railway Vehicle

NASA Astrophysics Data System (ADS)

Uyulan, Çağlar; Gokasan, Metin

2017-06-01

The nonlinear dynamic characteristics of a railway vehicle are checked into thoroughly by applying two different wheel-rail contact model: a heuristic nonlinear friction creepage model derived by using Kalker 's theory and Polach model including dead-zone clearance. This two models are matched with the quasi-static form of the LuGre model to obtain more realistic wheel-rail contact model. LuGre model parameters are determined using nonlinear optimization method, which it's objective is to minimize the error between the output of the Polach and Kalker model and quasi-static LuGre model for specific operating conditions. The symmetric/asymmetric bifurcation attitude and stable/unstable motion of the railway vehicle in the presence of nonlinearities which are yaw damping forces in the longitudinal suspension system are analyzed in great detail by changing the vehicle speed. Phase portraits of the lateral displacement of the leading wheelset of the railway vehicle are drawn below and on the critical speeds, where sub-critical Hopf bifurcation take place, for two wheel-rail contact model. Asymmetric periodic motions have been observed during the simulation in the lateral displacement of the wheelset under different vehicle speed range. The coexistence of multiple steady states cause bounces in the amplitude of vibrations, resulting instability problems of the railway vehicle. By using Lyapunov's indirect method, the critical hunting speeds are calculated with respect to the radius of the curved track parameter changes. Hunting, which is defined as the oscillation of the lateral displacement of wheelset with a large domain, is described by a limit cycle-type oscillation nature. The evaluated accuracy of the LuGre model adopted from Kalker's model results for prediction of critical speed is higher than the results of the LuGre model adopted from Polach's model. From the results of the analysis, the critical hunting speed must be resolved by investigating the track tests under various kind of excitations.

Analysis of friction and instability by the centre manifold theory for a non-linear sprag-slip model

NASA Astrophysics Data System (ADS)

Sinou, J.-J.; Thouverez, F.; Jezequel, L.

2003-08-01

This paper presents the research devoted to the study of instability phenomena in non-linear model with a constant brake friction coefficient. Indeed, the impact of unstable oscillations can be catastrophic. It can cause vehicle control problems and component degradation. Accordingly, complex stability analysis is required. This paper outlines stability analysis and centre manifold approach for studying instability problems. To put it more precisely, one considers brake vibrations and more specifically heavy trucks judder where the dynamic characteristics of the whole front axle assembly is concerned, even if the source of judder is located in the brake system. The modelling introduces the sprag-slip mechanism based on dynamic coupling due to buttressing. The non-linearity is expressed as a polynomial with quadratic and cubic terms. This model does not require the use of brake negative coefficient, in order to predict the instability phenomena. Finally, the centre manifold approach is used to obtain equations for the limit cycle amplitudes. The centre manifold theory allows the reduction of the number of equations of the original system in order to obtain a simplified system, without loosing the dynamics of the original system as well as the contributions of non-linear terms. The goal is the study of the stability analysis and the validation of the centre manifold approach for a complex non-linear model by comparing results obtained by solving the full system and by using the centre manifold approach. The brake friction coefficient is used as an unfolding parameter of the fundamental Hopf bifurcation point.

Hopf-link topological nodal-loop semimetals

NASA Astrophysics Data System (ADS)

Zhou, Yao; Xiong, Feng; Wan, Xiangang; An, Jin

2018-04-01

We construct a generic two-band model which can describe topological semimetals with multiple closed nodal loops. All the existing multi-nodal-loop semimetals, including the nodal-net, nodal-chain, and Hopf-link states, can be examined within the same framework. Based on a two-nodal-loop model, the corresponding drumhead surface states for these topologically different bulk states are studied and compared with each other. The connection of our model with Hopf insulators is also discussed. Furthermore, to identify experimentally these topologically different semimetal states, especially to distinguish the Hopf-link from unlinked ones, we also investigate their Landau levels. It is found that the Hopf-link state can be characterized by the existence of a quadruply degenerate zero-energy Landau band, regardless of the direction of the magnetic field.

Codimension-Two Bifurcation Analysis in DC Microgrids Under Droop Control

NASA Astrophysics Data System (ADS)

Lenz, Eduardo; Pagano, Daniel J.; Tahim, André P. N.

This paper addresses local and global bifurcations that may appear in electrical power systems, such as DC microgrids, which recently has attracted interest from the electrical engineering society. Most sources in these networks are voltage-type and operate in parallel. In such configuration, the basic technique for stabilizing the bus voltage is the so-called droop control. The main contribution of this work is a codimension-two bifurcation analysis of a small DC microgrid considering the droop control gain and the power processed by the load as bifurcation parameters. The codimension-two bifurcation set leads to practical rules for achieving a robust droop control design. Moreover, the bifurcation analysis also offers a better understanding of the dynamics involved in the problem and how to avoid possible instabilities. Simulation results are presented in order to illustrate the bifurcation analysis.

External Source of Infection and Nutritional Efficiency Control Chaos in a Predator-Prey Model with Disease in the Predator

NASA Astrophysics Data System (ADS)

Pada Das, Krishna; Roy, Prodip; Ghosh, Subhabrata; Maiti, Somnath

This paper deals with an eco-epidemiological approach with disease circulating through the predator species. Disease circulation in the predator species can be possible by contact as well as by external sources. Here, we try to discuss the role of external source of infection along with nutritional value on system dynamics. To establish our findings, we have worked out the local and global stability analysis of the equilibrium points with Hopf bifurcation analysis associated with interior equilibrium point. The ecological consequence by ecological basic reproduction number as well as the disease basic reproduction number or basic reproductive ratio are obtained and we have analyzed the community structure of the particular system with the help of ecological and disease basic reproduction numbers. Further we pay attention to the chaotic dynamics which is produced by disease circulating in predator species by contact. Our numerical simulations reveal that eco-epidemiological system without external source of infection induced chaotic dynamics for increasing force of infection due to contact, whereas in the presence of external source of infection, it exhibits stable solution. It is also observed that nutritional value can prevent chaotic dynamics. We conclude that chaotic dynamics can be controlled by the external source of infection as well as nutritional value. We apply basic tools of nonlinear dynamics such as Poincare section and maximum Lyapunov exponent to investigate chaotic behavior of the system.

Nonlinear analysis for dual-frequency concurrent energy harvesting

NASA Astrophysics Data System (ADS)

Yan, Zhimiao; Lei, Hong; Tan, Ting; Sun, Weipeng; Huang, Wenhu

2018-05-01

The dual-frequency responses of the hybrid energy harvester undergoing the base excitation and galloping were analyzed numerically. In this work, an approximate dual-frequency analytical method is proposed for the nonlinear analysis of such a system. To obtain the approximate analytical solutions of the full coupled distributed-parameter model, the forcing interactions is first neglected. Then, the electromechanical decoupled governing equation is developed using the equivalent structure method. The hybrid mechanical response is finally separated to be the self-excited and forced responses for deriving the analytical solutions, which are confirmed by the numerical simulations of the full coupled model. The forced response has great impacts on the self-excited response. The boundary of Hopf bifurcation is analytically determined by the onset wind speed to galloping, which is linearly increased by the electrical damping. Quenching phenomenon appears when the increasing base excitation suppresses the galloping. The theoretical quenching boundary depends on the forced mode velocity. The quenching region increases with the base acceleration and electrical damping, but decreases with the wind speed. Superior to the base-excitation-alone case, the existence of the aerodynamic force protects the hybrid energy harvester at resonance from damages caused by the excessive large displacement. From the view of the harvested power, the hybrid system surpasses the base-excitation-alone system or the galloping-alone system. This study advances our knowledge on intrinsic nonlinear dynamics of the dual-frequency energy harvesting system by taking advantage of the analytical solutions.

Multifractal analysis of a GCM climate

NASA Astrophysics Data System (ADS)

Carl, P.

2003-04-01

Multifractal analysis using the Wavelet Transform Modulus Maxima (WTMM) approach is being applied to the climate of a Mintz--Arakawa type, coarse resolution, two--layer AGCM. The model shows a backwards running period multiplication scenario throughout the northern summer, subsequent to a 'hard', subcritical Hopf bifurcation late in spring. This 'route out of chaos' (seen in cross sections of a toroidal phase space structure) is born in the planetary monsoon system which inflates the seasonal 'cycle' into these higher order structures and is blamed for the pronounced intraseasonal--to--centennial model climate variability. Previous analyses of the latter using advanced modal decompositions showed regularity based patterns in the time--frequency plane which are qualitatively similar to those obtained from the real world. The closer look here at the singularity structures, as a fundamental diagnostic supplement, aims at both more complete understanding (and quantification) of the model's qualitative dynamics and search for further tools of model intercomparison and verification in this respect. Analysing wavelet is the 10th derivative of the Gaussian which might suffice to suppress regular patterns in the data. Intraseasonal attractors, studied in time series of model precipitation over Central India, show shifting and braodening singularity spectra towards both more violent extreme events (premonsoon--monsoon transition) and weaker events (late summer to postmonsoon transition). Hints at a fractal basin boundary are found close to transition from period--2 to period--1 in the monsoon activity cycle. Interannual analyses are provided for runs with varied solar constants. To address the (in--)stationarity issue, first results are presented with a windowed multifractal analysis of longer--term runs ("singularity spectrogram").

Quantum Superalgebras at Roots of Unity and Topological Invariants of Three-manifolds

NASA Astrophysics Data System (ADS)

Blumen, Sacha C.

2006-01-01

The general method of Reshetikhin and Turaev is followed to develop topological invariants of closed, connected, orientable 3-manifolds from a new class of algebras called pseudo-modular Hopf algebras. Pseudo-modular Hopf algebras are a class of Z_2-graded ribbon Hopf algebras that generalise the concept of a modular Hopf algebra. The quantum superalgebra U_q(osp(1|2n)) over C is considered with q a primitive N^th root of unity for all integers N >= 3. For such a q, a certain left ideal I of U_q(osp(1|2n)) is also a two-sided Hopf ideal, and the quotient algebra U_q^(N)(osp(1|2n)) = U_q(osp(1|2n)) / I is a Z_2-graded ribbon Hopf algebra. For all n and all N >= 3, a finite collection of finite dimensional representations of U_q^(N)(osp(1|2n)) is defined. Each such representation of U_q^(N)(osp(1|2n)) is labelled by an integral dominant weight belonging to the truncated dominant Weyl chamber. Properties of these representations are considered: the quantum superdimension of each representation is calculated, each representation is shown to be self-dual, and more importantly, the decomposition of the tensor product of an arbitrary number of such representations is obtained for even N. It is proved that the quotient algebra U_q^(N)(osp(1|2n)), together with the set of finite dimensional representations discussed above, form a pseudo-modular Hopf algebra when N >= 6 is twice an odd number. Using this pseudo-modular Hopf algebra, we construct a topological invariant of 3-manifolds. This invariant is shown to be different to the topological invariants of 3-manifolds arising from quantum so(2n+1) at roots of unity.

Spin torque oscillator neuroanalog of von Neumann's microwave computer.

PubMed

Hoppensteadt, Frank

2015-10-01

Frequency and phase of neural activity play important roles in the behaving brain. The emerging understanding of these roles has been informed by the design of analog devices that have been important to neuroscience, among them the neuroanalog computer developed by O. Schmitt and A. Hodgkin in the 1930s. Later J. von Neumann, in a search for high performance computing using microwaves, invented a logic machine based on crystal diodes that can perform logic functions including binary arithmetic. Described here is an embodiment of his machine using nano-magnetics. Electrical currents through point contacts on a ferromagnetic thin film can create oscillations in the magnetization of the film. Under natural conditions these properties of a ferromagnetic thin film may be described by a nonlinear Schrödinger equation for the film's magnetization. Radiating solutions of this system are referred to as spin waves, and communication within the film may be by spin waves or by directed graphs of electrical connections. It is shown here how to formulate a STO logic machine, and by computer simulation how this machine can perform several computations simultaneously using multiplexing of inputs, that this system can evaluate iterated logic functions, and that spin waves may communicate frequency, phase and binary information. Neural tissue and the Schmitt-Hodgkin, von Neumann and STO devices share a common bifurcation structure, although these systems operate on vastly different space and time scales; namely, all may exhibit Andronov-Hopf bifurcations. This suggests that neural circuits may be capable of the computational functionality as described by von Neumann. Copyright © 2015 Elsevier Ireland Ltd. All rights reserved.

Subsystem design in aircraft power distribution systems using optimization

NASA Astrophysics Data System (ADS)

Chandrasekaran, Sriram

2000-10-01

The research reported in this dissertation focuses on the development of optimization tools for the design of subsystems in a modern aircraft power distribution system. The baseline power distribution system is built around a 270V DC bus. One of the distinguishing features of this power distribution system is the presence of regenerative power from the electrically driven flight control actuators and structurally integrated smart actuators back to the DC bus. The key electrical components of the power distribution system are bidirectional switching power converters, which convert, control and condition electrical power between the sources and the loads. The dissertation is divided into three parts. Part I deals with the formulation of an optimization problem for a sample system consisting of a regulated DC-DC buck converter preceded by an input filter. The individual subsystems are optimized first followed by the integrated optimization of the sample system. It is shown that the integrated optimization provides better results than that obtained by integrating the individually optimized systems. Part II presents a detailed study of piezoelectric actuators. This study includes modeling, optimization of the drive amplifier and the development of a current control law for piezoelectric actuators coupled to a simple mechanical structure. Linear and nonlinear methods to study subsystem interaction and stability are studied in Part III. A multivariable impedance ratio criterion applicable to three phase systems is proposed. Bifurcation methods are used to obtain global stability characteristics of interconnected systems. The application of a nonlinear design methodology, widely used in power systems, to incrementally improve the robustness of a system to Hopf bifurcation instability is discussed.

Global Nonlinear Analysis of Piezoelectric Energy Harvesting from Ambient and Aeroelastic Vibrations

NASA Astrophysics Data System (ADS)

Abdelkefi, Abdessattar

Converting vibrations to a usable form of energy has been the topic of many recent investigations. The ultimate goal is to convert ambient or aeroelastic vibrations to operate low-power consumption devices, such as microelectromechanical systems, heath monitoring sensors, wireless sensors or replacing small batteries that have a finite life span or would require hard and expensive maintenance. The transduction mechanisms used for transforming vibrations to electric power include: electromagnetic, electrostatic, and piezoelectric mechanisms. Because it can be used to harvest energy over a wide range of frequencies and because of its ease of application, the piezoelectric option has attracted significant interest. In this work, we investigate the performance of different types of piezoelectric energy harvesters. The objective is to design and enhance the performance of these harvesters. To this end, distributed-parameter and phenomenological models of these harvesters are developed. Global analysis of these models is then performed using modern methods of nonlinear dynamics. In the first part of this Dissertation, global nonlinear distributed-parameter models for piezoelectric energy harvesters under direct and parametric excitations are developed. The method of multiple scales is then used to derive nonlinear forms of the governing equations and associated boundary conditions, which are used to evaluate their performance and determine the effects of the nonlinear piezoelectric coefficients on their behavior in terms of softening or hardening. In the second part, we assess the influence of the linear and nonlinear parameters on the dynamic behavior of a wing-based piezoaeroelastic energy harvester. The system is composed of a rigid airfoil that is constrained to pitch and plunge and supported by linear and nonlinear torsional and flexural springs with a piezoelectric coupling attached to the plunge degree of freedom. Linear analysis is performed to determine the effects of the linear spring coefficients and electrical load resistance on the flutter speed. Then, the normal form of the Hopf bifurcation ( utter) is derived to characterize the type of instability and determine the effects of the aerodynamic nonlinearities and the nonlinear coefficients of the springs on the system's stability near the bifurcation. This is useful to characterize the effects of different parameters on the system's output and ensure that subcritical or "catastrophic" bifurcation does not take place. Both linear and nonlinear analyses are then used to design and enhance the performance of these harvesters. In the last part, the concept of energy harvesting from vortex-induced vibrations of a circular cylinder is investigated. The power levels that can be generated from these vibrations and the variations of these levels with the freestream velocity are determined. A mathematical model that accounts for the coupled lift force, cylinder motion and generated voltage is presented. Linear analysis of the electromechanical model is performed to determine the effects of the electrical load resistance on the natural frequency of the rigid cylinder and the onset of the synchronization region. The impacts of the nonlinearities on the cylinder's response and energy harvesting are then investigated.

Hopf solitons in the Nicole model

DOE Office of Scientific and Technical Information (OSTI.GOV)

Gillard, Mike; Sutcliffe, Paul

2010-12-15

The Nicole model is a conformal field theory in a three-dimensional space. It has topological soliton solutions classified by the integer-valued Hopf charge, and all currently known solitons are axially symmetric. A volume-preserving flow is used to construct soliton solutions numerically for all Hopf charges from 1 to 8. It is found that the known axially symmetric solutions are unstable for Hopf charges greater than 2 and new lower energy solutions are obtained that include knots and links. A comparison with the Skyrme-Faddeev model suggests many universal features, though there are some differences in the link types obtained in themore » two theories.« less

Cross-diffusion-induced subharmonic spatial resonances in a predator-prey system

NASA Astrophysics Data System (ADS)

Gambino, G.; Lombardo, M. C.; Sammartino, M.

2018-01-01

In this paper we investigate the complex dynamics originated by a cross-diffusion-induced subharmonic destabilization of the fundamental subcritical Turing mode in a predator-prey reaction-diffusion system. The model we consider consists of a two-species Lotka-Volterra system with linear diffusion and a nonlinear cross-diffusion term in the predator equation. The taxis term in the search strategy of the predator is responsible for the onset of complex dynamics. In fact, our model does not exhibit any Hopf or wave instability, and on the basis of the linear analysis one should only expect stationary patterns; nevertheless, the presence of the nonlinear cross-diffusion term is able to induce a secondary instability: due to a subharmonic spatial resonance, the stationary primary branch bifurcates to an out-of-phase oscillating solution. Noticeably, the strong resonance between the harmonic and the subharmonic is able to generate the oscillating pattern albeit the subharmonic is below criticality. We show that, as the control parameter is varied, the oscillating solution (sub T mode) can undergo a sequence of secondary instabilities, generating a transition toward chaotic dynamics. Finally, we investigate the emergence of sub T -mode solutions on two-dimensional domains: when the fundamental mode describes a square pattern, subharmonic resonance originates oscillating square patterns. In the case of subcritical Turing hexagon solutions, the internal interactions with a subharmonic mode are able to generate the so-called "twinkling-eyes" pattern.

Refuge-mediated predator-prey dynamics and biomass pyramids.

PubMed

Wang, Hao; Thanarajah, Silogini; Gaudreau, Philippe

2018-04-01

Refuge can greatly influence predator-prey dynamics by movements between the interior and the exterior of a refuge. The presence of refuge for prey decreases predation risk and can have important impacts on the sustainability of a predator-prey system. The principal purpose of this paper is to formulate and analyze a refuge-mediated predator-prey model when the refuge is available to protect a portion of prey from predation. We study the effect of the refuge size on the biomass ratio and extend our refuge model to incorporate fishing and predator migration separately. Our study suggests that decreasing the refuge size, increasing the predator fishing, and increasing the predator emigration stabilizes the system. Here, we investigate the dependence of Hopf bifurcation on refuge size in the presence of fishing or predator migration. Moreover, we discuss their effects on the biomass pyramid and establish a condition for the emergence of an inverted biomass pyramid. We perform numerical test and sensitivity analysis to check the robustness of our results and the relative importance of all parameters. We find that high fishing pressure may destroy the inverted biomass pyramid and thus decrease the resilience of reef ecosystems. In addition, increasing the emigration rate or decreasing the immigration rate decreases the predator-prey biomass ratio. An inverted biomass pyramid can occur in the presence of a stable limit cycle. Copyright © 2017 Elsevier Inc. All rights reserved.

Onset of chaos in helical vortex breakdown at low Reynolds number

NASA Astrophysics Data System (ADS)

Pasche, S.; Avellan, F.; Gallaire, F.

2018-06-01

The nonlinear dynamics of a swirling wake flow stemming from a Graboswksi-Berger vortex [Grabowski and Berger, J. Fluid Mech. 75, 525 (1976), 10.1017/S0022112076000360] in a semi-infinite domain is addressed at low Reynolds numbers for a fixed swirl number S =1.095 , defined as the ratio between the characteristic tangential velocity and the centerline axial velocity. In this system, only pure hydrodynamic instabilities develop and interact through the quadratic nonlinearities of the Navier-Stokes equations. Such interactions lead to the onset of chaos at a Reynolds value of Re=220 . This chaotic state is reached by following a Ruelle-Takens-Newhouse scenario, which is initiated by a Hopf bifurcation (the spiral vortex breakdown) as the Reynolds number increases. At larger Reynolds value, a frequency synchronization regime appears followed by a chaotic state again. This scenario is corroborated by nonlinear time series analyses. Stability analysis around the time-average flow and temporal-azimuthal Fourier decomposition of the nonlinear flow distributions both identify successfully the developing vortices and provide deeper insight into the development of the flow patterns leading to this route to chaos. Three single-helical vortices are involved: the primary spiral associated with the spiral vortex breakdown, a downstream spiral, and a near-wake spiral. As the Reynolds number increases, the frequencies of these vortices become closer, increasing their interactions by nonlinearity to eventually generate a strong chaotic axisymmetric oscillation.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Agarwala, Susama; Delaney, Colleen

This paper defines a generalization of the Connes-Moscovici Hopf algebra, H(1), that contains the entire Hopf algebra of rooted trees. A relationship between the former, a much studied object in non-commutative geometry, and the latter, a much studied object in perturbative quantum field theory, has been established by Connes and Kreimer. The results of this paper open the door to study the cohomology of the Hopf algebra of rooted trees.

Fractional flow reserve and coronary bifurcation anatomy: a novel quantitative model to assess and report the stenosis severity of bifurcation lesions.

PubMed

Tu, Shengxian; Echavarria-Pinto, Mauro; von Birgelen, Clemens; Holm, Niels R; Pyxaras, Stylianos A; Kumsars, Indulis; Lam, Ming Kai; Valkenburg, Ilona; Toth, Gabor G; Li, Yingguang; Escaned, Javier; Wijns, William; Reiber, Johan H C

2015-04-20

The aim of this study was to develop a new model for assessment of stenosis severity in a bifurcation lesion including its core. The diagnostic performance of this model, powered by 3-dimensional quantitative coronary angiography to predict the functional significance of obstructive bifurcation stenoses, was evaluated using fractional flow reserve (FFR) as the reference standard. Development of advanced quantitative models might help to establish a relationship between bifurcation anatomy and FFR. Patients who had undergone coronary angiography and interventions in 5 European cardiology centers were randomly selected and analyzed. Different bifurcation fractal laws, including Murray, Finet, and HK laws, were implemented in the bifurcation model, resulting in different degrees of stenosis severity. A total of 78 bifurcation lesions in 73 patients were analyzed. In 51 (65%) bifurcations, FFR was measured in the main vessel. A total of 34 (43.6%) interrogated vessels had an FFR≤0.80. Correlation between FFR and diameter stenosis was poor by conventional straight analysis (ρ=-0.23, p<0.001) but significantly improved by bifurcation analyses: the highest by the HK law (ρ=-0.50, p<0.001), followed by the Finet law (ρ=-0.49, p<0.001), and the Murray law (ρ=-0.41, p<0.001). The area under the receiver-operating characteristics curve for predicting FFR≤0.80 was significantly higher by bifurcation analysis compared with straight analysis: 0.72 (95% confidence interval: 0.61 to 0.82) versus 0.60 (95% confidence interval: 0.49 to 0.71; p=0.001). Applying a threshold of ≥50% diameter stenosis, as assessed by the bifurcation model, to predict FFR≤0.80 resulted in 23 true positives, 27 true negatives, 17 false positives, and 11 false negatives. The new bifurcation model provides a comprehensive assessment of bifurcation anatomy. Compared with straight analysis, identification of lesions with preserved FFR values in obstructive bifurcation stenoses was improved. Nevertheless, accuracy was limited by using solely anatomical parameters. Copyright © 2015 American College of Cardiology Foundation. Published by Elsevier Inc. All rights reserved.

Normal forms of Hopf-zero singularity

NASA Astrophysics Data System (ADS)

Gazor, Majid; Mokhtari, Fahimeh

2015-01-01

The Lie algebra generated by Hopf-zero classical normal forms is decomposed into two versal Lie subalgebras. Some dynamical properties for each subalgebra are described; one is the set of all volume-preserving conservative systems while the other is the maximal Lie algebra of nonconservative systems. This introduces a unique conservative-nonconservative decomposition for the normal form systems. There exists a Lie-subalgebra that is Lie-isomorphic to a large family of vector fields with Bogdanov-Takens singularity. This gives rise to a conclusion that the local dynamics of formal Hopf-zero singularities is well-understood by the study of Bogdanov-Takens singularities. Despite this, the normal form computations of Bogdanov-Takens and Hopf-zero singularities are independent. Thus, by assuming a quadratic nonzero condition, complete results on the simplest Hopf-zero normal forms are obtained in terms of the conservative-nonconservative decomposition. Some practical formulas are derived and the results implemented using Maple. The method has been applied on the Rössler and Kuramoto-Sivashinsky equations to demonstrate the applicability of our results.

Spirals in a reaction-diffusion system: Dependence of wave dynamics on excitability.

PubMed

Mahanta, Dhriti; Das, Nirmali Prabha; Dutta, Sumana

2018-02-01

A detailed study of the effects of excitability of the Belousov-Zhabotinsky (BZ) reaction on spiral wave properties has been carried out. Using the Oregonator model, we explore the various regimes of wave activity, from sustained oscillations to wave damping, as the system undergoes a Hopf bifurcation, that is achieved by varying the excitability parameter, ε. We also discover a short range of parameter values where random oscillations are observed. With an increase in the value of ε, the frequency of the wave decreases exponentially, as the dimension of the spiral core expands. These numerical results are confirmed by carrying out experiments in thin layers of the BZ system, where the excitability is changed by varying the concentrations of the reactant species. Effect of reactant concentrations on wave properties like time period and wavelength are also explored in detail. Drifting and meandering spirals are found in the parameter space under investigation, with the excitability affecting the tip trajectory in a way predicted by the numerical studies. This study acts as a quantitative evidence of the relationship between the excitability parameter, ε, and the substrate concentrations.

An intermittency route to global instability in low-density jets

NASA Astrophysics Data System (ADS)

Murugesan, Meenatchidevi; Zhu, Yuanhang; Li, Larry K. B.

2017-11-01

Above a critical Reynolds number (Re), a low-density jet can become globally unstable, transitioning from a steady state (i.e. a fixed point) to a self-excited oscillatory state (i.e. a limit cycle) via a Hopf bifurcation. In this experimental study, we show that this transition can sometimes involve intermittency. When Re is just slightly above the critical point, intermittent bursts of high-amplitude periodic oscillations emerge amidst a background of low-amplitude aperiodic fluctuations. As Re increases further, these intermittent bursts persist longer in time until they dominate the overall dynamics, causing the jet to transition fully to a periodic limit cycle. We identify this as Type-II Pomeau-Manneville intermittency by quantifying the statistical distribution of the duration of the aperiodic fluctuations at the onset of intermittency. This study shows that the transition to global instability in low-density jets is not always abrupt but can involve an intermediate state with characteristics of both the initial fixed point and the final limit cycle. This work was supported by the Research Grants Council of Hong Kong (Project No. 16235716 and 26202815).

Neural dynamics in reconfigurable silicon.

PubMed

Basu, A; Ramakrishnan, S; Petre, C; Koziol, S; Brink, S; Hasler, P E

2010-10-01

A neuromorphic analog chip is presented that is capable of implementing massively parallel neural computations while retaining the programmability of digital systems. We show measurements from neurons with Hopf bifurcations and integrate and fire neurons, excitatory and inhibitory synapses, passive dendrite cables, coupled spiking neurons, and central pattern generators implemented on the chip. This chip provides a platform for not only simulating detailed neuron dynamics but also uses the same to interface with actual cells in applications such as a dynamic clamp. There are 28 computational analog blocks (CAB), each consisting of ion channels with tunable parameters, synapses, winner-take-all elements, current sources, transconductance amplifiers, and capacitors. There are four other CABs which have programmable bias generators. The programmability is achieved using floating gate transistors with on-chip programming control. The switch matrix for interconnecting the components in CABs also consists of floating-gate transistors. Emphasis is placed on replicating the detailed dynamics of computational neural models. Massive computational area efficiency is obtained by using the reconfigurable interconnect as synaptic weights, resulting in more than 50 000 possible 9-b accurate synapses in 9 mm(2).

Spirals in a reaction-diffusion system: Dependence of wave dynamics on excitability

NASA Astrophysics Data System (ADS)

Mahanta, Dhriti; Das, Nirmali Prabha; Dutta, Sumana

2018-02-01

A detailed study of the effects of excitability of the Belousov-Zhabotinsky (BZ) reaction on spiral wave properties has been carried out. Using the Oregonator model, we explore the various regimes of wave activity, from sustained oscillations to wave damping, as the system undergoes a Hopf bifurcation, that is achieved by varying the excitability parameter, ɛ . We also discover a short range of parameter values where random oscillations are observed. With an increase in the value of ɛ , the frequency of the wave decreases exponentially, as the dimension of the spiral core expands. These numerical results are confirmed by carrying out experiments in thin layers of the BZ system, where the excitability is changed by varying the concentrations of the reactant species. Effect of reactant concentrations on wave properties like time period and wavelength are also explored in detail. Drifting and meandering spirals are found in the parameter space under investigation, with the excitability affecting the tip trajectory in a way predicted by the numerical studies. This study acts as a quantitative evidence of the relationship between the excitability parameter, ɛ , and the substrate concentrations.

Translational Symmetry-Breaking for Spiral Waves

NASA Astrophysics Data System (ADS)

LeBlanc, V. G.; Wulff, C.

2000-10-01

Spiral waves are observed in numerous physical situations, ranging from Belousov-Zhabotinsky (BZ) chemical reactions, to cardiac tissue, to slime-mold aggregates. Mathematical models with Euclidean symmetry have recently been developed to describe the dynamic behavior (for example, meandering) of spiral waves in excitable media. However, no physical experiment is ever infinite in spatial extent, so the Euclidean symmetry is only approximate. Experiments on spiral waves show that inhomogeneities can anchor spirals and that boundary effects (for example, boundary drifting) become very important when the size of the spiral core is comparable to the size of the reacting medium. Spiral anchoring and boundary drifting cannot be explained by the Euclidean model alone. In this paper, we investigate the effects on spiral wave dynamics of breaking the translation symmetry while keeping the rotation symmetry. This is accomplished by introducing a small perturbation in the five-dimensional center bundle equations (describing Hopf bifurcation from one-armed spiral waves) which is SO(2)-equivariant but not equivariant under translations. We then study the effects of this perturbation on rigid spiral rotation, on quasi-periodic meandering and on drifting.

An oscillating dynamic model of collective cells in a monolayer

NASA Astrophysics Data System (ADS)

Lin, Shao-Zhen; Xue, Shi-Lei; Li, Bo; Feng, Xi-Qiao

2018-03-01

Periodic oscillations of collective cells occur in the morphogenesis and organogenesis of various tissues and organs. In this paper, an oscillating cytodynamic model is presented by integrating the chemomechanical interplay between the RhoA effector signaling pathway and cell deformation. We show that both an isolated cell and a cell aggregate can undergo spontaneous oscillations as a result of Hopf bifurcation, upon which the system evolves into a limit cycle of chemomechanical oscillations. The dynamic characteristics are tailored by the mechanical properties of cells (e.g., elasticity, contractility, and intercellular tension) and the chemical reactions involved in the RhoA effector signaling pathway. External forces are found to modulate the oscillation intensity of collective cells in the monolayer and to polarize their oscillations along the direction of external tension. The proposed cytodynamic model can recapitulate the prominent features of cell oscillations observed in a variety of experiments, including both isolated cells (e.g., spreading mouse embryonic fibroblasts, migrating amoeboid cells, and suspending 3T3 fibroblasts) and multicellular systems (e.g., Drosophila embryogenesis and oogenesis).

Oscillatory dynamics in rock-paper-scissors games with mutations.

PubMed

Mobilia, Mauro

2010-05-07

We study the oscillatory dynamics in the generic three-species rock-paper-scissors games with mutations. In the mean-field limit, different behaviors are found: (a) for high mutation rate, there is a stable interior fixed point with coexistence of all species; (b) for low mutation rates, there is a region of the parameter space characterized by a limit cycle resulting from a Hopf bifurcation; (c) in the absence of mutations, there is a region where heteroclinic cycles yield oscillations of large amplitude (not robust against noise). After a discussion on the main properties of the mean-field dynamics, we investigate the stochastic version of the model within an individual-based formulation. Demographic fluctuations are therefore naturally accounted and their effects are studied using a diffusion theory complemented by numerical simulations. It is thus shown that persistent erratic oscillations (quasi-cycles) of large amplitude emerge from a noise-induced resonance phenomenon. We also analytically and numerically compute the average escape time necessary to reach a (quasi-)cycle on which the system oscillates at a given amplitude. Copyright (c) 2010 Elsevier Ltd. All rights reserved.

A Mathematical Model of Anthrax Transmission in Animal Populations.

PubMed

Saad-Roy, C M; van den Driessche, P; Yakubu, Abdul-Aziz

2017-02-01

A general mathematical model of anthrax (caused by Bacillus anthracis) transmission is formulated that includes live animals, infected carcasses and spores in the environment. The basic reproduction number [Formula: see text] is calculated, and existence of a unique endemic equilibrium is established for [Formula: see text] above the threshold value 1. Using data from the literature, elasticity indices for [Formula: see text] and type reproduction numbers are computed to quantify anthrax control measures. Including only herbivorous animals, anthrax is eradicated if [Formula: see text]. For these animals, oscillatory solutions arising from Hopf bifurcations are numerically shown to exist for certain parameter values with [Formula: see text] and to have periodicity as observed from anthrax data. Including carnivores and assuming no disease-related death, anthrax again goes extinct below the threshold. Local stability of the endemic equilibrium is established above the threshold; thus, periodic solutions are not possible for these populations. It is shown numerically that oscillations in spore growth may drive oscillations in animal populations; however, the total number of infected animals remains about the same as with constant spore growth.

Quantum walks, deformed relativity and Hopf algebra symmetries

PubMed Central

2016-01-01

We show how the Weyl quantum walk derived from principles in D'Ariano & Perinotti (D'Ariano & Perinotti 2014 Phys. Rev. A 90, 062106. (doi:10.1103/PhysRevA.90.062106)), enjoying a nonlinear Lorentz symmetry of dynamics, allows one to introduce Hopf algebras for position and momentum of the emerging particle. We focus on two special models of Hopf algebras–the usual Poincaré and the κ-Poincaré algebras. PMID:27091171

Quantum walks, deformed relativity and Hopf algebra symmetries.

PubMed

Bisio, Alessandro; D'Ariano, Giacomo Mauro; Perinotti, Paolo

2016-05-28

We show how the Weyl quantum walk derived from principles in D'Ariano & Perinotti (D'Ariano & Perinotti 2014Phys. Rev. A90, 062106. (doi:10.1103/PhysRevA.90.062106)), enjoying a nonlinear Lorentz symmetry of dynamics, allows one to introduce Hopf algebras for position and momentum of the emerging particle. We focus on two special models of Hopf algebras-the usual Poincaré and theκ-Poincaré algebras. © 2016 The Author(s).

Bifurcation and Stability Analysis of the Equilibrium States in Thermodynamic Systems in a Small Vicinity of the Equilibrium Values of Parameters

NASA Astrophysics Data System (ADS)

Barsuk, Alexandr A.; Paladi, Florentin

2018-04-01

The dynamic behavior of thermodynamic system, described by one order parameter and one control parameter, in a small neighborhood of ordinary and bifurcation equilibrium values of the system parameters is studied. Using the general methods of investigating the branching (bifurcations) of solutions for nonlinear equations, we performed an exhaustive analysis of the order parameter dependences on the control parameter in a small vicinity of the equilibrium values of parameters, including the stability analysis of the equilibrium states, and the asymptotic behavior of the order parameter dependences on the control parameter (bifurcation diagrams). The peculiarities of the transition to an unstable state of the system are discussed, and the estimates of the transition time to the unstable state in the neighborhood of ordinary and bifurcation equilibrium values of parameters are given. The influence of an external field on the dynamic behavior of thermodynamic system is analyzed, and the peculiarities of the system dynamic behavior are discussed near the ordinary and bifurcation equilibrium values of parameters in the presence of external field. The dynamic process of magnetization of a ferromagnet is discussed by using the general methods of bifurcation and stability analysis presented in the paper.

Control of clustered action potential firing in a mathematical model of entorhinal cortex stellate cells.

PubMed

Tait, Luke; Wedgwood, Kyle; Tsaneva-Atanasova, Krasimira; Brown, Jon T; Goodfellow, Marc

2018-07-14

The entorhinal cortex is a crucial component of our memory and spatial navigation systems and is one of the first areas to be affected in dementias featuring tau pathology, such as Alzheimer's disease and frontotemporal dementia. Electrophysiological recordings from principle cells of medial entorhinal cortex (layer II stellate cells, mEC-SCs) demonstrate a number of key identifying properties including subthreshold oscillations in the theta (4-12 Hz) range and clustered action potential firing. These single cell properties are correlated with network activity such as grid firing and coupling between theta and gamma rhythms, suggesting they are important for spatial memory. As such, experimental models of dementia have revealed disruption of organised dorsoventral gradients in clustered action potential firing. To better understand the mechanisms underpinning these different dynamics, we study a conductance based model of mEC-SCs. We demonstrate that the model, driven by extrinsic noise, can capture quantitative differences in clustered action potential firing patterns recorded from experimental models of tau pathology and healthy animals. The differential equation formulation of our model allows us to perform numerical bifurcation analyses in order to uncover the dynamic mechanisms underlying these patterns. We show that clustered dynamics can be understood as subcritical Hopf/homoclinic bursting in a fast-slow system where the slow sub-system is governed by activation of the persistent sodium current and inactivation of the slow A-type potassium current. In the full system, we demonstrate that clustered firing arises via flip bifurcations as conductance parameters are varied. Our model analyses confirm the experimentally suggested hypothesis that the breakdown of clustered dynamics in disease occurs via increases in AHP conductance. Copyright © 2018 The Authors. Published by Elsevier Ltd.. All rights reserved.

Numerical Simulation of Unsteady Blood Flow through Capillary Networks.

PubMed

Davis, J M; Pozrikidis, C

2011-08-01

A numerical method is implemented for computing unsteady blood flow through a branching capillary network. The evolution of the discharge hematocrit along each capillary segment is computed by integrating in time a one-dimensional convection equation using a finite-difference method. The convection velocity is determined by the local and instantaneous effective capillary blood viscosity, while the tube to discharge hematocrit ratio is deduced from available correlations. Boundary conditions for the discharge hematocrit at divergent bifurcations arise from the partitioning law proposed by Klitzman and Johnson involving a dimensionless exponent, q≥1. When q=1, the cells are partitioned in proportion to the flow rate; as q tends to infinity, the cells are channeled into the branch with the highest flow rate. Simulations are performed for a tree-like, perfectly symmetric or randomly perturbed capillary network with m generations. When the tree involves more than a few generations, a supercritical Hopf bifurcation occurs at a critical value of q, yielding spontaneous self-sustained oscillations in the absence of external forcing. A phase diagram in the m-q plane is presented to establish conditions for unsteady flow, and the effect of various geometrical and physical parameters is examined. For a given network tree order, m, oscillations can be induced for a sufficiently high value of q by increasing the apparent intrinsic viscosity, decreasing the ratio of the vessel diameter from one generation to the next, or by decreasing the diameter of the terminal vessels. With other parameters fixed, oscillations are inhibited by increasing m. The results of the continuum model are in excellent agreement with the predictions of a discrete model where the motion of individual cells is followed from inlet to outlet.

The Hindmarsh-Rose neuron model: bifurcation analysis and piecewise-linear approximations.

PubMed

Storace, Marco; Linaro, Daniele; de Lange, Enno

2008-09-01

This paper provides a global picture of the bifurcation scenario of the Hindmarsh-Rose model. A combination between simulations and numerical continuations is used to unfold the complex bifurcation structure. The bifurcation analysis is carried out by varying two bifurcation parameters and evidence is given that the structure that is found is universal and appears for all combinations of bifurcation parameters. The information about the organizing principles and bifurcation diagrams are then used to compare the dynamics of the model with that of a piecewise-linear approximation, customized for circuit implementation. A good match between the dynamical behaviors of the models is found. These results can be used both to design a circuit implementation of the Hindmarsh-Rose model mimicking the diversity of neural response and as guidelines to predict the behavior of the model as well as its circuit implementation as a function of parameters. (c) 2008 American Institute of Physics.

Polarization of light and hopf fibration

NASA Astrophysics Data System (ADS)

Jurčo, B.

1987-09-01

A set of polarization states of quasi-monochromatic light is described geometrically in terms of the Hopf fibration. Several associated alternative polarization parametrizations are given explicitly, including the Stokes parameters.

Bayesian sensitivity analysis of bifurcating nonlinear models

NASA Astrophysics Data System (ADS)

Becker, W.; Worden, K.; Rowson, J.

2013-01-01

Sensitivity analysis allows one to investigate how changes in input parameters to a system affect the output. When computational expense is a concern, metamodels such as Gaussian processes can offer considerable computational savings over Monte Carlo methods, albeit at the expense of introducing a data modelling problem. In particular, Gaussian processes assume a smooth, non-bifurcating response surface. This work highlights a recent extension to Gaussian processes which uses a decision tree to partition the input space into homogeneous regions, and then fits separate Gaussian processes to each region. In this way, bifurcations can be modelled at region boundaries and different regions can have different covariance properties. To test this method, both the treed and standard methods were applied to the bifurcating response of a Duffing oscillator and a bifurcating FE model of a heart valve. It was found that the treed Gaussian process provides a practical way of performing uncertainty and sensitivity analysis on large, potentially-bifurcating models, which cannot be dealt with by using a single GP, although an open problem remains how to manage bifurcation boundaries that are not parallel to coordinate axes.

Improved nonlinear plasmonic slot waveguide: a full study

NASA Astrophysics Data System (ADS)

Elsawy, Mahmoud M. R.; Nazabal, Virginie; Chauvet, Mathieu; Renversez, Gilles

2016-04-01

We present a full study of an improved nonlinear plasmonic slot waveguides (NPSWs) in which buffer linear dielectric layers are added between the Kerr type nonlinear dielectric core and the two semi-infinite metal regions. Our approach computes the stationary solutions using the fixed power algorithm, in which for a given structure the wave power is an input parameter and the outputs are the propagation constant and the corresponding field components. For TM polarized waves, the inclusion of these supplementary layers have two consequences. First, they reduced the overall losses. Secondly, they modify the types of solutions that propagate in the NPSWs adding new profiles enlarging the possibilities offered by these nonlinear waveguides. In addition to the symmetric linear plasmonic profile obtained in the simple plasmonic structure with linear core such that its effective index is above the linear core refractive index, we obtained a new field profile which is more localized in the core with an effective index below the core linear refractive index. In the nonlinear case, if the effective index of the symmetric linear mode is above the core linear refractive index, the mode field profiles now exhibit a spatial transition from a plasmonic type profile to a solitonic type one. Our structure also provides longer propagation length due to the decrease of the losses compared to the simple nonlinear slot waveguide and exhibits, for well-chosen refractive index or thickness of the buffer layer, a spatial transition of its main modes that can be controlled by the power. We provide a full phase diagram of the TM wave operating regimes of these improved NPSWs. The stability of the main TM modes is then demonstrated numerically using the FDTD. We also demonstrate the existence of TE waves for both linear and nonlinear cases (for some configurations) in which the maximum intensity is located in the middle of the waveguide. We indicate the bifurcation of the nonlinear asymmetric TE mode from the symmetric nonlinear one through the Hopf bifurcation. This kind of bifurcation is similar to the ones already obtained in TM case for our improved structure, and also for the simple NPSWs. At high power, above the bifurcation threshold, the fundamental symmetric nonlinear TE mode moves gradually to new nonlinear mode in which the soliton peak displays two peaks in the core. The losses of the TE modes decrease with the power for all the cases. This kind of structures could be fabricated and characterized experimentally due to the realistic parameters chosen to model them.

High femoral artery bifurcation predicts contralateral high bifurcation: implications for complex percutaneous cardiovascular procedures requiring large caliber and/or dual access.

PubMed

Gupta, Vipul; Feng, Kent; Cheruvu, Pavan; Boyer, Nathan; Yeghiazarians, Yerem; Ports, Thomas A; Zimmet, Jeffrey; Shunk, Kendrick; Boyle, Andrew J

2014-09-01

Recent advances in technology have led to an increase in the use of bilateral femoral artery access and the requirement for large-bore access. Optimal access is in the common femoral artery (CFA), rather than higher (in the external iliac artery) or lower (in one of the branches of the CFA). However, there is a paucity of data in the literature about the relationship between bifurcation level of one CFA and the contralateral CFA. To define the prevalence of high bifurcation of the CFA and the relationship between bifurcation level on both sides, we performed a retrospective analysis of all patients with bilateral femoral angiography. From 4880 femoral angiograms performed at UCSF cardiac catheterization laboratory between 2005-2013, a total of 273 patients had bilateral femoral angiograms. The prevalence of low/normal, high, and very-high femoral bifurcations was 70%, 26%, and 4%, respectively, with no difference between sides. A high or very-high bifurcation significantly increased the likelihood of a high bifurcation on the contralateral side (odds ratio >3.0). Multivariable logistic regression analysis revealed age, gender, self-reported race, height, weight, and body mass index were not predictive of high or very-high bifurcations on either side. In conclusion, high femoral artery bifurcations are common and increase the likelihood of a high bifurcation of the contralateral femoral artery.

Topological transformations of Hopf solitons in chiral ferromagnets and liquid crystals.

PubMed

Tai, Jung-Shen B; Ackerman, Paul J; Smalyukh, Ivan I

2018-01-30

Liquid crystals are widely known for their facile responses to external fields, which forms a basis of the modern information display technology. However, switching of molecular alignment field configurations typically involves topologically trivial structures, although singular line and point defects often appear as short-lived transient states. Here, we demonstrate electric and magnetic switching of nonsingular solitonic structures in chiral nematic and ferromagnetic liquid crystals. These topological soliton structures are characterized by Hopf indices, integers corresponding to the numbers of times that closed-loop-like spatial regions (dubbed "preimages") of two different single orientations of rod-like molecules or magnetization are linked with each other. We show that both dielectric and ferromagnetic response of the studied material systems allow for stabilizing a host of topological solitons with different Hopf indices. The field transformations during such switching are continuous when Hopf indices remain unchanged, even when involving transformations of preimages, but discontinuous otherwise.

Numerical continuation and bifurcation analysis in aircraft design: an industrial perspective.

PubMed

Sharma, Sanjiv; Coetzee, Etienne B; Lowenberg, Mark H; Neild, Simon A; Krauskopf, Bernd

2015-09-28

Bifurcation analysis is a powerful method for studying the steady-state nonlinear dynamics of systems. Software tools exist for the numerical continuation of steady-state solutions as parameters of the system are varied. These tools make it possible to generate 'maps of solutions' in an efficient way that provide valuable insight into the overall dynamic behaviour of a system and potentially to influence the design process. While this approach has been employed in the military aircraft control community to understand the effectiveness of controllers, the use of bifurcation analysis in the wider aircraft industry is yet limited. This paper reports progress on how bifurcation analysis can play a role as part of the design process for passenger aircraft. © 2015 The Author(s).

Bifurcation analysis of a heterogeneous traffic flow model

NASA Astrophysics Data System (ADS)

Wang, Yu-Qing; Yan, Bo-Wen; Zhou, Chao-Fan; Li, Wei-Kang; Jia, Bin

2018-03-01

In this work, a heterogeneous traffic flow model coupled with the periodic boundary condition is proposed. Based on the previous models, a heterogeneous system composed of more than one kind of vehicles is considered. By bifurcation analysis, bifurcation patterns of the heterogeneous system are discussed in three situations in detail and illustrated by diagrams of bifurcation patterns. Besides, the stability analysis of the heterogeneous system is performed to test its anti-interference ability. The relationship between the number of vehicles and the stability is obtained. Furthermore, the attractor analysis is applied to investigate the nature of the heterogeneous system near its steady-state neighborhood. Phase diagrams of the process of the heterogeneous system from initial state to equilibrium state are intuitively presented.

Beer bottle whistling: a stochastic Hopf bifurcation

NASA Astrophysics Data System (ADS)

Boujo, Edouard; Bourquard, Claire; Xiong, Yuan; Noiray, Nicolas

2017-11-01

Blowing in a bottle to produce sound is a popular and yet intriguing entertainment. We reproduce experimentally the common observation that the bottle ``whistles'', i.e. produces a distinct tone, for large enough blowing velocity and over a finite interval of blowing angle. For a given set of parameters, the whistling frequency stays constant over time while the acoustic pressure amplitude fluctuates. Transverse oscillations of the shear layer in the bottle's neck are clearly identified with time-resolved particle image velocimetry (PIV) and proper orthogonal decomposition (POD). To account for these observations, we develop an analytical model of linear acoustic oscillator (the air in the bottle) subject to nonlinear stochastic forcing (the turbulent jet impacting the bottle's neck). We derive a stochastic differential equation and, from the associated Fokker-Planck equation and the measured acoustic pressure signals, we identify the model's parameters with an adjoint optimization technique. Results are further validated experimentally, and allow us to explain (i) the occurrence of whistling in terms of linear instability, and (ii) the amplitude of the limit cycle as a competition between linear growth rate, noise intensity, and nonlinear saturation. E. B. and N. N. acknowledge support by Repower and the ETH Zurich Foundation.

A Mathematical Model for the Control of Infectious Diseases: Effects of TV and Radio Advertisements

NASA Astrophysics Data System (ADS)

Misra, A. K.; Rai, Rajanish Kumar

The broadcast of awareness programs through TV and radio advertisements (ads) makes people aware and brings behavioral changes among the individuals regarding the risk of infection and its control mechanisms. In this paper, we propose and analyze a nonlinear mathematical model for the control of infectious diseases due to impact of TV and radio advertisements. It is assumed that susceptible individuals are vulnerable to infection as well as information through TV and radio ads and they contract infection via direct contact with infected individuals. In the model formulation, it is also assumed that the growth rates in cumulative number of TV and radio ads are proportional to the number of infected individuals with decreasing function of aware individuals. Further, it is assumed that awareness among susceptible individuals induces behavioral changes and they form separate aware classes, which are fully protected from infection as they use precautionary measures for their protection during the infection period. The feasibility of equilibria and their stability properties are discussed. It is shown that the augmentation in dissemination rate of awareness among susceptible individuals due to TV and radio ads may cause stability switches through Hopf-bifurcation. The analytical findings are supported through numerical simulations.

Advantage of straight walk instability in turning maneuver of multilegged locomotion: a robotics approach

PubMed Central

Aoi, Shinya; Tanaka, Takahiro; Fujiki, Soichiro; Funato, Tetsuro; Senda, Kei; Tsuchiya, Kazuo

2016-01-01

Multilegged locomotion improves the mobility of terrestrial animals and artifacts. Using many legs has advantages, such as the ability to avoid falling and to tolerate leg malfunction. However, many intrinsic degrees of freedom make the motion planning and control difficult, and many contact legs can impede the maneuverability during locomotion. The underlying mechanism for generating agile locomotion using many legs remains unclear from biological and engineering viewpoints. The present study used a centipede-like multilegged robot composed of six body segments and twelve legs. The body segments are passively connected through yaw joints with torsional springs. The dynamic stability of the robot walking in a straight line changes through a supercritical Hopf bifurcation due to the body axis flexibility. We focused on a quick turning task of the robot and quantitatively investigated the relationship between stability and maneuverability in multilegged locomotion by using a simple control strategy. Our experimental results show that the straight walk instability does help the turning maneuver. We discuss the importance and relevance of our findings for biological systems and propose a design principle for a simple control scheme to create maneuverable locomotion of multilegged robots. PMID:27444746

Transonic Shock Oscillations and Wing Flutter Calculated with an Interactive Boundary Layer Coupling Method

NASA Technical Reports Server (NTRS)

Edwards, John W.

1996-01-01

A viscous-inviscid interactive coupling method is used for the computation of unsteady transonic flows involving separation and reattachment. A lag-entrainment integral boundary layer method is used with the transonic small disturbance potential equation in the CAP-TSDV (Computational Aeroelasticity Program - Transonic Small Disturbance) code. Efficient and robust computations of steady and unsteady separated flows, including steady separation bubbles and self-excited shock-induced oscillations are presented. The buffet onset boundary for the NACA 0012 airfoil is accurately predicted and shown computationally to be a Hopf bifurcation. Shock-induced oscillations are also presented for the 18 percent circular arc airfoil. The oscillation onset boundaries and frequencies are accurately predicted, as is the experimentally observed hysteresis of the oscillations with Mach number. This latter stability boundary is identified as a jump phenomenon. Transonic wing flutter boundaries are also shown for a thin swept wing and for a typical business jet wing, illustrating viscous effects on flutter and the effect of separation onset on the wing response at flutter. Calculations for both wings show limit cycle oscillations at transonic speeds in the vicinity of minimum flutter speed indices.

Onset of nonlinearity in a stochastic model for auto-chemotactic advancing epithelia

NASA Astrophysics Data System (ADS)

Ben Amar, Martine; Bianca, Carlo

2016-09-01

We investigate the role of auto-chemotaxis in the growth and motility of an epithelium advancing on a solid substrate. In this process, cells create their own chemoattractant allowing communications among neighbours, thus leading to a signaling pathway. As known, chemotaxis provokes the onset of cellular density gradients and spatial inhomogeneities mostly at the front, a phenomenon able to predict some features revealed in in vitro experiments. A continuous model is proposed where the coupling between the cellular proliferation, the friction on the substrate and chemotaxis is investigated. According to our results, the friction and proliferation stabilize the front whereas auto-chemotaxis is a factor of destabilization. This antagonist role induces a fingering pattern with a selected wavenumber k0. However, in the planar front case, the translational invariance of the experimental set-up gives also a mode at k = 0 and the coupling between these two modes in the nonlinear regime is responsible for the onset of a Hopf-bifurcation. The time-dependent oscillations of patterns observed experimentally can be predicted simply in this continuous non-linear approach. Finally the effects of noise are also investigated below the instability threshold.

Dynamics of a Delayed HIV-1 Infection Model with Saturation Incidence Rate and CTL Immune Response

NASA Astrophysics Data System (ADS)

Guo, Ting; Liu, Haihong; Xu, Chenglin; Yan, Fang

2016-12-01

In this paper, we investigate the dynamics of a five-dimensional virus model incorporating saturation incidence rate, CTL immune response and three time delays which represent the latent period, virus production period and immune response delay, respectively. We begin this model by proving the positivity and boundedness of the solutions. Our model admits three possible equilibrium solutions, namely the infection-free equilibrium E0, the infectious equilibrium without immune response E1 and the infectious equilibrium with immune response E2. Moreover, by analyzing corresponding characteristic equations, the local stability of each of the feasible equilibria and the existence of Hopf bifurcation at the equilibrium point E2 are established, respectively. Further, by using fluctuation lemma and suitable Lyapunov functionals, it is shown that E0 is globally asymptotically stable when the basic reproductive numbers for viral infection R0 is less than unity. When the basic reproductive numbers for immune response R1 is less than unity and R0 is greater than unity, the equilibrium point E1 is globally asymptotically stable. Finally, some numerical simulations are carried out for illustrating the theoretical results.

Effects of delay and noise in a negative feedback regulatory motif

NASA Astrophysics Data System (ADS)

Palassini, Matteo; Dies, Marta

2009-03-01

The small copy number of the molecules involved in gene regulation can induce nontrivial stochastic phenomena such as noise-induced oscillations. An often neglected aspect of regulation dynamics are the delays involved in transcription and translation. Delays introduce analytical and computational complications because the dynamics is non-Markovian. We study the interplay of noise and delays in a negative feedback model of the p53 core regulatory network. Recent experiments have found pronounced oscillations in the concentrations of proteins p53 and Mdm2 in individual cells subjected to DNA damage. Similar oscillations occur in the Hes-1 and NK-kB systems, and in circadian rhythms. Several mechanisms have been proposed to explain this oscillatory behaviour, such as deterministic limit cycles, with and without delay, or noise-induced excursions in excitable models. We consider a generic delayed Master Equation incorporating the activation of Mdm2 by p53 and the Mdm2-promoted degradation of p53. In the deterministic limit and for large delays, the model shows a Hopf bifurcation. Via exact stochastic simulations, we find strong noise-induced oscillations well outside the limit-cycle region. We propose that this may be a generic mechanism for oscillations in gene regulatory systems.

Hopf-algebraic structure of combinatorial objects and differential operators

NASA Technical Reports Server (NTRS)

Grossman, Robert; Larson, Richard G.

1989-01-01

A Hopf-algebraic structure on a vector space which has as basis a family of trees is described. Some applications of this structure to combinatorics and to differential operators are surveyed. Some possible future directions for this work are indicated.

Probe Knots and Hopf Insulators with Ultracold Atoms

NASA Astrophysics Data System (ADS)

Deng, Dong-Ling; Wang, Sheng-Tao; Sun, Kai; Duan, L.-M.

2018-01-01

Knots and links are fascinating and intricate topological objects. Their influence spans from DNA and molecular chemistry to vortices in superfluid helium, defects in liquid crystals and cosmic strings in the early universe. Here we find that knotted structures also exist in a peculiar class of three-dimensional topological insulators—the Hopf insulators. In particular, we demonstrate that the momentum-space spin textures of Hopf insulators are twisted in a nontrivial way, which implies the presence of various knot and link structures. We further illustrate that the knots and nontrivial spin textures can be probed via standard time-of-flight images in cold atoms as preimage contours of spin orientations in stereographic coordinates. The extracted Hopf invariants, knots, and links are validated to be robust to typical experimental imperfections. Our work establishes the existence of knotted structures in Hopf insulators, which may have potential applications in spintronics and quantum information processing. D.L.D., S.T.W. and L.M.D. are supported by the ARL, the IARPA LogiQ program, and the AFOSR MURI program, and supported by Tsinghua University for their visits. K.S. acknowledges the support from NSF under Grant No. PHY1402971. D.L.D. is also supported by JQI-NSF-PFC and LPS-MPO-CMTC at the final stage of this paper.

A Practice-Oriented Bifurcation Analysis for Pulse Energy Converters. Part 2: An Operating Regime

NASA Astrophysics Data System (ADS)

Kolokolov, Yury; Monovskaya, Anna

The paper continues the discussion on bifurcation analysis for applications in practice-oriented solutions for pulse energy conversion systems (PEC-systems). Since a PEC-system represents a nonlinear object with a variable structure, then the description of its dynamics evolution involves bifurcation analysis conceptions. This means the necessity to resolve the conflict-of-units between the notions used to describe natural evolution (i.e. evolution of the operating process towards nonoperating processes and vice versa) and the notions used to describe a desirable artificial regime (i.e. an operating regime). We consider cause-effect relations in the following sequence: nonlinear dynamics-output signal-operating characteristics, where these characteristics include stability and performance. Then regularities of nonlinear dynamics should be translated into regularities of the output signal dynamics, and, after, into an evolutional picture of each operating characteristic. In order to make the translation without losses, we first take into account heterogeneous properties within the structures of the operating process in the parametrical (P-) and phase (X-) spaces, and analyze regularities of the operating stability and performance on the common basis by use of the modified bifurcation diagrams built in joint PX-space. Then, the correspondence between causes (degradation of the operating process stability) and effects (changes of the operating characteristics) is decomposed into three groups of abnormalities: conditionally unavoidable abnormalities (CU-abnormalities); conditionally probable abnormalities (CP-abnormalities); conditionally regular abnormalities (CR-abnormalities). Within each of these groups the evolutional homogeneity is retained. After, the resultant evolution of each operating characteristic is naturally aggregated through the superposition of cause-effect relations in accordance with each of the abnormalities. We demonstrate that the practice-oriented bifurcation analysis has fundamentally specific purposes and tools, like for the computer-based bifurcation analysis and the experimental bifurcation analysis. That is why, from our viewpoint, it seems to be a rather novel direction in the general context of bifurcation analysis conceptions. We believe that the discussion could be interesting to pioneer research intended for the design of promising systems of pulse energy conversion.

Quantum Mechanics of the Einstein-Hopf Model.

ERIC Educational Resources Information Center

Milonni, P. W.

1981-01-01

The Einstein-Hopf model for the thermodynamic equilibrium between the electromagnetic field and dipole oscillators is considered within the framework of quantum mechanics. Both the wave and particle aspects of the Einstein fluctuation formula are interpreted in terms of the fundamental absorption and emission processes. (Author/SK)

Towards Cohomology of Renormalization: Bigrading the Combinatorial Hopf Algebra of Rooted Trees

NASA Astrophysics Data System (ADS)

Broadhurst, D. J.; Kreimer, D.

The renormalization of quantum field theory twists the antipode of a noncocommutative Hopf algebra of rooted trees, decorated by an infinite set of primitive divergences. The Hopf algebra of undecorated rooted trees, ℌR, generated by a single primitive divergence, solves a universal problem in Hochschild cohomology. It has two nontrivial closed Hopf subalgebras: the cocommutative subalgebra ℌladder of pure ladder diagrams and the Connes-Moscovici noncocommutative subalgebra ℌCM of noncommutative geometry. These three Hopf algebras admit a bigrading by n, the number of nodes, and an index k that specifies the degree of primitivity. In each case, we use iterations of the relevant coproduct to compute the dimensions of subspaces with modest values of n and k and infer a simple generating procedure for the remainder. The results for ℌladder are familiar from the theory of partitions, while those for ℌCM involve novel transforms of partitions. Most beautiful is the bigrading of ℌR, the largest of the three. Thanks to Sloane's superseeker, we discovered that it saturates all possible inequalities. We prove this by using the universal Hochschild-closed one-cocycle B+, which plugs one set of divergences into another, and by generalizing the concept of natural growth beyond that entailed by the Connes-Moscovici case. We emphasize the yet greater challenge of handling the infinite set of decorations of realistic quantum field theory.

Bifurcation theory applied to aircraft motions

NASA Technical Reports Server (NTRS)

Hui, W. H.; Tobak, M.

1985-01-01

Bifurcation theory is used to analyze the nonlinear dynamic stability characteristics of single-degree-of-freedom motions of an aircraft or a flap about a trim position. The bifurcation theory analysis reveals that when the bifurcation parameter, e.g., the angle of attack, is increased beyond a critical value at which the aerodynamic damping vanishes, a new solution representing finite-amplitude periodic motion bifurcates from the previously stable steady motion. The sign of a simple criterion, cast in terms of aerodynamic properties, determines whether the bifurcating solution is stable (supercritical) or unstable (subcritical). For the pitching motion of a flap-plate airfoil flying at supersonic/hypersonic speed, and for oscillation of a flap at transonic speed, the bifurcation is subcritical, implying either that exchanges of stability between steady and periodic motion are accompanied by hysteresis phenomena, or that potentially large aperiodic departures from steady motion may develop. On the other hand, for the rolling oscillation of a slender delta wing in subsonic flight (wing rock), the bifurcation is found to be supercritical. This and the predicted amplitude of the bifurcation periodic motion are in good agreement with experiments.

Bifurcation theory applied to aircraft motions

NASA Technical Reports Server (NTRS)

Hui, W. H.; Tobak, M.

1985-01-01

The bifurcation theory is used to analyze the nonlinear dynamic stability characteristics of single-degree-of-freedom motions of an aircraft or a flap about a trim position. The bifurcation theory analysis reveals that when the bifurcation parameter, e.g., the angle of attack, is increased beyond a critical value at which the aerodynamic damping vanishes, a new solution representing finite-amplitude periodic motion bifurcates from the previously stable steady motion. The sign of a simple criterion, cast in terms of aerodynamic properties, determines whether the bifurcating solution is stable (supercritical) or unstable (critical). For the pitching motion of a flap-plate airfoil flying at supersonic/hypersonic speed, and for oscillation of a flap at transonic speed, the bifurcation is subcritical, implying either that exchanges of stability between steady and periodic motion are accompanied by hysteresis phenomena, or that potentially large aperiodic departures from steady motion may develop. On the other hand, for the rolling oscillation of a slender delta wing in subsonic flight (wing rock), the bifurcation is found to be supercritical. This and the predicted amplitude of the bifurcation periodic motion are in good agreement with the experiments.

Bifurcations and stability of standing waves in the nonlinear Schrödinger equation on the tadpole graph

NASA Astrophysics Data System (ADS)

Noja, Diego; Pelinovsky, Dmitry; Shaikhova, Gaukhar

2015-07-01

We develop a detailed analysis of edge bifurcations of standing waves in the nonlinear Schrödinger (NLS) equation on a tadpole graph (a ring attached to a semi-infinite line subject to the Kirchhoff boundary conditions at the junction). It is shown in the recent work [7] by using explicit Jacobi elliptic functions that the cubic NLS equation on a tadpole graph admits a rich structure of standing waves. Among these, there are different branches of localized waves bifurcating from the edge of the essential spectrum of an associated Schrödinger operator. We show by using a modified Lyapunov-Schmidt reduction method that the bifurcation of localized standing waves occurs for every positive power nonlinearity. We distinguish a primary branch of never vanishing standing waves bifurcating from the trivial solution and an infinite sequence of higher branches with oscillating behavior in the ring. The higher branches bifurcate from the branches of degenerate standing waves with vanishing tail outside the ring. Moreover, we analyze stability of bifurcating standing waves. Namely, we show that the primary branch is composed by orbitally stable standing waves for subcritical power nonlinearities, while all nontrivial higher branches are linearly unstable near the bifurcation point. The stability character of the degenerate branches remains inconclusive at the analytical level, whereas heuristic arguments based on analysis of embedded eigenvalues of negative Krein signatures support the conjecture of their linear instability at least near the bifurcation point. Numerical results for the cubic NLS equation show that this conjecture is valid and that the degenerate branches become spectrally stable far away from the bifurcation point.

Hopf solitons on compact manifolds

NASA Astrophysics Data System (ADS)

Ward, R. S.

2018-02-01

Hopf solitons in the Skyrme-Faddeev system on R3 typically have a complicated structure, in particular when the Hopf number Q is large. By contrast, if we work on a compact 3-manifold M, and the energy functional consists only of the Skyrme term (the strong-coupling limit), then the picture simplifies. There is a topological lower bound E ≥ Q on the energy, and the local minima of E can look simple even for large Q. The aim here is to describe and investigate some of these solutions, when M is S3, T3, or S2 × S1. In addition, we review the more elementary baby-Skyrme system, with M being S2 or T2.

Bacterial effector HopF2 interacts with AvrPto and suppresses Arabidopsis innate immunity at the plasma membrane

USDA-ARS?s Scientific Manuscript database

Plant pathogenic bacteria inject a cocktail of effector proteins into host plant cells to modulate the host immune response, thereby promoting pathogenicity. How or whether these effectors work cooperatively is largely unknown. The Pseudomonas syringae DC3000 effector HopF2 suppresses the host plan...

Normal forms for Hopf-Zero singularities with nonconservative nonlinear part

NASA Astrophysics Data System (ADS)

Gazor, Majid; Mokhtari, Fahimeh; Sanders, Jan A.

In this paper we are concerned with the simplest normal form computation of the systems x˙=2xf(x,y2+z2), y˙=z+yf(x,y2+z2), z˙=-y+zf(x,y2+z2), where f is a formal function with real coefficients and without any constant term. These are the classical normal forms of a larger family of systems with Hopf-Zero singularity. Indeed, these are defined such that this family would be a Lie subalgebra for the space of all classical normal form vector fields with Hopf-Zero singularity. The simplest normal forms and simplest orbital normal forms of this family with nonzero quadratic part are computed. We also obtain the simplest parametric normal form of any non-degenerate perturbation of this family within the Lie subalgebra. The symmetry group of the simplest normal forms is also discussed. This is a part of our results in decomposing the normal forms of Hopf-Zero singular systems into systems with a first integral and nonconservative systems.

Parameterizing Coefficients of a POD-Based Dynamical System

NASA Technical Reports Server (NTRS)

Kalb, Virginia L.

2010-01-01

A method of parameterizing the coefficients of a dynamical system based of a proper orthogonal decomposition (POD) representing the flow dynamics of a viscous fluid has been introduced. (A brief description of POD is presented in the immediately preceding article.) The present parameterization method is intended to enable construction of the dynamical system to accurately represent the temporal evolution of the flow dynamics over a range of Reynolds numbers. The need for this or a similar method arises as follows: A procedure that includes direct numerical simulation followed by POD, followed by Galerkin projection to a dynamical system has been proven to enable representation of flow dynamics by a low-dimensional model at the Reynolds number of the simulation. However, a more difficult task is to obtain models that are valid over a range of Reynolds numbers. Extrapolation of low-dimensional models by use of straightforward Reynolds-number-based parameter continuation has proven to be inadequate for successful prediction of flows. A key part of the problem of constructing a dynamical system to accurately represent the temporal evolution of the flow dynamics over a range of Reynolds numbers is the problem of understanding and providing for the variation of the coefficients of the dynamical system with the Reynolds number. Prior methods do not enable capture of temporal dynamics over ranges of Reynolds numbers in low-dimensional models, and are not even satisfactory when large numbers of modes are used. The basic idea of the present method is to solve the problem through a suitable parameterization of the coefficients of the dynamical system. The parameterization computations involve utilization of the transfer of kinetic energy between modes as a function of Reynolds number. The thus-parameterized dynamical system accurately predicts the flow dynamics and is applicable to a range of flow problems in the dynamical regime around the Hopf bifurcation. Parameter-continuation software can be used on the parameterized dynamical system to derive a bifurcation diagram that accurately predicts the temporal flow behavior.

Continuous model for the rock-scissors-paper game between bacteriocin producing bacteria.

PubMed

Neumann, Gunter; Schuster, Stefan

2007-06-01

In this work, important aspects of bacteriocin producing bacteria and their interplay are elucidated. Various attempts to model the resistant, producer and sensitive Escherichia coli strains in the so-called rock-scissors-paper (RSP) game had been made in the literature. The question arose whether there is a continuous model with a cyclic structure and admitting an oscillatory dynamics as observed in various experiments. The May-Leonard system admits a Hopf bifurcation, which is, however, degenerate and hence inadequate. The traditional differential equation model of the RSP-game cannot be applied either to the bacteriocin system because it involves positive interaction terms. In this paper, a plausible competitive Lotka-Volterra system model of the RSP game is presented and the dynamics generated by that model is analyzed. For the first time, a continuous, spatially homogeneous model that describes the competitive interaction between bacteriocin-producing, resistant and sensitive bacteria is established. The interaction terms have negative coefficients. In some experiments, for example, in mice cultures, migration seemed to be essential for the reinfection in the RSP cycle. Often statistical and spatial effects such as migration and mutation are regarded to be essential for periodicity. Our model gives rise to oscillatory dynamics in the RSP game without such effects. Here, a normal form description of the limit cycle and conditions for its stability are derived. The toxicity of the bacteriocin is used as a bifurcation parameter. Exact parameter ranges are obtained for which a stable (robust) limit cycle and a stable heteroclinic cycle exist in the three-species game. These parameters are in good accordance with the observed relations for the E. coli strains. The roles of growth rate and growth yield of the three strains are discussed. Numerical calculations show that the sensitive, which might be regarded as the weakest, can have the longest sojourn times.

Twist for Snyder space

NASA Astrophysics Data System (ADS)

Meljanac, Daniel; Meljanac, Stjepan; Mignemi, Salvatore; Pikutić, Danijel; Štrajn, Rina

2018-03-01

We construct the twist operator for the Snyder space. Our starting point is a non-associative star product related to a Hermitian realisation of the noncommutative coordinates originally introduced by Snyder. The corresponding coproduct of momenta is non-coassociative. The twist is constructed using a general definition of the star product in terms of a bi-differential operator in the Hopf algebroid approach. The result is given by a closed analytical expression. We prove that this twist reproduces the correct coproducts of the momenta and the Lorentz generators. The twisted Poincaré symmetry is described by a non-associative Hopf algebra, while the twisted Lorentz symmetry is described by the undeformed Hopf algebra. This new twist might be important in the construction of different types of field theories on Snyder space.

Analysis of spontaneous oscillations for a three-state power-stroke model.

PubMed

Washio, Takumi; Hisada, Toshiaki; Shintani, Seine A; Higuchi, Hideo

2017-02-01

Our study considers the mechanism of the spontaneous oscillations of molecular motors that are driven by the power stroke principle by applying linear stability analysis around the stationary solution. By representing the coupling equation of microscopic molecular motor dynamics and mesoscopic sarcomeric dynamics by a rank-1 updated matrix system, we derived the analytical representations of the eigenmodes of the Jacobian matrix that cause the oscillation. Based on these analytical representations, we successfully derived the essential conditions for the oscillation in terms of the rate constants of the power stroke and the reversal stroke transitions of the molecular motor. Unlike the two-state model, in which the dependence of the detachment rates on the motor coordinates or the applied forces on the motors plays a key role for the oscillation, our three-state power stroke model demonstrates that the dependence of the rate constants of the power and reversal strokes on the strains in the elastic elements in the motor molecules plays a key role, where these rate constants are rationally determined from the free energy available for the power stroke, the stiffness of the elastic element in the molecular motor, and the working stroke size. By applying the experimentally confirmed values to the free energy, the stiffness, and the working stroke size, our numerical model reproduces well the experimentally observed oscillatory behavior. Furthermore, our analysis shows that two eigenmodes with real positive eigenvalues characterize the oscillatory behavior, where the eigenmode with the larger eigenvalue indicates the transient of the system of the quick sarcomeric lengthening induced by the collective reversal strokes, and the smaller eigenvalue correlates with the speed of sarcomeric shortening, which is much slower than lengthening. Applying the perturbation analyses with primal physical parameters, we find that these two real eigenvalues occur on two branches derived from a merge point of a pair of complex-conjugate eigenvalues generated by Hopf bifurcation.

Experiment Evaluation of Bifurcation in Sands

NASA Technical Reports Server (NTRS)

Alshibi, Khalid A.; Sture, Stein

2000-01-01

The basic principles of bifurcation analysis have been established by several investigators, however several issues remain unresolved, specifically how do stress level, grain size distribution, and boundary conditions affect general bifurcation phenomena in pressure sensitive and dilatant materials. General geometrical and kinematics conditions for moving surfaces of discontinuity was derived and applied to problems of instability of solids. In 1962, the theoretical framework of bifurcation by studying the acceleration waves in elasto-plastic (J2) solids were presented. Bifurcation analysis for more specific forms of constitutive behavior was examined by studying localization in pressure-sensitive, dilatant materials, however, analyses were restricted to plane deformation states only. Bifurcation analyses were presented and applied to predict shear band formations in sand under plane strain condition. The properties of discontinuous bifurcation solutions for elastic-plastic solids under axisymmetric and plane strain loading conditions were studied. The study focused on theory, but also references and comparisons to experiments were made. The current paper includes a presentation of a summary of bifurcation analyses for biaxial and triaxial (axisymmetric) loading conditions. The Coulomb model is implemented using incremental piecewise scheme to predict the constitutive relations and shear band inclination angles. Then, a comprehensive evaluation of bifurcation phenomena is presented based on data from triaxial experiments performed under microgravity conditions aboard the Space Shuttle under very low effective confining pressure (0.05 to 1.30 kPa), in which very high peak friction angles (47 to 75 degrees) and dilatancy angles (30 to 31 degrees) were measured. The evaluation will be extended to include biaxial experiments performed on the same material under low (10 kPa) and moderate (100 kPa) confining pressures. A comparison between the behavior under biaxial and triaxial loading conditions will be presented, and related issues concerning influence of confining pressure will be discussed.

Invariants, Attractors and Bifurcation in Two Dimensional Maps with Polynomial Interaction

NASA Astrophysics Data System (ADS)

Hacinliyan, Avadis Simon; Aybar, Orhan Ozgur; Aybar, Ilknur Kusbeyzi

This work will present an extended discrete-time analysis on maps and their generalizations including iteration in order to better understand the resulting enrichment of the bifurcation properties. The standard concepts of stability analysis and bifurcation theory for maps will be used. Both iterated maps and flows are used as models for chaotic behavior. It is well known that when flows are converted to maps by discretization, the equilibrium points remain the same but a richer bifurcation scheme is observed. For example, the logistic map has a very simple behavior as a differential equation but as a map fold and period doubling bifurcations are observed. A way to gain information about the global structure of the state space of a dynamical system is investigating invariant manifolds of saddle equilibrium points. Studying the intersections of the stable and unstable manifolds are essential for understanding the structure of a dynamical system. It has been known that the Lotka-Volterra map and systems that can be reduced to it or its generalizations in special cases involving local and polynomial interactions admit invariant manifolds. Bifurcation analysis of this map and its higher iterates can be done to understand the global structure of the system and the artifacts of the discretization by comparing with the corresponding results from the differential equation on which they are based.

Allosteric regulation of phosphofructokinase controls the emergence of glycolytic oscillations in isolated yeast cells.

PubMed

Gustavsson, Anna-Karin; van Niekerk, David D; Adiels, Caroline B; Kooi, Bob; Goksör, Mattias; Snoep, Jacky L

2014-06-01

Oscillations are widely distributed in nature and synchronization of oscillators has been described at the cellular level (e.g. heart cells) and at the population level (e.g. fireflies). Yeast glycolysis is the best known oscillatory system, although it has been studied almost exclusively at the population level (i.e. limited to observations of average behaviour in synchronized cultures). We studied individual yeast cells that were positioned with optical tweezers in a microfluidic chamber to determine the precise conditions for autonomous glycolytic oscillations. Hopf bifurcation points were determined experimentally in individual cells as a function of glucose and cyanide concentrations. The experiments were analyzed in a detailed mathematical model and could be interpreted in terms of an oscillatory manifold in a three-dimensional state-space; crossing the boundaries of the manifold coincides with the onset of oscillations and positioning along the longitudinal axis of the volume sets the period. The oscillatory manifold could be approximated by allosteric control values of phosphofructokinase for ATP and AMP. The mathematical models described here have been submitted to the JWS Online Cellular Systems Modelling Database and can be accessed at http://jjj.mib.ac.uk/webMathematica/UItester.jsp?modelName=gustavsson5. [Database section added 14 May 2014 after original online publication]. © 2014 FEBS.

Size-dependent diffusion promotes the emergence of spatiotemporal patterns

NASA Astrophysics Data System (ADS)

Zhang, Lai; Thygesen, Uffe Høgsbro; Banerjee, Malay

2014-07-01

Spatiotemporal patterns, indicating the spatiotemporal variability of individual abundance, are a pronounced scenario in ecological interactions. Most of the existing models for spatiotemporal patterns treat species as homogeneous groups of individuals with average characteristics by ignoring intraspecific physiological variations at the individual level. Here we explore the impacts of size variation within species resulting from individual ontogeny, on the emergence of spatiotemporal patterns in a fully size-structured population model. We found that size dependency of animal's diffusivity greatly promotes the formation of spatiotemporal patterns, by creating regular spatiotemporal patterns out of temporal chaos. We also found that size-dependent diffusion can substitute large-amplitude base harmonics with spatiotemporal patterns with lower amplitude oscillations but with enriched harmonics. Finally, we found that the single-generation cycle is more likely to drive spatiotemporal patterns compared to predator-prey cycles, meaning that the mechanism of Hopf bifurcation might be more common than hitherto appreciated since the former cycle is more widespread than the latter in case of interacting populations. Due to the ubiquity of individual ontogeny in natural ecosystems we conclude that diffusion variability within populations is a significant driving force for the emergence of spatiotemporal patterns. Our results offer a perspective on self-organized phenomena, and pave a way to understand such phenomena in systems organized as complex ecological networks.

The interaction evolution model of mass incidents with delay in a social network

NASA Astrophysics Data System (ADS)

Huo, Liang'an; Ma, Chenyang

2017-10-01

Recent years have witnessed rapid development of information technology. Today, modern media is widely used for the purpose of spreading information rapidly and widely. In particular, through micro-blog promotions, individuals tend to express their viewpoints and spread information on the internet, which could easily lead to public opinions. Moreover, government authorities also disseminate official information to guide public opinion and eliminate any incorrect conjecture. In this paper, a dynamical model with two delays is investigated to exhibit the interaction evolution between the public and official opinion fields in network mass incidents. Based on the theory of differential equations, the interaction mechanism between two public opinion fields in a micro-blog environment is analyzed. Two delays are proposed in the model to depict the response delays of public and official opinion fields. Some stable conditions are obtained, which shows that Hopf bifurcation can occur as delays cross critical values. Further, some numerical simulations are carried out to verify theoretical results. Our model indicates that there exists a golden time for government intervention, which should be emphasized given the impact of modern media and inaccurate rumors. If the government releases official information during the golden time, mass incidents on the internet can be controlled effectively.

Order out of Randomness: Self-Organization Processes in Astrophysics

NASA Astrophysics Data System (ADS)

Aschwanden, Markus J.; Scholkmann, Felix; Béthune, William; Schmutz, Werner; Abramenko, Valentina; Cheung, Mark C. M.; Müller, Daniel; Benz, Arnold; Chernov, Guennadi; Kritsuk, Alexei G.; Scargle, Jeffrey D.; Melatos, Andrew; Wagoner, Robert V.; Trimble, Virginia; Green, William H.

2018-03-01

Self-organization is a property of dissipative nonlinear processes that are governed by a global driving force and a local positive feedback mechanism, which creates regular geometric and/or temporal patterns, and decreases the entropy locally, in contrast to random processes. Here we investigate for the first time a comprehensive number of (17) self-organization processes that operate in planetary physics, solar physics, stellar physics, galactic physics, and cosmology. Self-organizing systems create spontaneous " order out of randomness", during the evolution from an initially disordered system to an ordered quasi-stationary system, mostly by quasi-periodic limit-cycle dynamics, but also by harmonic (mechanical or gyromagnetic) resonances. The global driving force can be due to gravity, electromagnetic forces, mechanical forces (e.g., rotation or differential rotation), thermal pressure, or acceleration of nonthermal particles, while the positive feedback mechanism is often an instability, such as the magneto-rotational (Balbus-Hawley) instability, the convective (Rayleigh-Bénard) instability, turbulence, vortex attraction, magnetic reconnection, plasma condensation, or a loss-cone instability. Physical models of astrophysical self-organization processes require hydrodynamic, magneto-hydrodynamic (MHD), plasma, or N-body simulations. Analytical formulations of self-organizing systems generally involve coupled differential equations with limit-cycle solutions of the Lotka-Volterra or Hopf-bifurcation type.

The Trade-Off Mechanism in Mammalian Circadian Clock Model with Two Time Delays

NASA Astrophysics Data System (ADS)

Yan, Jie; Kang, Xiaxia; Yang, Ling

Circadian clock is an autonomous oscillator which orchestrates the daily rhythms of physiology and behaviors. This study is devoted to explore how a positive feedback loop affects the dynamics of mammalian circadian clock. We simplify an experimentally validated mathematical model in our previous work, to a nonlinear differential equation with two time delays. This simplified mathematical model incorporates the pacemaker of mammalian circadian clock, a negative primary feedback loop, and a critical positive auxiliary feedback loop, Rev-erbα/Cry1 loop. We perform analytical studies of the system. Delay-dependent conditions for the asymptotic stability of the nontrivial positive steady state of the model are investigated. We also prove the existence of Hopf bifurcation, which leads to self-sustained oscillation of mammalian circadian clock. Our theoretical analyses show that the oscillatory regime is reduced upon the participation of the delayed positive auxiliary loop. However, further simulations reveal that the auxiliary loop can enable the circadian clock gain widely adjustable amplitudes and robust period. Thus, the positive auxiliary feedback loop may provide a trade-off mechanism, to use the small loss in the robustness of oscillation in exchange for adaptable flexibility in mammalian circadian clock. The results obtained from the model may gain new insights into the dynamics of biological oscillators with interlocked feedback loops.

Coiling of elastic rods on rigid substrates

PubMed Central

Jawed, Mohammad K.; Da, Fang; Joo, Jungseock; Grinspun, Eitan; Reis, Pedro M.

2014-01-01

We investigate the deployment of a thin elastic rod onto a rigid substrate and study the resulting coiling patterns. In our approach, we combine precision model experiments, scaling analyses, and computer simulations toward developing predictive understanding of the coiling process. Both cases of deposition onto static and moving substrates are considered. We construct phase diagrams for the possible coiling patterns and characterize them as a function of the geometric and material properties of the rod, as well as the height and relative speeds of deployment. The modes selected and their characteristic length scales are found to arise from a complex interplay between gravitational, bending, and twisting energies of the rod, coupled to the geometric nonlinearities intrinsic to the large deformations. We give particular emphasis to the first sinusoidal mode of instability, which we find to be consistent with a Hopf bifurcation, and analyze the meandering wavelength and amplitude. Throughout, we systematically vary natural curvature of the rod as a control parameter, which has a qualitative and quantitative effect on the pattern formation, above a critical value that we determine. The universality conferred by the prominent role of geometry in the deformation modes of the rod suggests using the gained understanding as design guidelines, in the original applications that motivated the study. PMID:25267649

On the bistable zone of milling processes

PubMed Central

Dombovari, Zoltan; Stepan, Gabor

2015-01-01

A modal-based model of milling machine tools subjected to time-periodic nonlinear cutting forces is introduced. The model describes the phenomenon of bistability for certain cutting parameters. In engineering, these parameter domains are referred to as unsafe zones, where steady-state milling may switch to chatter for certain perturbations. In mathematical terms, these are the parameter domains where the periodic solution of the corresponding nonlinear, time-periodic delay differential equation is linearly stable, but its domain of attraction is limited due to the existence of an unstable quasi-periodic solution emerging from a secondary Hopf bifurcation. A semi-numerical method is presented to identify the borders of these bistable zones by tracking the motion of the milling tool edges as they might leave the surface of the workpiece during the cutting operation. This requires the tracking of unstable quasi-periodic solutions and the checking of their grazing to a time-periodic switching surface in the infinite-dimensional phase space. As the parameters of the linear structural behaviour of the tool/machine tool system can be obtained by means of standard modal testing, the developed numerical algorithm provides efficient support for the design of milling processes with quick estimates of those parameter domains where chatter can still appear in spite of setting the parameters into linearly stable domains. PMID:26303918

Bifurcation and Spike Adding Transition in Chay-Keizer Model

NASA Astrophysics Data System (ADS)

Lu, Bo; Liu, Shenquan; Liu, Xuanliang; Jiang, Xiaofang; Wang, Xiaohui

Electrical bursting is an activity which is universal in excitable cells such as neurons and various endocrine cells, and it encodes rich physiological information. As burst delay identifies that the signal integration has reached the threshold at which it can generate an action potential, the number of spikes in a burst may have essential physiological implications, and the transition of bursting in excitable cells is associated with the bifurcation phenomenon closely. In this paper, we focus on the transition of the spike count per burst of the pancreatic β-cells within a mathematical model and bifurcation phenomenon in the Chay-Keizer model, which is utilized to simulate the pancreatic β-cells. By the fast-slow dynamical bifurcation analysis and the bi-parameter bifurcation analysis, the local dynamics of the Chay-Keizer system around the Bogdanov-Takens bifurcation is illustrated. Then the variety of the number of spikes per burst is discussed by changing the settings of a single parameter and bi-parameter. Moreover, results on the number of spikes within a burst are summarized in ISIs (interspike intervals) sequence diagrams, maximum and minimum, and the number of spikes under bi-parameter value changes.

Bifurcation of finitely deformed thick-walled electroelastic cylindrical tubes subject to a radial electric field

NASA Astrophysics Data System (ADS)

Melnikov, Andrey; Ogden, Ray W.

2018-06-01

This paper is concerned with the bifurcation analysis of a pressurized electroelastic circular cylindrical tube with closed ends and compliant electrodes on its curved boundaries. The theory of small incremental electroelastic deformations superimposed on a finitely deformed electroelastic tube is used to determine those underlying configurations for which the superimposed deformations do not maintain the perfect cylindrical shape of the tube. First, prismatic bifurcations are examined and solutions are obtained which show that for a neo-Hookean electroelastic material prismatic modes of bifurcation become possible under inflation. This result contrasts with that for the purely elastic case for which prismatic bifurcation modes were found only for an externally pressurized tube. Second, axisymmetric bifurcations are analyzed, and results for both neo-Hookean and Mooney-Rivlin electroelastic energy functions are obtained. The solutions show that in the presence of a moderate electric field the electroelastic tube becomes more susceptible to bifurcation, i.e., for fixed values of the axial stretch axisymmetric bifurcations become possible at lower values of the circumferential stretches than in the corresponding problems in the absence of an electric field. As the magnitude of the electric field increases, however, the possibility of bifurcation under internal pressure becomes restricted to a limited range of values of the axial stretch and is phased out completely for sufficiently large electric fields. Then, axisymmetric bifurcation is only possible under external pressure.

A bifurcation study to guide the design of a landing gear with a combined uplock/downlock mechanism.

PubMed

Knowles, James A C; Lowenberg, Mark H; Neild, Simon A; Krauskopf, Bernd

2014-12-08

This paper discusses the insights that a bifurcation analysis can provide when designing mechanisms. A model, in the form of a set of coupled steady-state equations, can be derived to describe the mechanism. Solutions to this model can be traced through the mechanism's state versus parameter space via numerical continuation, under the simultaneous variation of one or more parameters. With this approach, crucial features in the response surface, such as bifurcation points, can be identified. By numerically continuing these points in the appropriate parameter space, the resulting bifurcation diagram can be used to guide parameter selection and optimization. In this paper, we demonstrate the potential of this technique by considering an aircraft nose landing gear, with a novel locking strategy that uses a combined uplock/downlock mechanism. The landing gear is locked when in the retracted or deployed states. Transitions between these locked states and the unlocked state (where the landing gear is a mechanism) are shown to depend upon the positions of two fold point bifurcations. By performing a two-parameter continuation, the critical points are traced to identify operational boundaries. Following the variation of the fold points through parameter space, a minimum spring stiffness is identified that enables the landing gear to be locked in the retracted state. The bifurcation analysis also shows that the unlocking of a retracted landing gear should use an unlock force measure, rather than a position indicator, to de-couple the effects of the retraction and locking actuators. Overall, the study demonstrates that bifurcation analysis can enhance the understanding of the influence of design choices over a wide operating range where nonlinearity is significant.

Onset of normal and inverse homoclinic bifurcation in a double plasma system near a plasma fireball

DOE Office of Scientific and Technical Information (OSTI.GOV)

Mitra, Vramori; Sarma, Bornali; Sarma, Arun

Plasma fireballs are generated due to a localized discharge and appear as a luminous glow with a sharp boundary, which suggests the presence of a localized electric field such as electrical sheath or double layer structure. The present work reports the observation of normal and inverse homoclinic bifurcation phenomena in plasma oscillations that are excited in the presence of fireball in a double plasma device. The controlling parameters for these observations are the ratio of target to source chamber (n{sub T}/n{sub S}) densities and applied electrode voltage. Homoclinic bifurcation is noticed in the plasma potential fluctuations as the system evolvesmore » from narrow to long time period oscillations and vice versa with the change of control parameter. The dynamical transition in plasma fireball is demonstrated by spectral analysis, recurrence quantification analysis (RQA), and statistical measures, viz., skewness and kurtosis. The increasing trend of normalized variance reflects that enhancing n{sub T}/n{sub S} induces irregularity in plasma dynamics. The exponential growth of the time period is strongly indicative of homoclinic bifurcation in the system. The gradual decrease of skewness and increase of kurtosis with the increase of n{sub T}/n{sub S} also reflect growing complexity in the system. The visual change of recurrence plot and gradual enhancement of RQA variables DET, L{sub max}, and ENT reflects the bifurcation behavior in the dynamics. The combination of RQA and spectral analysis is a clear evidence that homoclinic bifurcation occurs due to the presence of plasma fireball with different density ratios. However, inverse bifurcation takes place due to the change of fireball voltage. Some of the features observed in the experiment are consistent with a model that describes the dynamics of ionization instabilities.« less

A bifurcation study to guide the design of a landing gear with a combined uplock/downlock mechanism

PubMed Central

Knowles, James A. C.; Lowenberg, Mark H.; Neild, Simon A.; Krauskopf, Bernd

2014-01-01

This paper discusses the insights that a bifurcation analysis can provide when designing mechanisms. A model, in the form of a set of coupled steady-state equations, can be derived to describe the mechanism. Solutions to this model can be traced through the mechanism's state versus parameter space via numerical continuation, under the simultaneous variation of one or more parameters. With this approach, crucial features in the response surface, such as bifurcation points, can be identified. By numerically continuing these points in the appropriate parameter space, the resulting bifurcation diagram can be used to guide parameter selection and optimization. In this paper, we demonstrate the potential of this technique by considering an aircraft nose landing gear, with a novel locking strategy that uses a combined uplock/downlock mechanism. The landing gear is locked when in the retracted or deployed states. Transitions between these locked states and the unlocked state (where the landing gear is a mechanism) are shown to depend upon the positions of two fold point bifurcations. By performing a two-parameter continuation, the critical points are traced to identify operational boundaries. Following the variation of the fold points through parameter space, a minimum spring stiffness is identified that enables the landing gear to be locked in the retracted state. The bifurcation analysis also shows that the unlocking of a retracted landing gear should use an unlock force measure, rather than a position indicator, to de-couple the effects of the retraction and locking actuators. Overall, the study demonstrates that bifurcation analysis can enhance the understanding of the influence of design choices over a wide operating range where nonlinearity is significant. PMID:25484601

Arctic melt ponds and bifurcations in the climate system

NASA Astrophysics Data System (ADS)

Sudakov, I.; Vakulenko, S. A.; Golden, K. M.

2015-05-01

Understanding how sea ice melts is critical to climate projections. In the Arctic, melt ponds that develop on the surface of sea ice floes during the late spring and summer largely determine their albedo - a key parameter in climate modeling. Here we explore the possibility of a conceptual sea ice climate model passing through a bifurcation point - an irreversible critical threshold as the system warms, by incorporating geometric information about melt pond evolution. This study is based on a bifurcation analysis of the energy balance climate model with ice-albedo feedback as the key mechanism driving the system to bifurcation points.

Bifurcation of solutions to Hamiltonian boundary value problems

NASA Astrophysics Data System (ADS)

McLachlan, R. I.; Offen, C.

2018-06-01

A bifurcation is a qualitative change in a family of solutions to an equation produced by varying parameters. In contrast to the local bifurcations of dynamical systems that are often related to a change in the number or stability of equilibria, bifurcations of boundary value problems are global in nature and may not be related to any obvious change in dynamical behaviour. Catastrophe theory is a well-developed framework which studies the bifurcations of critical points of functions. In this paper we study the bifurcations of solutions of boundary-value problems for symplectic maps, using the language of (finite-dimensional) singularity theory. We associate certain such problems with a geometric picture involving the intersection of Lagrangian submanifolds, and hence with the critical points of a suitable generating function. Within this framework, we then study the effect of three special cases: (i) some common boundary conditions, such as Dirichlet boundary conditions for second-order systems, restrict the possible types of bifurcations (for example, in generic planar systems only the A-series beginning with folds and cusps can occur); (ii) integrable systems, such as planar Hamiltonian systems, can exhibit a novel periodic pitchfork bifurcation; and (iii) systems with Hamiltonian symmetries or reversing symmetries can exhibit restricted bifurcations associated with the symmetry. This approach offers an alternative to the analysis of critical points in function spaces, typically used in the study of bifurcation of variational problems, and opens the way to the detection of more exotic bifurcations than the simple folds and cusps that are often found in examples.

Approximate solution to the Hopf Phi equation for isotropic homogeneous fluid turbulence

NASA Technical Reports Server (NTRS)

Rosen, G.

1982-01-01

Consistent with the observed t to the -n decay laws for isotropic homogeneous turbulence and the form of the longitudinal correlation function f(r, t) for small r, the Hopf Phi equation is shown to be satisfied approximately by an asymptotic power series in t to the -n. This solution features a self-similar universal equilibrium functional which manifests Kolmogoroff-type scaling.

The Hopf algebra structure of the h-deformed Z3-graded quantum supergroup GLh,j(1|1)

NASA Astrophysics Data System (ADS)

Yasar, Ergün

2016-07-01

In this work, we define a new proper singular g matrix to construct a Z3-graded calculus on the h-deformed quantum superplane. Using the obtained calculus, we construct a new h-deformed Z3-graded quantum supergroup and give some features of it. Finally, we build up the Hopf algebra structure of this supergroup.

Overdetermined elliptic problems in topological disks

NASA Astrophysics Data System (ADS)

Mira, Pablo

2018-06-01

We introduce a method, based on the Poincaré-Hopf index theorem, to classify solutions to overdetermined problems for fully nonlinear elliptic equations in domains diffeomorphic to a closed disk. Applications to some well-known nonlinear elliptic PDEs are provided. Our result can be seen as the analogue of Hopf's uniqueness theorem for constant mean curvature spheres, but for the general analytic context of overdetermined elliptic problems.

An equation-free approach to agent-based computation: Bifurcation analysis and control of stationary states

NASA Astrophysics Data System (ADS)

Siettos, C. I.; Gear, C. W.; Kevrekidis, I. G.

2012-08-01

We show how the equation-free approach can be exploited to enable agent-based simulators to perform system-level computations such as bifurcation, stability analysis and controller design. We illustrate these tasks through an event-driven agent-based model describing the dynamic behaviour of many interacting investors in the presence of mimesis. Using short bursts of appropriately initialized runs of the detailed, agent-based simulator, we construct the coarse-grained bifurcation diagram of the (expected) density of agents and investigate the stability of its multiple solution branches. When the mimetic coupling between agents becomes strong enough, the stable stationary state loses its stability at a coarse turning point bifurcation. We also demonstrate how the framework can be used to design a wash-out dynamic controller that stabilizes open-loop unstable stationary states even under model uncertainty.

Analysis of stability and bifurcations of fixed points and periodic solutions of a lumped model of neocortex with two delays

PubMed Central

2012-01-01

A lumped model of neural activity in neocortex is studied to identify regions of multi-stability of both steady states and periodic solutions. Presence of both steady states and periodic solutions is considered to correspond with epileptogenesis. The model, which consists of two delay differential equations with two fixed time lags is mainly studied for its dependency on varying connection strength between populations. Equilibria are identified, and using linear stability analysis, all transitions are determined under which both trivial and non-trivial fixed points lose stability. Periodic solutions arising at some of these bifurcations are numerically studied with a two-parameter bifurcation analysis. PMID:22655859

Bifurcation Analysis and Application for Impulsive Systems with Delayed Impulses

NASA Astrophysics Data System (ADS)

Church, Kevin E. M.; Liu, Xinzhi

In this article, we present a systematic approach to bifurcation analysis of impulsive systems with autonomous or periodic right-hand sides that may exhibit delayed impulse terms. Methods include Lyapunov-Schmidt reduction and center manifold reduction. Both methods are presented abstractly in the context of the stroboscopic map associated to a given impulsive system, and are illustrated by way of two in-depth examples: the analysis of a SIR model of disease transmission with seasonality and unevenly distributed moments of treatment, and a scalar logistic differential equation with a delayed census impulsive harvesting effort. It is proven that in some special cases, the logistic equation can exhibit a codimension two bifurcation at a 1:1 resonance point.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Lee, Cheng-Hung; Department of Mechanical Engineering, Chang Gung University, Tao-Yuan, Taiwan; Jhong, Guan-Heng

The deployment of metallic stents during percutaneous coronary intervention has become common in the treatment of coronary bifurcation lesions. However, restenosis occurs mostly at the bifurcation area even in present era of drug-eluting stents. To achieve adequate deployment, physicians may unintentionally apply force to the strut of the stents through balloon, guiding catheters, or other devices. This force may deform the struts and impose excessive mechanical stresses on the arterial vessels, resulting in detrimental outcomes. This study investigated the relationship between the distribution of stress in a stent and bifurcation angle using finite element analysis. The unintentionally applied force followingmore » stent implantation was measured using a force sensor that was made in the laboratory. Geometrical information on the coronary arteries of 11 subjects was extracted from contrast-enhanced computed tomography scan data. The numerical results reveal that the application of force by physicians generated significantly higher mechanical stresses in the arterial bifurcation than in the proximal and distal parts of the stent (post hoc P < 0.01). The maximal stress on the vessels was significantly higher at bifurcation angle <70° than at angle ≧70° (P < 0.05). The maximal stress on the vessels was negatively correlated with bifurcation angle (P < 0.01). Stresses at the bifurcation ostium may cause arterial wall injury and restenosis, especially at small bifurcation angles. These finding highlight the effect of force-induced mechanical stress at coronary artery bifurcation stenting, and potential mechanisms of in-stent restenosis, along with their relationship with bifurcation angle.« less

Uniqueness Results for Weak Leray-Hopf Solutions of the Navier-Stokes System with Initial Values in Critical Spaces

NASA Astrophysics Data System (ADS)

Barker, T.

2018-03-01

The main subject of this paper concerns the establishment of certain classes of initial data, which grant short time uniqueness of the associated weak Leray-Hopf solutions of the three dimensional Navier-Stokes equations. In particular, our main theorem that this holds for any solenodial initial data, with finite L_2(R^3) norm, that also belongs to certain subsets of {it{VMO}}^{-1}(R^3). As a corollary of this, we obtain the same conclusion for any solenodial u0 belonging to L2(R^3)\\cap \\dot{B}^{-1+3/p}_{p,∞}(R^3), for any 3

Impact adding bifurcation in an autonomous hybrid dynamical model of church bell

NASA Astrophysics Data System (ADS)

Brzeski, P.; Chong, A. S. E.; Wiercigroch, M.; Perlikowski, P.

2018-05-01

In this paper we present the bifurcation analysis of the yoke-bell-clapper system which corresponds to the biggest bell "Serce Lodzi" mounted in the Cathedral Basilica of St Stanislaus Kostka, Lodz, Poland. The mathematical model of the system considered in this work has been derived and verified based on measurements of dynamics of the real bell. We perform numerical analysis both by direct numerical integration and path-following method using toolbox ABESPOL (Chong, 2016). By introducing the active yoke the position of the bell-clapper system with respect to the yoke axis of rotation can be easily changed and it can be used to probe the system dynamics. We found a wide variety of periodic and non-periodic solutions, and examined the ranges of coexistence of solutions and transitions between them via different types of bifurcations. Finally, a new type of bifurcation induced by a grazing event - an "impact adding bifurcation" has been proposed. When it occurs, the number of impacts between the bell and the clapper is increasing while the period of the system's motion stays the same.

Mappings of Least Dirichlet Energy and their Hopf Differentials

NASA Astrophysics Data System (ADS)

Iwaniec, Tadeusz; Onninen, Jani

2013-08-01

The paper is concerned with mappings {h \\colon {X}} {{begin{array}{ll} onto \\ longrightarrow }} {{Y}} between planar domains having least Dirichlet energy. The existence and uniqueness (up to a conformal change of variables in {{X}}) of the energy-minimal mappings is established within the class {overline{fancyscript{H}}_2({X}, {Y})} of strong limits of homeomorphisms in the Sobolev space {fancyscript{W}^{1,2}({X}, {Y})} , a result of considerable interest in the mathematical models of nonlinear elasticity. The inner variation of the independent variable in {{X}} leads to the Hopf differential {hz overline{h_{bar{z}}} dz ⊗ dz} and its trajectories. For a pair of doubly connected domains, in which {{X}} has finite conformal modulus, we establish the following principle: A mapping {h in overline{fancyscript{H}}2 ({X}, {Y})} is energy-minimal if and only if its Hopf-differential is analytic in {{X}} and real along {partial {X}} . In general, the energy-minimal mappings may not be injective, in which case one observes the occurrence of slits in {{X}} (cognate with cracks). Slits are triggered by points of concavity of {{Y}} . They originate from {partial {X}} and advance along vertical trajectories of the Hopf differential toward {{X}} where they eventually terminate, so no crosscuts are created.

Bifurcations and degenerate periodic points in a three dimensional chaotic fluid flow

DOE Office of Scientific and Technical Information (OSTI.GOV)

Smith, L. D., E-mail: lachlan.smith@monash.edu; CSIRO Mineral Resources, Clayton, Victoria 3800; Rudman, M.

2016-05-15

Analysis of the periodic points of a conservative periodic dynamical system uncovers the basic kinematic structure of the transport dynamics and identifies regions of local stability or chaos. While elliptic and hyperbolic points typically govern such behaviour in 3D systems, degenerate (parabolic) points also play an important role. These points represent a bifurcation in local stability and Lagrangian topology. In this study, we consider the ramifications of the two types of degenerate periodic points that occur in a model 3D fluid flow. (1) Period-tripling bifurcations occur when the local rotation angle associated with elliptic points is reversed, creating a reversalmore » in the orientation of associated Lagrangian structures. Even though a single unstable point is created, the bifurcation in local stability has a large influence on local transport and the global arrangement of manifolds as the unstable degenerate point has three stable and three unstable directions, similar to hyperbolic points, and occurs at the intersection of three hyperbolic periodic lines. The presence of period-tripling bifurcation points indicates regions of both chaos and confinement, with the extent of each depending on the nature of the associated manifold intersections. (2) The second type of bifurcation occurs when periodic lines become tangent to local or global invariant surfaces. This bifurcation creates both saddle–centre bifurcations which can create both chaotic and stable regions, and period-doubling bifurcations which are a common route to chaos in 2D systems. We provide conditions for the occurrence of these tangent bifurcations in 3D conservative systems, as well as constraints on the possible types of tangent bifurcation that can occur based on topological considerations.« less

Dynamics of the seasonal variation of the North Equatorial Current bifurcation

NASA Astrophysics Data System (ADS)

Chen, Zhaohui; Wu, Lixin

2011-02-01

The dynamics of the seasonal variation of the North Equatorial Current (NEC) bifurcation is studied using a 1.5-layer nonlinear reduced-gravity Pacific basin model and a linear, first-mode baroclinic Rossby wave model. The model-simulated bifurcation latitude exhibits a distinct seasonal cycle with the southernmost latitude in June and the northernmost latitude in November, consistent with observational analysis. It is found that the seasonal migration of the NEC bifurcation latitude (NBL) not only is determined by wind locally in the tropics, as suggested in previous studies, but is also significantly intensified by the extratropical wind through coastal Kelvin waves. The model further demonstrates that the amplitude of the NEC bifurcation is also associated with stratification. A strong (weak) stratification leads to a fast (slow) phase speed of first-mode baroclinic Rossby waves, and thus large (small) annual range of the bifurcation latitude. Therefore, it is expected that in a warm climate the NBL should have a large range of annual migration.

Walking dynamics of the passive compass-gait model under OGY-based control: Emergence of bifurcations and chaos

NASA Astrophysics Data System (ADS)

Gritli, Hassène; Belghith, Safya

2017-06-01

An analysis of the passive dynamic walking of a compass-gait biped model under the OGY-based control approach using the impulsive hybrid nonlinear dynamics is presented in this paper. We describe our strategy for the development of a simplified analytical expression of a controlled hybrid Poincaré map and then for the design of a state-feedback control. Our control methodology is based mainly on the linearization of the impulsive hybrid nonlinear dynamics around a desired nominal one-periodic hybrid limit cycle. Our analysis of the controlled walking dynamics is achieved by means of bifurcation diagrams. Some interesting nonlinear phenomena are displayed, such as the period-doubling bifurcation, the cyclic-fold bifurcation, the period remerging, the period bubbling and chaos. A comparison between the raised phenomena in the impulsive hybrid nonlinear dynamics and the hybrid Poincaré map under control was also presented.

Cellular instability in rapid directional solidification - Bifurcation theory

NASA Technical Reports Server (NTRS)

Braun, R. J.; Davis, S. H.

1992-01-01

Merchant and Davis performed a linear stability analysis on a model for the directional solidification of a dilute binary alloy valid for all speeds. The analysis revealed that nonequilibrium segregation effects modify the Mullins and Sekerka cellular mode, whereas attachment kinetics has no effect on these cells. In this paper, the nonlinear stability of the steady cellular mode is analyzed. A Landau equation is obtained that determines the amplitude of the cells. The Landau coefficient here depends on both nonequilibrium segregation effects and attachment kinetics. This equation gives the ranges of parameters for subcritical bifurcation (jump transition) or supercritical bifurcation (smooth transition) to cells.

Evolution of supersonic corner vortex in a hypersonic inlet/isolator model

NASA Astrophysics Data System (ADS)

Huang, He-Xia; Tan, Hui-Jun; Sun, Shu; Ling, Yu

2016-12-01

There are complex corner vortex flows in a rectangular hypersonic inlet/isolator. The corner vortex propagates downstream and interacts with the shocks and expansion waves in the isolator repeatedly. The supersonic corner vortex in a generic hypersonic inlet/isolator model is theoretically and numerically analyzed at a freestream Mach number of 4.92. The cross-flow topology of the corner vortex flow is found to obey Zhang's theory ["Analytical analysis of subsonic and supersonic vortex formation," Acta Aerodyn. Sin. 13, 259-264 (1995)] strictly, except for the short process with the vortex core situated in a subsonic flow which is surrounded by a supersonic flow. In general, the evolution history of the corner vortex under the influence of the background waves in the hypersonic inlet/isolator model can be classified into two types, namely, from the adverse pressure gradient region to the favorable pressure gradient region and the reversed one. For type 1, the corner vortex is a one-celled vortex with the cross-sectional streamlines spiraling inwards at first. Then the Hopf bifurcation occurs and the streamlines in the outer part of the limit cycle switch to spiraling outwards, yielding a two-celled vortex. The limit cycle shrinks gradually and finally vanishes with the streamlines of the entire corner vortex spiraling outwards. For type 2, the cross-sectional streamlines of the corner vortex spiral outwards first. Then a stable limit cycle is formed, yielding a two-celled vortex. The short-lived limit cycle forces the streamlines in the corner vortex to change the spiraling trends rapidly. Although it is found in this paper that there are some defects on the theoretical proof of the limit cycle, Zhang's theory is proven useful for the prediction and qualitative analysis of the complex corner vortex in a hypersonic inlet/isolator. In addition, three conservation laws inside the limit cycle are obtained.

Adler-Kostant-Symes scheme for face and Calogero-Moser-Sutherland-type models

NASA Astrophysics Data System (ADS)

Jurčo, Branislav; Schupp, Peter

1998-07-01

We give the construction of quantum Lax equations for IRF models and the difference version of the Calogero-Moser-Sutherland model introduced by Ruijsenaars. We solve the equations using factorization properties of the underlying face Hopf algebras/elliptic quantum groups. This construction is in the spirit of the Adler-Kostant-Symes method and generalizes our previous work to the case of face Hopf algebras/elliptic quantum groups with dynamical R matrices.

Spacetime from locality of interactions in deformations of special relativity: The example of κ -Poincaré Hopf algebra

NASA Astrophysics Data System (ADS)

Carmona, J. M.; Cortés, J. L.; Relancio, J. J.

2018-03-01

A new proposal for the notion of spacetime in a relativistic generalization of special relativity based on a modification of the composition law of momenta is presented. Locality of interactions is the principle which defines the spacetime structure for a system of particles. The formulation based on κ -Poincaré Hopf algebra is shown to be contained in this framework as a particular example.

Bifurcation analysis of an automatic dynamic balancing mechanism for eccentric rotors

NASA Astrophysics Data System (ADS)

Green, K.; Champneys, A. R.; Lieven, N. J.

2006-04-01

We present a nonlinear bifurcation analysis of the dynamics of an automatic dynamic balancing mechanism for rotating machines. The principle of operation is to deploy two or more masses that are free to travel around a race at a fixed distance from the hub and, subsequently, balance any eccentricity in the rotor. Mathematically, we start from a Lagrangian description of the system. It is then shown how under isotropic conditions a change of coordinates into a rotating frame turns the problem into a regular autonomous dynamical system, amenable to a full nonlinear bifurcation analysis. Using numerical continuation techniques, curves are traced of steady states, limit cycles and their bifurcations as parameters are varied. These results are augmented by simulations of the system trajectories in phase space. Taking the case of a balancer with two free masses, broad trends are revealed on the existence of a stable, dynamically balanced steady-state solution for specific rotation speeds and eccentricities. However, the analysis also reveals other potentially attracting states—non-trivial steady states, limit cycles, and chaotic motion—which are not in balance. The transient effects which lead to these competing states, which in some cases coexist, are investigated.

Bifurcation Analysis of a DC-DC Bidirectional Power Converter Operating with Constant Power Loads

NASA Astrophysics Data System (ADS)

Cristiano, Rony; Pagano, Daniel J.; Benadero, Luis; Ponce, Enrique

Direct current (DC) microgrids (MGs) are an emergent option to satisfy new demands for power quality and integration of renewable resources in electrical distribution systems. This work addresses the large-signal stability analysis of a DC-DC bidirectional converter (DBC) connected to a storage device in an islanding MG. This converter is responsible for controlling the balance of power (load demand and generation) under constant power loads (CPLs). In order to control the DC bus voltage through a DBC, we propose a robust sliding mode control (SMC) based on a washout filter. Dynamical systems techniques are exploited to assess the quality of this switching control strategy. In this sense, a bifurcation analysis is performed to study the nonlinear stability of a reduced model of this system. The appearance of different bifurcations when load parameters and control gains are changed is studied in detail. In the specific case of Teixeira Singularity (TS) bifurcation, some experimental results are provided, confirming the mathematical predictions. Both a deeper insight in the dynamic behavior of the controlled system and valuable design criteria are obtained.

Double versus single stenting for coronary bifurcation lesions: a meta-analysis.

PubMed

Katritsis, Demosthenes G; Siontis, George C M; Ioannidis, John P A

2009-10-01

Several trials have addressed whether bifurcation lesions require stenting of both the main vessel and side branch, but uncertainty remains on the benefits of such double versus single stenting of the main vessel only. We have conducted a meta-analysis of randomized trials including patients with coronary bifurcation lesions who were randomly selected to undergo percutaneous coronary intervention by either double or single stenting. Six studies (n=1642 patients) were eligible. There was increased risk of myocardial infarction with double stenting (risk ratio, 1.78; P=0.001 by fixed effects; risk ratio, 1.49 with Bayesian meta-analysis). The summary point estimate suggested also an increased risk of stent thrombosis with double stenting, but the difference was not nominally significant given the sparse data (risk ratio, 1.85; P=0.19). No obvious difference was seen for death (risk ratio, 0.81; P=0.66) and target lesion revascularization (risk ratio, 1.09; P=0.67). Stenting of both the main vessel and side branch in bifurcation lesions may increase myocardial infarction and stent thrombosis risk compared with stenting of the main vessel only.

Chaotic Behavior of a Generalized Sprott E Differential System

NASA Astrophysics Data System (ADS)

Oliveira, Regilene; Valls, Claudia

A chaotic system with only one equilibrium, a stable node-focus, was introduced by Wang and Chen [2012]. This system was found by adding a nonzero constant b to the Sprott E system [Sprott, 1994]. The coexistence of three types of attractors in this autonomous system was also considered by Braga and Mello [2013]. Adding a second parameter to the Sprott E differential system, we get the autonomous system ẋ = ayz + b,ẏ = x2 - y,ż = 1 - 4x, where a,b ∈ ℝ are parameters and a≠0. In this paper, we consider theoretically some global dynamical aspects of this system called here the generalized Sprott E differential system. This polynomial differential system is relevant because it is the first polynomial differential system in ℝ3 with two parameters exhibiting, besides the point attractor and chaotic attractor, coexisting stable limit cycles, demonstrating that this system is truly complicated and interesting. More precisely, we show that for b sufficiently small this system can exhibit two limit cycles emerging from the classical Hopf bifurcation at the equilibrium point p = (1/4, 1/16, 0). We also give a complete description of its dynamics on the Poincaré sphere at infinity by using the Poincaré compactification of a polynomial vector field in ℝ3, and we show that it has no first integrals in the class of Darboux functions.

A numerical study of coarsening in the two-dimensional complex Ginzburg-Landau equation

NASA Astrophysics Data System (ADS)

Liu, Weigang; Tauber, Uwe

The complex Ginzburg-Landau equation with additive noise is a stochastic partial differential equation that describes a remarkably wide range of physical systems: coupled non-linear oscillators subject to external noise near a Hopf bifurcation instability; spontaneous structure formation in non-equilibrium systems, e.g., in cyclically competing populations; and driven-dissipative Bose-Einstein condensation, realized in open systems on the interface of quantum optics and many-body physics. We employ a finite-difference method to numerically solve the noisy complex Ginzburg-Landau equation on a two-dimensional domain with the goal to investigate the coarsening dynamics following a quench from a strongly fluctuating defect turbulence phase to a long-range ordered phase. We start from a simplified amplitude equation, solve it numerically, and then study the spatio-temporal behavior characterized by the spontaneous creation and annihilation of topological defects (spiral waves). We check our simulation results against the known dynamical phase diagram in this non-equilibrium system, tentatively analyze the coarsening kinetics following sudden quenches, and characterize the ensuing aging scaling behavior. In addition, we aim to use Voronoi triangulation to study the cellular structure in the phase turbulence and frozen states. This research is supported by the U. S. Department of Energy, Office of Basic Energy Sciences, Division of Materials Science and Engineering under Award DE-FG02-09ER46613.

Oscillations in a simple climate-vegetation model

NASA Astrophysics Data System (ADS)

Rombouts, J.; Ghil, M.

2015-05-01

We formulate and analyze a simple dynamical systems model for climate-vegetation interaction. The planet we consider consists of a large ocean and a land surface on which vegetation can grow. The temperature affects vegetation growth on land and the amount of sea ice on the ocean. Conversely, vegetation and sea ice change the albedo of the planet, which in turn changes its energy balance and hence the temperature evolution. Our highly idealized, conceptual model is governed by two nonlinear, coupled ordinary differential equations, one for global temperature, the other for vegetation cover. The model exhibits either bistability between a vegetated and a desert state or oscillatory behavior. The oscillations arise through a Hopf bifurcation off the vegetated state, when the death rate of vegetation is low enough. These oscillations are anharmonic and exhibit a sawtooth shape that is characteristic of relaxation oscillations, as well as suggestive of the sharp deglaciations of the Quaternary. Our model's behavior can be compared, on the one hand, with the bistability of even simpler, Daisyworld-style climate-vegetation models. On the other hand, it can be integrated into the hierarchy of models trying to simulate and explain oscillatory behavior in the climate system. Rigorous mathematical results are obtained that link the nature of the feedbacks with the nature and the stability of the solutions. The relevance of model results to climate variability on various timescales is discussed.

Oscillations in a simple climate-vegetation model

NASA Astrophysics Data System (ADS)

Rombouts, J.; Ghil, M.

2015-02-01

We formulate and analyze a simple dynamical systems model for climate-vegetation interaction. The planet we consider consists of a large ocean and a land surface on which vegetation can grow. The temperature affects vegetation growth on land and the amount of sea ice on the ocean. Conversely, vegetation and sea ice change the albedo of the planet, which in turn changes its energy balance and hence the temperature evolution. Our highly idealized, conceptual model is governed by two nonlinear, coupled ordinary differential equations, one for global temperature, the other for vegetation cover. The model exhibits either bistability between a vegetated and a desert state or oscillatory behavior. The oscillations arise through a Hopf bifurcation off the vegetated state, when the death rate of vegetation is low enough. These oscillations are anharmonic and exhibit a sawtooth shape that is characteristic of relaxation oscillations, as well as suggestive of the sharp deglaciations of the Quaternary. Our model's behavior can be compared, on the one hand, with the bistability of even simpler, Daisyworld-style climate-vegetation models. On the other hand, it can be integrated into the hierarchy of models trying to simulate and explain oscillatory behavior in the climate system. Rigorous mathematical results are obtained that link the nature of the feedbacks with the nature and the stability of the solutions. The relevance of model results to climate variability on various time scales is discussed.

Late-time behaviour of the tilted Bianchi type VIh models

NASA Astrophysics Data System (ADS)

Hervik, S.; van den Hoogen, R. J.; Lim, W. C.; Coley, A. A.

2007-08-01

We study tilted perfect fluid cosmological models with a constant equation of state parameter in spatially homogeneous models of Bianchi type VIh using dynamical systems methods and numerical experimentation, with an emphasis on their future asymptotic evolution. We determine all of the equilibrium points of the type VIh state space (which correspond to exact self-similar solutions of the Einstein equations, some of which are new), and their stability is investigated. We find that there are vacuum plane-wave solutions that act as future attractors. In the parameter space, a 'loophole' is shown to exist in which there are no stable equilibrium points. We then show that a Hopf-bifurcation can occur resulting in a stable closed orbit (which we refer to as the Mussel attractor) corresponding to points both inside the loophole and points just outside the loophole; in the former case the closed curves act as late-time attractors while in the latter case these attracting curves will co-exist with attracting equilibrium points. In the special Bianchi type III case, centre manifold theory is required to determine the future attractors. Comprehensive numerical experiments are carried out to complement and confirm the analytical results presented. We note that the Bianchi type VIh case is of particular interest in that it contains many different subcases which exhibit many of the different possible future asymptotic behaviours of Bianchi cosmological models.

Impact of leakage delay on bifurcation in high-order fractional BAM neural networks.

PubMed

Huang, Chengdai; Cao, Jinde

2018-02-01

The effects of leakage delay on the dynamics of neural networks with integer-order have lately been received considerable attention. It has been confirmed that fractional neural networks more appropriately uncover the dynamical properties of neural networks, but the results of fractional neural networks with leakage delay are relatively few. This paper primarily concentrates on the issue of bifurcation for high-order fractional bidirectional associative memory(BAM) neural networks involving leakage delay. The first attempt is made to tackle the stability and bifurcation of high-order fractional BAM neural networks with time delay in leakage terms in this paper. The conditions for the appearance of bifurcation for the proposed systems with leakage delay are firstly established by adopting time delay as a bifurcation parameter. Then, the bifurcation criteria of such system without leakage delay are successfully acquired. Comparative analysis wondrously detects that the stability performance of the proposed high-order fractional neural networks is critically weakened by leakage delay, they cannot be overlooked. Numerical examples are ultimately exhibited to attest the efficiency of the theoretical results. Copyright © 2017 Elsevier Ltd. All rights reserved.

Critical evaluation of the unsteady aerodynamics approach to dynamic stability at high angles of attack

NASA Technical Reports Server (NTRS)

Hui, W. H.

1985-01-01

Bifurcation theory is used to analyze the nonlinear dynamic stability characteristics of an aircraft subject to single-degree-of-freedom. The requisite moment of the aerodynamic forces in the equations of motion is shown to be representable in a form equivalent to the response to finite amplitude oscillations. It is shown how this information can be deduced from the case of infinitesimal-amplitude oscillations. The bifurcation theory analysis reveals that when the bifurcation parameter is increased beyond a critical value at which the aerodynamic damping vanishes, new solutions representing finite amplitude periodic motions bifurcate from the previously stable steady motion. The sign of a simple criterion, cast in terms of aerodynamic properties, determines whether the bifurcating solutions are stable or unstable. For the pitching motion of flat-plate airfoils flying at supersonic/hypersonic speed and for oscillation of flaps at transonic speed, the bifurcation is subcritical, implying either the exchanges of stability between steady and periodic motion are accompanied by hysteresis phenomena, or that potentially large aperiodic departures from steady motion may develop.

Von Bertalanffy's dynamics under a polynomial correction: Allee effect and big bang bifurcation

NASA Astrophysics Data System (ADS)

Leonel Rocha, J.; Taha, A. K.; Fournier-Prunaret, D.

2016-02-01

In this work we consider new one-dimensional populational discrete dynamical systems in which the growth of the population is described by a family of von Bertalanffy's functions, as a dynamical approach to von Bertalanffy's growth equation. The purpose of introducing Allee effect in those models is satisfied under a correction factor of polynomial type. We study classes of von Bertalanffy's functions with different types of Allee effect: strong and weak Allee's functions. Dependent on the variation of four parameters, von Bertalanffy's functions also includes another class of important functions: functions with no Allee effect. The complex bifurcation structures of these von Bertalanffy's functions is investigated in detail. We verified that this family of functions has particular bifurcation structures: the big bang bifurcation of the so-called “box-within-a-box” type. The big bang bifurcation is associated to the asymptotic weight or carrying capacity. This work is a contribution to the study of the big bang bifurcation analysis for continuous maps and their relationship with explosion birth and extinction phenomena.

Renormalization in Quantum Field Theory and the Riemann-Hilbert Problem I: The Hopf Algebra Structure of Graphs and the Main Theorem

NASA Astrophysics Data System (ADS)

Connes, Alain; Kreimer, Dirk

This paper gives a complete selfcontained proof of our result announced in [6] showing that renormalization in quantum field theory is a special instance of a general mathematical procedure of extraction of finite values based on the Riemann-Hilbert problem. We shall first show that for any quantum field theory, the combinatorics of Feynman graphs gives rise to a Hopf algebra which is commutative as an algebra. It is the dual Hopf algebra of the enveloping algebra of a Lie algebra whose basis is labelled by the one particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed from insertions of one graph in the other and vice versa. The corresponding Lie group G is the group of characters of . We shall then show that, using dimensional regularization, the bare (unrenormalized) theory gives rise to a loop where C is a small circle of complex dimensions around the integer dimension D of space-time. Our main result is that the renormalized theory is just the evaluation at z=D of the holomorphic part γ+ of the Birkhoff decomposition of γ. We begin to analyse the group G and show that it is a semi-direct product of an easily understood abelian group by a highly non-trivial group closely tied up with groups of diffeomorphisms. The analysis of this latter group as well as the interpretation of the renormalization group and of anomalous dimensions are the content of our second paper with the same overall title.

TQ-bifurcations in discrete dynamical systems: Analysis of qualitative rearrangements of the oscillation mode

DOE Office of Scientific and Technical Information (OSTI.GOV)

Makarenko, A. V., E-mail: avm.science@mail.ru

A new class of bifurcations is defined in discrete dynamical systems, and methods for their diagnostics and the analysis of their properties are presented. The TQ-bifurcations considered are implemented in discrete mappings and are related to the qualitative rearrangement of the shape of trajectories in an extended space of states. Within the demonstration of the main capabilities of the toolkit, an analysis is carried out of a logistic mapping in a domain to the right of the period-doubling limit point. Five critical values of the parameter are found for which the geometric structure of the trajectories of the mapping experiencesmore » a qualitative rearrangement. In addition, an analysis is carried out of the so-called “trace map,” which arises in the problems of quantum-mechanical description of various properties of discrete crystalline and quasicrystalline lattices.« less

Bifurcation and response analysis of a nonlinear flexible rotating disc immersed in bounded compressible fluid

NASA Astrophysics Data System (ADS)

Remigius, W. Dheelibun; Sarkar, Sunetra; Gupta, Sayan

2017-03-01

Use of heavy gases in centrifugal compressors for enhanced oil extraction have made the impellers susceptible to failures through acousto-elastic instabilities. This study focusses on understanding the dynamical behavior of such systems by considering the effects of the bounded fluid housed in a casing on a rotating disc. First, a mathematical model is developed that incorporates the interaction between the rotating impeller - modelled as a flexible disc - and the bounded compressible fluid medium in which it is immersed. The nonlinear effects arising due to large deformations of the disc have been included in the formulation so as to capture the post flutter behavior. A bifurcation analysis is carried out with the disc rotational speed as the bifurcation parameter to investigate the dynamical behavior of the coupled system and estimate the stability boundaries. Parametric studies reveal that the relative strengths of the various dissipation mechanisms in the coupled system play a significant role that affect the bifurcation route and the post flutter behavior in the acousto-elastic system.

Analysis of chaotic saddles in a nonlinear vibro-impact system

NASA Astrophysics Data System (ADS)

Feng, Jinqian

2017-07-01

In this paper, a computational investigation of chaotic saddles in a nonlinear vibro-impact system is presented. For a classical Duffing vibro-impact oscillator, we employ the bisection procedure and an improved stagger-and-step method to present evidence of visual chaotic saddles on the fractal basin boundary and in the internal basin, respectively. The results show that the period saddles play an important role in the evolution of chaotic saddle. The dynamics mechanics of three types of bifurcation such as saddle-node bifurcation, chaotic saddle crisis bifurcation and interior chaotic crisis bifurcation are discussed. The results reveal that the period saddle created at saddle-node bifurcation is responsible for the switch of the internal chaotic saddle to the boundary chaotic saddle. At chaotic saddle crisis bifurcation, a large chaotic saddle can divide into two different chaotic saddle connected by a period saddle. The intersection points between stable and unstable manifolds of this period saddle supply access for chaotic orbits from one chaotic saddle to another and eventually induce the coupling of these two chaotic saddle. Interior chaotic crisis bifurcation is associated with the intersection of stable and unstable manifolds of the period saddle connecting two chaotic invariant sets. In addition, the gaps in chaotic saddle is responsible for the fractal structure.

The physics of hearing: fluid mechanics and the active process of the inner ear.

PubMed

Reichenbach, Tobias; Hudspeth, A J

2014-07-01

Most sounds of interest consist of complex, time-dependent admixtures of tones of diverse frequencies and variable amplitudes. To detect and process these signals, the ear employs a highly nonlinear, adaptive, real-time spectral analyzer: the cochlea. Sound excites vibration of the eardrum and the three miniscule bones of the middle ear, the last of which acts as a piston to initiate oscillatory pressure changes within the liquid-filled chambers of the cochlea. The basilar membrane, an elastic band spiraling along the cochlea between two of these chambers, responds to these pressures by conducting a largely independent traveling wave for each frequency component of the input. Because the basilar membrane is graded in mass and stiffness along its length, however, each traveling wave grows in magnitude and decreases in wavelength until it peaks at a specific, frequency-dependent position: low frequencies propagate to the cochlear apex, whereas high frequencies culminate at the base. The oscillations of the basilar membrane deflect hair bundles, the mechanically sensitive organelles of the ear's sensory receptors, the hair cells. As mechanically sensitive ion channels open and close, each hair cell responds with an electrical signal that is chemically transmitted to an afferent nerve fiber and thence into the brain. In addition to transducing mechanical inputs, hair cells amplify them by two means. Channel gating endows a hair bundle with negative stiffness, an instability that interacts with the motor protein myosin-1c to produce a mechanical amplifier and oscillator. Acting through the piezoelectric membrane protein prestin, electrical responses also cause outer hair cells to elongate and shorten, thus pumping energy into the basilar membrane's movements. The two forms of motility constitute an active process that amplifies mechanical inputs, sharpens frequency discrimination, and confers a compressive nonlinearity on responsiveness. These features arise because the active process operates near a Hopf bifurcation, the generic properties of which explain several key features of hearing. Moreover, when the gain of the active process rises sufficiently in ultraquiet circumstances, the system traverses the bifurcation and even a normal ear actually emits sound. The remarkable properties of hearing thus stem from the propagation of traveling waves on a nonlinear and excitable medium.

The physics of hearing: fluid mechanics and the active process of the inner ear

NASA Astrophysics Data System (ADS)

Reichenbach, Tobias; Hudspeth, A. J.

2014-07-01

Most sounds of interest consist of complex, time-dependent admixtures of tones of diverse frequencies and variable amplitudes. To detect and process these signals, the ear employs a highly nonlinear, adaptive, real-time spectral analyzer: the cochlea. Sound excites vibration of the eardrum and the three miniscule bones of the middle ear, the last of which acts as a piston to initiate oscillatory pressure changes within the liquid-filled chambers of the cochlea. The basilar membrane, an elastic band spiraling along the cochlea between two of these chambers, responds to these pressures by conducting a largely independent traveling wave for each frequency component of the input. Because the basilar membrane is graded in mass and stiffness along its length, however, each traveling wave grows in magnitude and decreases in wavelength until it peaks at a specific, frequency-dependent position: low frequencies propagate to the cochlear apex, whereas high frequencies culminate at the base. The oscillations of the basilar membrane deflect hair bundles, the mechanically sensitive organelles of the ear's sensory receptors, the hair cells. As mechanically sensitive ion channels open and close, each hair cell responds with an electrical signal that is chemically transmitted to an afferent nerve fiber and thence into the brain. In addition to transducing mechanical inputs, hair cells amplify them by two means. Channel gating endows a hair bundle with negative stiffness, an instability that interacts with the motor protein myosin-1c to produce a mechanical amplifier and oscillator. Acting through the piezoelectric membrane protein prestin, electrical responses also cause outer hair cells to elongate and shorten, thus pumping energy into the basilar membrane's movements. The two forms of motility constitute an active process that amplifies mechanical inputs, sharpens frequency discrimination, and confers a compressive nonlinearity on responsiveness. These features arise because the active process operates near a Hopf bifurcation, the generic properties of which explain several key features of hearing. Moreover, when the gain of the active process rises sufficiently in ultraquiet circumstances, the system traverses the bifurcation and even a normal ear actually emits sound. The remarkable properties of hearing thus stem from the propagation of traveling waves on a nonlinear and excitable medium.

Symmetry-Breaking Bifurcation in the Nonlinear Schrödinger Equation with Symmetric Potentials

NASA Astrophysics Data System (ADS)

Kirr, E.; Kevrekidis, P. G.; Pelinovsky, D. E.

2011-12-01

We consider the focusing (attractive) nonlinear Schrödinger (NLS) equation with an external, symmetric potential which vanishes at infinity and supports a linear bound state. We prove that the symmetric, nonlinear ground states must undergo a symmetry breaking bifurcation if the potential has a non-degenerate local maxima at zero. Under a generic assumption we show that the bifurcation is either a subcritical or supercritical pitchfork. In the particular case of double-well potentials with large separation, the power of nonlinearity determines the subcritical or supercritical character of the bifurcation. The results are obtained from a careful analysis of the spectral properties of the ground states at both small and large values for the corresponding eigenvalue parameter.

a Triangular Deformation of the Two-Dimensional POINCARÉ Algebra

NASA Astrophysics Data System (ADS)

Khorrami, M.; Shariati, A.; Abolhassani, M. R.; Aghamohammadi, A.

Contracting the h-deformation of SL(2, ℝ), we construct a new deformation of two-dimensional Poincaré's algebra, the algebra of functions on its group and its differential structure. It is seen that these dual Hopf algebras are isomorphic to each other. It is also shown that the Hopf algebra is triangular, and its universal R-matrix is also constructed explicitly. We then find a deformation map for the universal enveloping algebra, and at the end, give the deformed mass shells and Lorentz transformation.

Defining Electron Bifurcation in the Electron-Transferring Flavoprotein Family.

PubMed

Garcia Costas, Amaya M; Poudel, Saroj; Miller, Anne-Frances; Schut, Gerrit J; Ledbetter, Rhesa N; Fixen, Kathryn R; Seefeldt, Lance C; Adams, Michael W W; Harwood, Caroline S; Boyd, Eric S; Peters, John W

2017-11-01

Electron bifurcation is the coupling of exergonic and endergonic redox reactions to simultaneously generate (or utilize) low- and high-potential electrons. It is the third recognized form of energy conservation in biology and was recently described for select electron-transferring flavoproteins (Etfs). Etfs are flavin-containing heterodimers best known for donating electrons derived from fatty acid and amino acid oxidation to an electron transfer respiratory chain via Etf-quinone oxidoreductase. Canonical examples contain a flavin adenine dinucleotide (FAD) that is involved in electron transfer, as well as a non-redox-active AMP. However, Etfs demonstrated to bifurcate electrons contain a second FAD in place of the AMP. To expand our understanding of the functional variety and metabolic significance of Etfs and to identify amino acid sequence motifs that potentially enable electron bifurcation, we compiled 1,314 Etf protein sequences from genome sequence databases and subjected them to informatic and structural analyses. Etfs were identified in diverse archaea and bacteria, and they clustered into five distinct well-supported groups, based on their amino acid sequences. Gene neighborhood analyses indicated that these Etf group designations largely correspond to putative differences in functionality. Etfs with the demonstrated ability to bifurcate were found to form one group, suggesting that distinct conserved amino acid sequence motifs enable this capability. Indeed, structural modeling and sequence alignments revealed that identifying residues occur in the NADH- and FAD-binding regions of bifurcating Etfs. Collectively, a new classification scheme for Etf proteins that delineates putative bifurcating versus nonbifurcating members is presented and suggests that Etf-mediated bifurcation is associated with surprisingly diverse enzymes. IMPORTANCE Electron bifurcation has recently been recognized as an electron transfer mechanism used by microorganisms to maximize energy conservation. Bifurcating enzymes couple thermodynamically unfavorable reactions with thermodynamically favorable reactions in an overall spontaneous process. Here we show that the electron-transferring flavoprotein (Etf) enzyme family exhibits far greater diversity than previously recognized, and we provide a phylogenetic analysis that clearly delineates bifurcating versus nonbifurcating members of this family. Structural modeling of proteins within these groups reveals key differences between the bifurcating and nonbifurcating Etfs. Copyright © 2017 American Society for Microbiology.

Defining Electron Bifurcation in the Electron-Transferring Flavoprotein Family

PubMed Central

Garcia Costas, Amaya M.; Poudel, Saroj; Miller, Anne-Frances; Schut, Gerrit J.; Ledbetter, Rhesa N.; Seefeldt, Lance C.; Adams, Michael W. W.

2017-01-01

ABSTRACT Electron bifurcation is the coupling of exergonic and endergonic redox reactions to simultaneously generate (or utilize) low- and high-potential electrons. It is the third recognized form of energy conservation in biology and was recently described for select electron-transferring flavoproteins (Etfs). Etfs are flavin-containing heterodimers best known for donating electrons derived from fatty acid and amino acid oxidation to an electron transfer respiratory chain via Etf-quinone oxidoreductase. Canonical examples contain a flavin adenine dinucleotide (FAD) that is involved in electron transfer, as well as a non-redox-active AMP. However, Etfs demonstrated to bifurcate electrons contain a second FAD in place of the AMP. To expand our understanding of the functional variety and metabolic significance of Etfs and to identify amino acid sequence motifs that potentially enable electron bifurcation, we compiled 1,314 Etf protein sequences from genome sequence databases and subjected them to informatic and structural analyses. Etfs were identified in diverse archaea and bacteria, and they clustered into five distinct well-supported groups, based on their amino acid sequences. Gene neighborhood analyses indicated that these Etf group designations largely correspond to putative differences in functionality. Etfs with the demonstrated ability to bifurcate were found to form one group, suggesting that distinct conserved amino acid sequence motifs enable this capability. Indeed, structural modeling and sequence alignments revealed that identifying residues occur in the NADH- and FAD-binding regions of bifurcating Etfs. Collectively, a new classification scheme for Etf proteins that delineates putative bifurcating versus nonbifurcating members is presented and suggests that Etf-mediated bifurcation is associated with surprisingly diverse enzymes. IMPORTANCE Electron bifurcation has recently been recognized as an electron transfer mechanism used by microorganisms to maximize energy conservation. Bifurcating enzymes couple thermodynamically unfavorable reactions with thermodynamically favorable reactions in an overall spontaneous process. Here we show that the electron-transferring flavoprotein (Etf) enzyme family exhibits far greater diversity than previously recognized, and we provide a phylogenetic analysis that clearly delineates bifurcating versus nonbifurcating members of this family. Structural modeling of proteins within these groups reveals key differences between the bifurcating and nonbifurcating Etfs. PMID:28808132

Topological characterization and early detection of bifurcations and chaos in complex systems using persistent homology.

PubMed

Mittal, Khushboo; Gupta, Shalabh

2017-05-01

Early detection of bifurcations and chaos and understanding their topological characteristics are essential for safe and reliable operation of various electrical, chemical, physical, and industrial processes. However, the presence of non-linearity and high-dimensionality in system behavior makes this analysis a challenging task. The existing methods for dynamical system analysis provide useful tools for anomaly detection (e.g., Bendixson-Dulac and Poincare-Bendixson criteria can detect the presence of limit cycles); however, they do not provide a detailed topological understanding about system evolution during bifurcations and chaos, such as the changes in the number of subcycles and their positions, lifetimes, and sizes. This paper addresses this research gap by using topological data analysis as a tool to study system evolution and develop a mathematical framework for detecting the topological changes in the underlying system using persistent homology. Using the proposed technique, topological features (e.g., number of relevant k-dimensional holes, etc.) are extracted from nonlinear time series data which are useful for deeper analysis of the system behavior and early detection of bifurcations and chaos. When applied to a Logistic map, a Duffing oscillator, and a real life Op-amp based Jerk circuit, these features are shown to accurately characterize the system dynamics and detect the onset of chaos.

Bifurcation of small limit cycles in cubic integrable systems using higher-order analysis

NASA Astrophysics Data System (ADS)

Tian, Yun; Yu, Pei

2018-05-01

In this paper, we present a method of higher-order analysis on bifurcation of small limit cycles around an elementary center of integrable systems under perturbations. This method is equivalent to higher-order Melinikov function approach used for studying bifurcation of limit cycles around a center but simpler. Attention is focused on planar cubic polynomial systems and particularly it is shown that the system studied by Żoła¸dek (1995) [24] can indeed have eleven limit cycles under perturbations at least up to 7th order. Moreover, the pattern of numbers of limit cycles produced near the center is discussed up to 39th-order perturbations, and no more than eleven limit cycles are found.

Complexity and chaos control in a discrete-time prey-predator model

NASA Astrophysics Data System (ADS)

Din, Qamar

2017-08-01

We investigate the complex behavior and chaos control in a discrete-time prey-predator model. Taking into account the Leslie-Gower prey-predator model, we propose a discrete-time prey-predator system with predator partially dependent on prey and investigate the boundedness, existence and uniqueness of positive equilibrium and bifurcation analysis of the system by using center manifold theorem and bifurcation theory. Various feedback control strategies are implemented for controlling the bifurcation and chaos in the system. Numerical simulations are provided to illustrate theoretical discussion.

Hydroelastic effects in the aorta bifurcation zone

NASA Technical Reports Server (NTRS)

Volmir, A. S.; Gersheyn, M. S.; Purinya, B. A.

1980-01-01

The mechanical behavior of the vessels and blood is mathematically analyzed at the point of aortic bifurcation using a homogeneous single layer channel as a model of the aorta. Allowance is made for the fact that the aortic intima is considerably less rigid than the other layers. For analysis of blood flow in the major arteries, the blood is treated as a viscous Newtonian fluid whose movements are described by Navier-Stokes equations and a continuity equation. Blood flow dynamics at the aortic bifurcation are discussed on the basis of the results.

DISCO Interacting Protein 2 regulates axonal bifurcation and guidance of Drosophila mushroom body neurons.

PubMed

Nitta, Yohei; Yamazaki, Daisuke; Sugie, Atsushi; Hiroi, Makoto; Tabata, Tetsuya

2017-01-15

Axonal branching is one of the key processes within the enormous complexity of the nervous system to enable a single neuron to send information to multiple targets. However, the molecular mechanisms that control branch formation are poorly understood. In particular, previous studies have rarely addressed the mechanisms underlying axonal bifurcation, in which axons form new branches via splitting of the growth cone. We demonstrate that DISCO Interacting Protein 2 (DIP2) is required for precise axonal bifurcation in Drosophila mushroom body (MB) neurons by suppressing ectopic bifurcation and regulating the guidance of sister axons. We also found that DIP2 localize to the plasma membrane. Domain function analysis revealed that the AMP-synthetase domains of DIP2 are essential for its function, which may involve exerting a catalytic activity that modifies fatty acids. Genetic analysis and subsequent biochemical analysis suggested that DIP2 is involved in the fatty acid metabolization of acyl-CoA. Taken together, our results reveal a function of DIP2 in the developing nervous system and provide a potential functional relationship between fatty acid metabolism and axon morphogenesis. Copyright © 2016 Elsevier Inc. All rights reserved.

Analysis and design of nonlinear resonances via singularity theory

NASA Astrophysics Data System (ADS)

Cirillo, G. I.; Habib, G.; Kerschen, G.; Sepulchre, R.

2017-03-01

Bifurcation theory and continuation methods are well-established tools for the analysis of nonlinear mechanical systems subject to periodic forcing. We illustrate the added value and the complementary information provided by singularity theory with one distinguished parameter. While tracking bifurcations reveals the qualitative changes in the behaviour, tracking singularities reveals how structural changes are themselves organised in parameter space. The complementarity of that information is demonstrated in the analysis of detached resonance curves in a two-degree-of-freedom system.

Do Dogs Know Bifurcations?

ERIC Educational Resources Information Center

Minton, Roland; Pennings, Timothy J.

2007-01-01

When a dog (in this case, Tim Pennings' dog Elvis) is in the water and a ball is thrown downshore, it must choose to swim directly to the ball or first swim to shore. The mathematical analysis of this problem leads to the computation of bifurcation points at which the optimal strategy changes.

Numerical simulation of magnetic nanoparticles targeting in a bifurcation vessel

NASA Astrophysics Data System (ADS)

Larimi, M. M.; Ramiar, A.; Ranjbar, A. A.

2014-08-01

Guiding magnetic iron oxide nanoparticles with the help of an external magnetic field to its target is the principle behind the development of super paramagnetic iron oxide nanoparticles (SPIONs) as novel drug delivery vehicles. The present paper is devoted to study on MDT (Magnetic Drug Targeting) technique by particle tracking in the presence of magnetic field in a bifurcation vessel. The blood flow in bifurcation is considered incompressible, unsteady and Newtonian. The flow analysis applies the time dependent, two dimensional, incompressible Navier-Stokes equations for Newtonian fluids. The Lagrangian particle tracking is performed to estimate particle behavior under influence of imposed magnetic field gradients along the bifurcation. According to the results, the magnetic field increased the volume fraction of particle in target region, but in vessels with high Reynolds number, the efficiency of MDT technique is very low. Also the results showed that in the bifurcation vessels with lower angles, wall shear stress is higher and consequently the risk of the vessel wall rupture increases.

Nondimensional Parameters and Equations for Nonlinear and Bifurcation Analyses of Thin Anisotropic Quasi-Shallow Shells

NASA Technical Reports Server (NTRS)

Nemeth, Michael P.

2010-01-01

A comprehensive development of nondimensional parameters and equations for nonlinear and bifurcations analyses of quasi-shallow shells, based on the Donnell-Mushtari-Vlasov theory for thin anisotropic shells, is presented. A complete set of field equations for geometrically imperfect shells is presented in terms general of lines-of-curvature coordinates. A systematic nondimensionalization of these equations is developed, several new nondimensional parameters are defined, and a comprehensive stress-function formulation is presented that includes variational principles for equilibrium and compatibility. Bifurcation analysis is applied to the nondimensional nonlinear field equations and a comprehensive set of bifurcation equations are presented. An extensive collection of tables and figures are presented that show the effects of lamina material properties and stacking sequence on the nondimensional parameters.

Simple diffusion can support the pitchfork, the flip bifurcations, and the chaos

NASA Astrophysics Data System (ADS)

Meng, Lili; Li, Xinfu; Zhang, Guang

2017-12-01

In this paper, a discrete rational fration population model with the Dirichlet boundary conditions will be considered. According to the discrete maximum principle and the sub- and supper-solution method, the necessary and sufficient conditions of uniqueness and existence of positive steady state solutions will be obtained. In addition, the dynamical behavior of a special two patch metapopulation model is investigated by using the bifurcation method, the center manifold theory, the bifurcation diagrams and the largest Lyapunov exponent. The results show that there exist the pitchfork, the flip bifurcations, and the chaos. Clearly, these phenomena are caused by the simple diffusion. The theoretical analysis of chaos is very imortant, unfortunately, there is not any results in this hand. However, some open problems are given.

Bifurcations in a discrete time model composed of Beverton-Holt function and Ricker function.

PubMed

Shang, Jin; Li, Bingtuan; Barnard, Michael R

2015-05-01

We provide rigorous analysis for a discrete-time model composed of the Ricker function and Beverton-Holt function. This model was proposed by Lewis and Li [Bull. Math. Biol. 74 (2012) 2383-2402] in the study of a population in which reproduction occurs at a discrete instant of time whereas death and competition take place continuously during the season. We show analytically that there exists a period-doubling bifurcation curve in the model. The bifurcation curve divides the parameter space into the region of stability and the region of instability. We demonstrate through numerical bifurcation diagrams that the regions of periodic cycles are intermixed with the regions of chaos. We also study the global stability of the model. Copyright © 2015 Elsevier Inc. All rights reserved.

Interaction of pulsating and spinning waves in condensed phase combustion

DOE Office of Scientific and Technical Information (OSTI.GOV)

Booty, M.R.; Margolis, S.B.; Matkowsky, B.J.

1986-10-01

The authors employ a nonlinear stability analysis in the neighborhood of a multiple bifurcation point to describe the interaction of pulsating and spinning modes of condensed phase combustion. Such phenomena occur in the synthesis of refractory materials. In particular, they consider the propagation of combustion waves in a long thermally insulated cylindrical sample and show that steady, planar combustion is stable for a modified activation energy/melting parameter less than a critical value. Above this critical value primary bifurcation states, corresponding to time-periodic pulsating and spinning modes of combustion, emanate from the steadily propagating solution. By varying the sample radius, themore » authors split a multiple bifurcation point to obtain bifurcation diagrams which exhibit secondary, tertiary, and quarternary branching to various types of quasi-periodic combustion waves.« less

Bifurcation Analysis Using Rigorous Branch and Bound Methods

NASA Technical Reports Server (NTRS)

Smith, Andrew P.; Crespo, Luis G.; Munoz, Cesar A.; Lowenberg, Mark H.

2014-01-01

For the study of nonlinear dynamic systems, it is important to locate the equilibria and bifurcations occurring within a specified computational domain. This paper proposes a new approach for solving these problems and compares it to the numerical continuation method. The new approach is based upon branch and bound and utilizes rigorous enclosure techniques to yield outer bounding sets of both the equilibrium and local bifurcation manifolds. These sets, which comprise the union of hyper-rectangles, can be made to be as tight as desired. Sufficient conditions for the existence of equilibrium and bifurcation points taking the form of algebraic inequality constraints in the state-parameter space are used to calculate their enclosures directly. The enclosures for the bifurcation sets can be computed independently of the equilibrium manifold, and are guaranteed to contain all solutions within the computational domain. A further advantage of this method is the ability to compute a near-maximally sized hyper-rectangle of high dimension centered at a fixed parameter-state point whose elements are guaranteed to exclude all bifurcation points. This hyper-rectangle, which requires a global description of the bifurcation manifold within the computational domain, cannot be obtained otherwise. A test case, based on the dynamics of a UAV subject to uncertain center of gravity location, is used to illustrate the efficacy of the method by comparing it with numerical continuation and to evaluate its computational complexity.

Determination of Phonation Instability Pressure and Phonation Pressure Range in Excised Larynges

ERIC Educational Resources Information Center

Zhang, Yu; Reynders, William J.; Jiang, Jack J.; Tateya, Ichiro

2007-01-01

Purpose: The present study was a methodological study designed to reveal the dynamic mechanisms of phonation instability pressure (PIP) using bifurcation analysis. Phonation pressure range (PPR) was also proposed for assessing the pressure range of normal vocal fold vibrations. Method: The authors first introduced the concept of bifurcation on the…

Explicit Solutions and Bifurcations for a Class of Generalized Boussinesq Wave Equation

NASA Astrophysics Data System (ADS)

Ma, Zhi-Min; Sun, Yu-Huai; Liu, Fu-Sheng

2013-03-01

In this paper, the generalized Boussinesq wave equation utt — uxx + a(um)xx + buxxxx = 0 is investigated by using the bifurcation theory and the method of phase portraits analysis. Under the different parameter conditions, the exact explicit parametric representations for solitary wave solutions and periodic wave solutions are obtained.

Pest control through viral disease: mathematical modeling and analysis.

PubMed

Bhattacharyya, S; Bhattacharya, D K

2006-01-07

This paper deals with the mathematical modeling of pest management under viral infection (i.e. using viral pesticide) and analysis of its essential mathematical features. As the viral infection induces host lysis which releases more virus into the environment, on the average 'kappa' viruses per host, kappain(1,infinity), the 'virus replication parameter' is chosen as the main parameter on which the dynamics of the infection depends. We prove that there exists a threshold value kappa(0) beyond which the endemic equilibrium bifurcates from the free disease one. Still for increasing kappa values, the endemic equilibrium bifurcates towards a periodic solution. We further analyse the orbital stability of the periodic orbits arising from bifurcation by applying Poor's condition. A concluding discussion with numerical simulation of the model is then presented.

Wiener-Hopf Approaches to Regulator, Filter/Observer, and Optimal Coupler Problems.

DTIC Science & Technology

1980-01-01

AD-A081 489 SYSTEMS TECHNOLOGY INC HAWTHORNE CALIF F/B 12/1 WIENER-HOPF APPROACHES TO REGULATOR, FILTER/OBSERVER, AND OPTTM--ETC(U) JAN 80 R F...821701 IN 1’ 41 Report onR-cR25-260-1 W R-EOPF APN9ACHES TO REGUATOR 7ILTR/OBSERVE, ɜAM OFZnAL COULR NDOM" Richard F. Whitbeck Az Systems Technology...NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT. TASK AREA S WORK UNIT NUMBERS Systems Technology, Inc. A/ 13766 S. Hawthorne Blvd. Hawthorne

Turing patterns in parabolic systems of conservation laws and numerically observed stability of periodic waves

NASA Astrophysics Data System (ADS)

Barker, Blake; Jung, Soyeun; Zumbrun, Kevin

2018-03-01

Turing patterns on unbounded domains have been widely studied in systems of reaction-diffusion equations. However, up to now, they have not been studied for systems of conservation laws. Here, we (i) derive conditions for Turing instability in conservation laws and (ii) use these conditions to find families of periodic solutions bifurcating from uniform states, numerically continuing these families into the large-amplitude regime. For the examples studied, numerical stability analysis suggests that stable periodic waves can emerge either from supercritical Turing bifurcations or, via secondary bifurcation as amplitude is increased, from subcritical Turing bifurcations. This answers in the affirmative a question of Oh-Zumbrun whether stable periodic solutions of conservation laws can occur. Determination of a full small-amplitude stability diagram - specifically, determination of rigorous Eckhaus-type stability conditions - remains an interesting open problem.

Analysis of non-Newtonian effects within an aorta-iliac bifurcation region.

PubMed

Iasiello, Marcello; Vafai, Kambiz; Andreozzi, Assunta; Bianco, Nicola

2017-11-07

The geometry of the arteries at or near arterial bifurcation influences the blood flow field, which is an important factor affecting arteriogenesis. The blood can act sometimes as a non-Newtonian fluid. However, many studies have argued that for large and medium arteries, the blood flow can be considered to be Newtonian. In this work a comprehensive investigation of non-Newtonian effects on the blood fluid dynamic behavior in an aorta-iliac bifurcation is presented. The aorta-iliac geometry is reconstructed with references to the values reported in Shah et al. (1978); the 3D geometrical model consists of three filleted cylinders of different diameters. Governing equations with the appropriate boundary conditions are solved with a finite-element code. Different rheological models are used for the blood flow through the lumen and detailed comparisons are presented for the aorta-iliac bifurcation. Results are presented in terms of the velocity profiles in the bifurcation zone and Wall Shear Stress (WSS) for different sides of the bifurcation both for male and female geometries, showing that the Newtonian fluid assumption can be made without any particular loss in terms of accuracy with respect to the other more complex rheological models. Copyright © 2017 Elsevier Ltd. All rights reserved.

Quantum field theory and coalgebraic logic in theoretical computer science.

PubMed

Basti, Gianfranco; Capolupo, Antonio; Vitiello, Giuseppe

2017-11-01

We suggest that in the framework of the Category Theory it is possible to demonstrate the mathematical and logical dual equivalence between the category of the q-deformed Hopf Coalgebras and the category of the q-deformed Hopf Algebras in quantum field theory (QFT), interpreted as a thermal field theory. Each pair algebra-coalgebra characterizes a QFT system and its mirroring thermal bath, respectively, so to model dissipative quantum systems in far-from-equilibrium conditions, with an evident significance also for biological sciences. Our study is in fact inspired by applications to neuroscience where the brain memory capacity, for instance, has been modeled by using the QFT unitarily inequivalent representations. The q-deformed Hopf Coalgebras and the q-deformed Hopf Algebras constitute two dual categories because characterized by the same functor T, related with the Bogoliubov transform, and by its contravariant application T op , respectively. The q-deformation parameter is related to the Bogoliubov angle, and it is effectively a thermal parameter. Therefore, the different values of q identify univocally, and label the vacua appearing in the foliation process of the quantum vacuum. This means that, in the framework of Universal Coalgebra, as general theory of dynamic and computing systems ("labelled state-transition systems"), the so labelled infinitely many quantum vacua can be interpreted as the Final Coalgebra of an "Infinite State Black-Box Machine". All this opens the way to the possibility of designing a new class of universal quantum computing architectures based on this coalgebraic QFT formulation, as its ability of naturally generating a Fibonacci progression demonstrates. Copyright © 2017 Elsevier Ltd. All rights reserved.

Period doubling cascades of limit cycles in cardiac action potential models as precursors to chaotic early Afterdepolarizations.

PubMed

Kügler, Philipp; Bulelzai, M A K; Erhardt, André H

2017-04-04

Early afterdepolarizations (EADs) are pathological voltage oscillations during the repolarization phase of cardiac action potentials (APs). EADs are caused by drugs, oxidative stress or ion channel disease, and they are considered as potential precursors to cardiac arrhythmias in recent attempts to redefine the cardiac drug safety paradigm. The irregular behaviour of EADs observed in experiments has been previously attributed to chaotic EAD dynamics under periodic pacing, made possible by a homoclinic bifurcation in the fast subsystem of the deterministic AP system of differential equations. In this article we demonstrate that a homoclinic bifurcation in the fast subsystem of the action potential model is neither a necessary nor a sufficient condition for the genesis of chaotic EADs. We rather argue that a cascade of period doubling (PD) bifurcations of limit cycles in the full AP system paves the way to chaotic EAD dynamics across a variety of models including a) periodically paced and spontaneously active cardiomyocytes, b) periodically paced and non-active cardiomyocytes as well as c) unpaced and spontaneously active cardiomyocytes. Furthermore, our bifurcation analysis reveals that chaotic EAD dynamics may coexist in a stable manner with fully regular AP dynamics, where only the initial conditions decide which type of dynamics is displayed. EADs are a potential source of cardiac arrhythmias and hence are of relevance both from the viewpoint of drug cardiotoxicity testing and the treatment of cardiomyopathies. The model-independent association of chaotic EADs with period doubling cascades of limit cycles introduced in this article opens novel opportunities to study chaotic EADs by means of bifurcation control theory and inverse bifurcation analysis. Furthermore, our results may shed new light on the synchronization and propagation of chaotic EADs in homogeneous and heterogeneous multicellular and cardiac tissue preparations.

Synchronization of oscillations in coupled multimode optoelectronic oscillators: bifurcation analysis

NASA Astrophysics Data System (ADS)

Balakin, M.; Gulyaev, A.; Kazaryan, A.; Yarovoy, O.

2018-04-01

We study influence of time delay in coupling on the dynamics of two coupled multimode optoelectronic oscillators. We reveal the structure of main synchronization region on the parameter plane and main bifurcations leading to synchronization and multistability formation. The dynamics of the system is studied in a wide range of values of control parameters.

DOE Office of Scientific and Technical Information (OSTI.GOV)

Saha, Debajyoti, E-mail: debajyoti.saha@saha.ac.in; Kumar Shaw, Pankaj; Janaki, M. S.

Order-chaos-order was observed in the relaxation oscillations of a glow discharge plasma with variation in the discharge voltage. The first transition exhibits an inverse homoclinic bifurcation followed by a homoclinic bifurcation in the second transition. For the two regimes of observations, a detailed analysis of correlation dimension, Lyapunov exponent, and Renyi entropy was carried out to explore the complex dynamics of the system.

Everolimus-eluting bioresorbable vascular scaffolds implanted in coronary bifurcation lesions: Impact of polymeric wide struts on side-branch impairment.

PubMed

De Paolis, Marcella; Felix, Cordula; van Ditzhuijzen, Nienke; Fam, Jiang Ming; Karanasos, Antonis; de Boer, Sanneke; van Mieghem, Nicolas M; Daemen, Joost; Costa, Francesco; Bergoli, Luis Carlos; Ligthart, Jurgen M R; Regar, Evelyn; de Jaegere, Peter P; Zijlstra, Felix; van Geuns, Robert Jan; Diletti, Roberto

2016-10-15

Limited data are available on bioresorbable vascular scaffolds (BVS) performance in bifurcations lesions and on the impact of BVS wider struts on side-branch impairment. Patients with at least one coronary bifurcation lesion involving a side-branch ≥2mm in diameter and treated with at least one BVS were examined. Procedural and angiographic data were collected and a dedicated methodology for off-line quantitative coronary angiography (QCA) in bifurcation was applied (eleven-segment model), to assess side-branch impairment occurring any time during the procedure. Two- and three-dimensional QCA were used. Optical coherence tomography (OCT) analysis was performed in a subgroup of patients and long-term clinical outcomes reported. A total of 102 patients with 107 lesions, were evaluated. Device- and procedural-successes were 99.1% and 94.3%, respectively. Side-branch impairment occurring any time during the procedure was reported in 13 bifurcations (12.1%) and at the end of the procedure in 6.5%. Side-branch minimal lumen diameter (Pre: 1.45±0.41mm vs Final: 1.48±0.42mm, p=0.587) %diameter-stenosis (Pre: 26.93±16.89% vs Final: 27.80±15.57%, p=0.904) and minimal lumen area (Pre: 1.97±0.89mm(2) vs Final: 2.17±1.09mm(2), p=0.334), were not significantly affected by BVS implantation. Mean malapposed struts at the bifurcation polygon-of-confluence were 0.63±1.11. The results of the present investigation suggest feasibility and relative safety of BVS implantation in coronary bifurcations. BVS wide struts have a low impact on side-branch impairment when considering bifurcations with side-branch diameter≥2mm. Copyright © 2016. Published by Elsevier Ireland Ltd.

Concentration and limit behaviors of stationary measures

NASA Astrophysics Data System (ADS)

Huang, Wen; Ji, Min; Liu, Zhenxin; Yi, Yingfei

2018-04-01

In this paper, we study limit behaviors of stationary measures of the Fokker-Planck equations associated with a system of ordinary differential equations perturbed by a class of multiplicative noise including additive white noise case. As the noises are vanishing, various results on the invariance and concentration of the limit measures are obtained. In particular, we show that if the noise perturbed systems admit a uniform Lyapunov function, then the stationary measures form a relatively sequentially compact set whose weak∗-limits are invariant measures of the unperturbed system concentrated on its global attractor. In the case that the global attractor contains a strong local attractor, we further show that there exists a family of admissible multiplicative noises with respect to which all limit measures are actually concentrated on the local attractor; and on the contrary, in the presence of a strong local repeller in the global attractor, there exists a family of admissible multiplicative noises with respect to which no limit measure can be concentrated on the local repeller. Moreover, we show that if there is a strongly repelling equilibrium in the global attractor, then limit measures with respect to typical families of multiplicative noises are always concentrated away from the equilibrium. As applications of these results, an example of stochastic Hopf bifurcation and an example with non-decomposable ω-limit sets are provided. Our study is closely related to the problem of noise stability of compact invariant sets and invariant measures of the unperturbed system.

Causes and consequences of complex population dynamics in an annual plant, Cardamine pensylvanica

DOE Office of Scientific and Technical Information (OSTI.GOV)

Crone, E.E.

1995-11-08

The relative importance of density-dependent and density-independent factors in determining the population dynamics of plants has been widely debated with little resolution. In this thesis, the author explores the effects of density-dependent population regulation on population dynamics in Cardamine pensylvanica, an annual plant. In the first chapter, she shows that experimental populations of C. pensylvanica cycled from high to low density in controlled constant-environment conditions. These cycles could not be explained by external environmental changes or simple models of direct density dependence (N{sub t+1} = f[N{sub t}]), but they could be explained by delayed density dependence (N{sub t+1} = f[N{submore » t}, N{sub t+1}]). In the second chapter, she shows that the difference in the stability properties of population growth models with and without delayed density dependence is due to the presence of Hopf as well as slip bifurcations from stable to chaotic population dynamics. She also measures delayed density dependence due to effects of parental density on offspring quality in C. pensylvanica and shows that this is large enough to be the cause of the population dynamics observed in C. pensylvanica. In the third chapter, the author extends her analyses of density-dependent population growth models to include interactions between competing species. In the final chapter, she compares the effects of fixed spatial environmental variation and variation in population size on the evolutionary response of C. pensylvanica populations.« less

A Mixed Mode Cochlear Amplifier Including Neural Feedback

NASA Astrophysics Data System (ADS)

Flax, Matthew R.; Holmes, W. Harvey

2011-11-01

The mixed mode cochlear amplifier (MMCA) model is derived from the physiology of the cochlea. It is comprised of three main elements of the peripheral hearing system: the cochlear mechanics, hair cell motility, and neurophysiology. This model expresses both active compression wave and active traveling wave modes of operation. The inclusion of a neural loop with a time delay, and a new paradigm for the mechanical response of the outer hair cells, are believed to be unique features of the MMCA. These elements combine to form an active feedback loop to constitute the cochlear amplifier, whose input is a passive traveling wave vibration. The result is a cycle-by-cycle amplifier with nonlinear response. This system can assume an infinite number of different operating states. The stable state and the first few amplitude-limited unstable (Hopf-bifurcated) states are significant in describing the operation of the peripheral hearing system. A hierarchy of models can be constructed from this concept, depending on the amount of detail included. The simplest model of the MMCA is a nonlinear delay line resonator. It was found that even this simple MMCA version can explain a large number of hearing phenomena, at least qualitatively. This paper concentrates on explaining the fractional octave shift from the living to postmortem response in terms of the new model. Other mechanical, hair cell and neurological phenomena can also be accounted for by the MMCA, including two-tone suppression behavior, distortion product responses, otoacoustic emissions and neural spontaneous rates.

Bifurcation analysis of the regulation of nociceptive neuronal activity

NASA Astrophysics Data System (ADS)

Dik, O. E.

2017-11-01

A model of the membrane of a nociceptive neuron from a rat dorsal ganglion has been used to address the problem of analyzing the regulation of nociceptive signals by 5-hydroxy-γ-pyrone-2-carboxylic acid, which is the active pharmaceutic ingredient of the analgesic Anoceptin. The study has applied bifurcation analysis to report the relationship between the values of model parameters and the type of problem solution before and after the parameters change in response to analgesic modulation.

High-resolution mapping of bifurcations in nonlinear biochemical circuits

NASA Astrophysics Data System (ADS)

Genot, A. J.; Baccouche, A.; Sieskind, R.; Aubert-Kato, N.; Bredeche, N.; Bartolo, J. F.; Taly, V.; Fujii, T.; Rondelez, Y.

2016-08-01

Analog molecular circuits can exploit the nonlinear nature of biochemical reaction networks to compute low-precision outputs with fewer resources than digital circuits. This analog computation is similar to that employed by gene-regulation networks. Although digital systems have a tractable link between structure and function, the nonlinear and continuous nature of analog circuits yields an intricate functional landscape, which makes their design counter-intuitive, their characterization laborious and their analysis delicate. Here, using droplet-based microfluidics, we map with high resolution and dimensionality the bifurcation diagrams of two synthetic, out-of-equilibrium and nonlinear programs: a bistable DNA switch and a predator-prey DNA oscillator. The diagrams delineate where function is optimal, dynamics bifurcates and models fail. Inverse problem solving on these large-scale data sets indicates interference from enzymatic coupling. Additionally, data mining exposes the presence of rare, stochastically bursting oscillators near deterministic bifurcations.

An Unexpected Detection of Bifurcated Blue Straggler Sequences in the Young Globular Cluster NGC 2173

NASA Astrophysics Data System (ADS)

Li, Chengyuan; Deng, Licai; de Grijs, Richard; Jiang, Dengkai; Xin, Yu

2018-03-01

The bifurcated patterns in the color–magnitude diagrams of blue straggler stars (BSSs) have attracted significant attention. This type of special (but rare) pattern of two distinct blue straggler sequences is commonly interpreted as evidence that cluster core-collapse-driven stellar collisions are an efficient formation mechanism. Here, we report the detection of a bifurcated blue straggler distribution in a young Large Magellanic Cloud cluster, NGC 2173. Because of the cluster’s low central stellar number density and its young age, dynamical analysis shows that stellar collisions alone cannot explain the observed BSSs. Therefore, binary evolution is instead the most viable explanation of the origin of these BSSs. However, the reason why binary evolution would render the color–magnitude distribution of BSSs bifurcated remains unclear. C. Li, L. Deng, and R. de Grijs jointly designed this project.

Control-based continuation: Bifurcation and stability analysis for physical experiments

NASA Astrophysics Data System (ADS)

Barton, David A. W.

2017-02-01

Control-based continuation is technique for tracking the solutions and bifurcations of nonlinear experiments. The idea is to apply the method of numerical continuation to a feedback-controlled physical experiment such that the control becomes non-invasive. Since in an experiment it is not (generally) possible to set the state of the system directly, the control target becomes a proxy for the state. Control-based continuation enables the systematic investigation of the bifurcation structure of a physical system, much like if it was numerical model. However, stability information (and hence bifurcation detection and classification) is not readily available due to the presence of stabilising feedback control. This paper uses a periodic auto-regressive model with exogenous inputs (ARX) to approximate the time-varying linearisation of the experiment around a particular periodic orbit, thus providing the missing stability information. This method is demonstrated using a physical nonlinear tuned mass damper.

Interaction of pulsating and spinning waves in nonadiabatic flame propagation

DOE Office of Scientific and Technical Information (OSTI.GOV)

Booty, M.R.; Margolis, S.B.; Matkowsky, B.J.

1987-12-01

The authors consider nonadiabatic premixed flame propagation in a long cylindrical channel. A steadily propagating planar flame exists for heat losses below a critical value. It is stable provided that the Lewis number and the volumetric heat loss coefficient are sufficiently small. At critical values of these parameters, bifurcated states, corresponding to time-periodic pulsating cellular flames, emanate from the steadily propagating solution. The authors analyze the problem in a neighborhood of a multiple primary bifurcation point. By varying the radius of the channel, they split the multiple bifurcation point and show that various types of stable periodic and quasi-periodic pulsatingmore » flames can arise as secondary, tertiary, and quaternary bifurcations. Their analysis describes several types of spinning and pulsating flame propagation which have been experimentally observed in nonadiabatic flames, and also describes additional quasi-periodic modes of burning which have yet to be documented experimentally.« less

Experimental Tracking of Limit-Point Bifurcations and Backbone Curves Using Control-Based Continuation

NASA Astrophysics Data System (ADS)

Renson, Ludovic; Barton, David A. W.; Neild, Simon A.

Control-based continuation (CBC) is a means of applying numerical continuation directly to a physical experiment for bifurcation analysis without the use of a mathematical model. CBC enables the detection and tracking of bifurcations directly, without the need for a post-processing stage as is often the case for more traditional experimental approaches. In this paper, we use CBC to directly locate limit-point bifurcations of a periodically forced oscillator and track them as forcing parameters are varied. Backbone curves, which capture the overall frequency-amplitude dependence of the system’s forced response, are also traced out directly. The proposed method is demonstrated on a single-degree-of-freedom mechanical system with a nonlinear stiffness characteristic. Results are presented for two configurations of the nonlinearity — one where it exhibits a hardening stiffness characteristic and one where it exhibits softening-hardening.

Resonant Mode Interaction in Rayleigh-Benard Convection: Hexagon Case

NASA Astrophysics Data System (ADS)

Moroz, Vadim

2002-11-01

Resonant interaction of the "main" convection mode (q) with the first harmonic (q=sqrt(3)*q) is studied in the regime where both modes are (close to being) linearly unstable. The bifurcations and the resulting family of solutions are classified using symmetry group analysis; dependence of coefficients in amplitude equations on the fluid properties is analyzed. Main focus is made on the bifurcation from hex pattern with wavevector q*sqrt(3) to hex pattern q. Particular experimental conditions under which the bifurcation could be observed (as well as existence of other instabilities which could preempt it) are stated. Helpful discussions with Prof. Hermann Riecke are gladly acknowledged.

Analysis of red blood cell partitioning at bifurcations in simulated microvascular networks

NASA Astrophysics Data System (ADS)

Balogh, Peter; Bagchi, Prosenjit

2018-05-01

Partitioning of red blood cells (RBCs) at vascular bifurcations has been studied over many decades using in vivo, in vitro, and theoretical models. These studies have shown that RBCs usually do not distribute to the daughter vessels with the same proportion as the blood flow. Such disproportionality occurs, whereby the cell distribution fractions are either higher or lower than the flow fractions and have been referred to as classical partitioning and reverse partitioning, respectively. The current work presents a study of RBC partitioning based on, for the first time, a direct numerical simulation (DNS) of a flowing cell suspension through modeled vascular networks that are comprised of multiple bifurcations and have topological similarity to microvasculature in vivo. The flow of deformable RBCs at physiological hematocrits is considered through the networks, and the 3D dynamics of each individual cell are accurately resolved. The focus is on the detailed analysis of the partitioning, based on the DNS data, as it develops naturally in successive bifurcations, and the underlying mechanisms. We find that while the time-averaged partitioning at a bifurcation manifests in one of two ways, namely, the classical or reverse partitioning, the time-dependent behavior can cycle between these two types. We identify and analyze four different cellular-scale mechanisms underlying the time-dependent partitioning. These mechanisms arise, in general, either due to an asymmetry in the RBC distribution in the feeding vessels caused by the events at an upstream bifurcation or due to a temporary increase in cell concentration near capillary bifurcations. Using the DNS results, we show that a positive skewness in the hematocrit profile in the feeding vessel is associated with the classical partitioning, while a negative skewness is associated with the reverse one. We then present a detailed analysis of the two components of disproportionate partitioning as identified in prior studies, namely, plasma skimming and cell screening. The plasma skimming component is shown to under-predict the disproportionality, leaving the cell screening component to make up for the difference. The crossing of the separation surface by the cells is observed to be a dominant mechanism underlying the cell screening, which is shown to mitigate extreme heterogeneity in RBC distribution across the networks.

The inner topological structure and defect control of magnetic skyrmions

NASA Astrophysics Data System (ADS)

Ren, Ji-Rong; Yu, Zhong-Xi

2017-10-01

We prove that the integrand of magnetic skyrmions can be expressed as curvature tensor of Wu-Yang potential. Taking the projection of the normalized magnetization vector on the 2-dim material surface, and according to Duan's decomposition theory of gauge potential, we reveal that every single skyrmion is just characterized by Hopf index and Brouwer degree at the zero point of this vector field. Our theory meet the results that experimental physicists have achieved by many technologies. The inner topological structure expression of skyrmion with Hopf index and Brouwer degree will be indispensable mathematical basis of skyrmion logic gates.

Towards an emergent model of solitonic particles from non-trivial vacuum structure

NASA Astrophysics Data System (ADS)

Gillard, Adam B.; Gresnigt, Niels G.

2017-12-01

We motivate and introduce what we refer to as the principles of Lie-stability and Hopf-stability and see what the physical theories must look like. Lie-stability is needed on the classical side and Hopf-stability is needed on the quantum side. We implement these two principles together with Lie-deformations consistent with basic constraints on the classical kinematical variables to arrive at the form of a theory that identifies standard model fermions with quantum solitonic trefoil knotted flux tubes which emerge from a flux tube vacuum network. Moreover, twisted unknot fluxtubes form natural dark matter candidates

Breakdown of a 2D Heteroclinic Connection in the Hopf-Zero Singularity (I)

NASA Astrophysics Data System (ADS)

Baldomá, I.; Castejón, O.; Seara, T. M.

2018-04-01

In this paper we study a beyond all orders phenomenon which appears in the analytic unfoldings of the Hopf-zero singularity. It consists in the breakdown of a two-dimensional heteroclinic surface which exists in the truncated normal form of this singularity at any order. The results in this paper are twofold: on the one hand, we give results for generic unfoldings which lead to sharp exponentially small upper bounds of the difference between these manifolds. On the other hand, we provide asymptotic formulas for this difference by means of the Melnikov function for some non-generic unfoldings.

Quantum Double of Yangian of strange Lie superalgebra Qn and multiplicative formula for universal R-matrix

NASA Astrophysics Data System (ADS)

Stukopin, Vladimir

2018-02-01

Main result is the multiplicative formula for universal R-matrix for Quantum Double of Yangian of strange Lie superalgebra Qn type. We introduce the Quantum Double of the Yangian of the strange Lie superalgebra Qn and define its PBW basis. We compute the Hopf pairing for the generators of the Yangian Double. From the Hopf pairing formulas we derive a factorized multiplicative formula for the universal R-matrix of the Yangian Double of the Lie superalgebra Qn . After them we obtain coefficients in this multiplicative formula for universal R-matrix.

An investigation of correlation between left coronary bifurcation angle and hemodynamic changes in coronary stenosis by coronary computed tomography angiography-derived computational fluid dynamics

PubMed Central

Chaichana, Thanapong

2017-01-01

Background To investigate the correlation between left coronary bifurcation angle and coronary stenosis as assessed by coronary computed tomography angiography (CCTA)-generated computational fluid dynamics (CFD) analysis when compared to the CCTA analysis of coronary lumen stenosis and plaque lesion length with invasive coronary angiography (ICA) as the reference method. Methods Thirty patients (22 males, mean age: 59±6.9 years) with calcified plaques at the left coronary artery were included in the study with all patients undergoing CCTA and ICA examinations. CFD simulation was performed to analyze hemodynamic changes to the left coronary artery models in terms of wall shear stress, wall pressure and flow velocity, with findings correlated to the coronary stenosis and degree of bifurcation angle. Calcified plaque length was measured in the left coronary artery with diagnostic value compared to that from coronary lumen and bifurcation angle assessments. Results Of 26 significant stenosis at left anterior descending (LAD) and 13 at left circumflex (LCx) on CCTA, only 14 and 5 of them were confirmed to be >50% stenosis at LAD and LCx respectively on ICA, resulting in sensitivity, specificity, positive predictive value (PPV) and negative predictive value (NPV) of 100%, 52%, 49% and 100%. The mean plaque length was measured 5.3±3.6 and 4.4±1.9 mm at LAD and LCx, respectively, with diagnostic sensitivity, specificity, PPV and NPV being 92.8%, 46.7%, 61.9% and 87.5% for extensively calcified plaques. The mean bifurcation angle was measured 83.9±13.6º and 83.8±13.3º on CCTA and ICA, respectively, with no significant difference (P=0.98). The corresponding sensitivity, specificity, PPV and NPV were 100%, 78.6%, 84.2% and 100% based on bifurcation angle measurement on CCTA, 100%, 73.3%, 78.9% and 100% based on bifurcation angle measurements on ICA, respectively. Wall shear stress was noted to increase in the LAD and LCx models with significant stenosis and wider angulation (>80º), but demonstrated little or no change in most of the coronary models with no significant stenosis and narrower angulation (<80º). Conclusions This study further clarifies the relationship between left coronary bifurcation angle and significant stenosis, with angulation measurement serving as a more accurate approach than coronary lumen assessment or plaque lesion length for determining significant coronary stenosis. Left coronary bifurcation angle is suggested to be incorporated into coronary artery disease (CAD) assessment when diagnosing significant CAD. PMID:29184766

Using CONTENT 1.5 to analyze an SIR model for childhood infectious diseases

NASA Astrophysics Data System (ADS)

Su, Rui; He, Daihai

2008-11-01

In this work, we introduce a standard software CONTENT 1.5 for analysis of dynamical systems. A simple model for childhood infectious diseases is used as an example. The detailed steps to obtain the bifurcation structures of the system are given. These bifurcation structures can be used to explain the observed dynamical transition in measles incidences.

Network inoculation: Heteroclinics and phase transitions in an epidemic model

NASA Astrophysics Data System (ADS)

Yang, Hui; Rogers, Tim; Gross, Thilo

2016-08-01

In epidemiological modelling, dynamics on networks, and, in particular, adaptive and heterogeneous networks have recently received much interest. Here, we present a detailed analysis of a previously proposed model that combines heterogeneity in the individuals with adaptive rewiring of the network structure in response to a disease. We show that in this model, qualitative changes in the dynamics occur in two phase transitions. In a macroscopic description, one of these corresponds to a local bifurcation, whereas the other one corresponds to a non-local heteroclinic bifurcation. This model thus provides a rare example of a system where a phase transition is caused by a non-local bifurcation, while both micro- and macro-level dynamics are accessible to mathematical analysis. The bifurcation points mark the onset of a behaviour that we call network inoculation. In the respective parameter region, exposure of the system to a pathogen will lead to an outbreak that collapses but leaves the network in a configuration where the disease cannot reinvade, despite every agent returning to the susceptible class. We argue that this behaviour and the associated phase transitions can be expected to occur in a wide class of models of sufficient complexity.

Numerical bifurcation analysis of immunological models with time delays

NASA Astrophysics Data System (ADS)

Luzyanina, Tatyana; Roose, Dirk; Bocharov, Gennady

2005-12-01

In recent years, a large number of mathematical models that are described by delay differential equations (DDEs) have appeared in the life sciences. To analyze the models' dynamics, numerical methods are necessary, since analytical studies can only give limited results. In turn, the availability of efficient numerical methods and software packages encourages the use of time delays in mathematical modelling, which may lead to more realistic models. We outline recently developed numerical methods for bifurcation analysis of DDEs and illustrate the use of these methods in the analysis of a mathematical model of human hepatitis B virus infection.

Delay differential equations via the matrix Lambert W function and bifurcation analysis: application to machine tool chatter.

PubMed

Yi, Sun; Nelson, Patrick W; Ulsoy, A Galip

2007-04-01

In a turning process modeled using delay differential equations (DDEs), we investigate the stability of the regenerative machine tool chatter problem. An approach using the matrix Lambert W function for the analytical solution to systems of delay differential equations is applied to this problem and compared with the result obtained using a bifurcation analysis. The Lambert W function, known to be useful for solving scalar first-order DDEs, has recently been extended to a matrix Lambert W function approach to solve systems of DDEs. The essential advantages of the matrix Lambert W approach are not only the similarity to the concept of the state transition matrix in lin ear ordinary differential equations, enabling its use for general classes of linear delay differential equations, but also the observation that we need only the principal branch among an infinite number of roots to determine the stability of a system of DDEs. The bifurcation method combined with Sturm sequences provides an algorithm for determining the stability of DDEs without restrictive geometric analysis. With this approach, one can obtain the critical values of delay, which determine the stability of a system and hence the preferred operating spindle speed without chatter. We apply both the matrix Lambert W function and the bifurcation analysis approach to the problem of chatter stability in turning, and compare the results obtained to existing methods. The two new approaches show excellent accuracy and certain other advantages, when compared to traditional graphical, computational and approximate methods.

Cutting Balloon Angioplasty in the Treatment of Short Infrapopliteal Bifurcation Disease.

PubMed

Iezzi, Roberto; Posa, Alessandro; Santoro, Marco; Nestola, Massimiliano; Contegiacomo, Andrea; Tinelli, Giovanni; Paolini, Alessandra; Flex, Andrea; Pitocco, Dario; Snider, Francesco; Bonomo, Lorenzo

2015-08-01

To evaluate the safety, feasibility, and effectiveness of cutting balloon angioplasty in the management of infrapopliteal bifurcation disease. Between November 2010 and March 2013, 23 patients (mean age 69.6±9.01 years, range 56-89; 16 men) suffering from critical limb ischemia were treated using cutting balloon angioplasty (single cutting balloon, T-shaped double cutting balloon, or double kissing cutting balloon technique) for 47 infrapopliteal artery bifurcation lesions (16 popliteal bifurcation and 9 tibioperoneal bifurcation) in 25 limbs. Follow-up consisted of clinical examination and duplex ultrasonography at 1 month and every 3 months thereafter. All treatments were technically successful. No 30-day death or adverse events needing treatment were registered. No flow-limiting dissection was observed, so no stent implantation was necessary. The mean postprocedure minimum lumen diameter and acute gain were 0.28±0.04 and 0.20±0.06 cm, respectively, with a residual stenosis of 0.04±0.02 cm. Primary and secondary patency rates were estimated as 89.3% and 93.5% at 6 months and 77.7% and 88.8% at 12 months, respectively; 1-year primary and secondary patency rates of the treated bifurcation were 74.2% and 87.0%, respectively. The survival rate estimated by Kaplan-Meier analysis was 82.5% at 1 year. Cutting balloon angioplasty seems to be a safe and effective tool in the routine treatment of short/ostial infrapopliteal bifurcation lesions, avoiding procedure-related complications, overcoming the limitations of conventional angioplasty, and improving the outcome of catheter-based therapy. © The Author(s) 2015.

Isotope effect in normal-to-local transition of acetylene bending modes

DOE PAGES

Ma, Jianyi; Xu, Dingguo; Guo, Hua; ...

2012-01-01

The normal-to-local transition for the bending modes of acetylene is considered a prelude to its isomerization to vinylidene. Here, such a transition in fully deuterated acetylene is investigated using a full-dimensional quantum model. It is found that the local benders emerge at much lower energies and bending quantum numbers than in the hydrogen isotopomer HCCH. This is accompanied by a transition to a second kind of bending mode called counter-rotator, again at lower energies and quantum numbers than in HCCH. These transitions are also investigated using bifurcation analysis of two empirical spectroscopic fitting Hamiltonians for pure bending modes, which helpsmore » to understand the origin of the transitions semiclassically as branchings or bifurcations out of the trans and normal bend modes when the latter become dynamically unstable. The results of the quantum model and the empirical bifurcation analysis are in very good agreement.« less

Mathematical model and stability analysis of fluttering and autorotation of an articulated plate into a flow

NASA Astrophysics Data System (ADS)

Rostami, Ali Bakhshandeh; Fernandes, Antonio Carlos

2018-03-01

This paper is dedicated to develop a mathematical model that can simulate nonlinear phenomena of a hinged plate which places into the fluid flow (1 DOF). These phenomena are fluttering (oscillation motion), autorotation (continuous rotation) and chaotic motion (combination of fluttering and autorotation). Two mathematical models are developed for 1 DOF problem using two eminent mathematical models which had been proposed for falling plates (3 DOF). The procedures of developing these models are elaborated and then these results are compared to experimental data. The best model in the simulation of the phenomena is chosen for stability and bifurcation analysis. Based on these analyses, this model shows a transcritical bifurcation and as a result, the stability diagram and threshold are presented. Moreover, an analytical expression is given for finding the boundary of bifurcation from the fluttering to the autorotation.

Compressive Failure of Fiber Composites under Multi-Axial Loading

NASA Technical Reports Server (NTRS)

Basu, Shiladitya; Waas, Anthony M.; Ambur, Damodar R.

2006-01-01

This paper examines the compressive strength of a fiber reinforced lamina under multi-axial stress states. An equilibrium analysis is carried out in which a kinked band of rotated fibers, described by two angles, is sandwiched between two regions in which the fibers are nominally straight. Proportional multi-axial stress states are examined. The analysis includes the possibility of bifurcation from the current equilibrium state. The compressive strength of the lamina is contingent upon either attaining a load maximum in the equilibrium response or satisfaction of a bifurcation condition, whichever occurs first. The results show that for uniaxial loading a non-zero kink band angle beta produces the minimum limit load. For multi-axial loading, different proportional loading paths show regimes of bifurcation dominated and limit load dominated behavior. The present results are able to capture the beneficial effect of transverse compression in raising the composite compressive strength as observed in experiments.

Theory of freezing in simple systems

DOE Office of Scientific and Technical Information (OSTI.GOV)

Cerjan, C.; Bagchi, B.

The transition parameters for the freezing of two one-component liquids into crystalline solids are evaluated by two theoretical approaches. The first system considered is liquid sodium which crystallizes into a body-centered-cubic (bcc) lattice; the second system is the freezing of adhesive hard spheres into a face-centered-cubic (fcc) lattice. Two related theoretical techniques are used in this evaluation: One is based upon a recently developed bifurcation analysis; the other is based upon the theory of freezing developed by Ramakrishnan and Yussouff. For liquid sodium, where experimental information is available, the predictions of the two theories agree well with experiment and eachmore » other. The adhesive-hard-sphere system, which displays a triple point and can be used to fit some liquids accurately, shows a temperature dependence of the freezing parameters which is similar to Lennard-Jones systems. At very low temperature, the fractional density change on freezing shows a dramatic increase as a function of temperature indicating the importance of all the contributions due to the triplet direction correlation function. Also, we consider the freezing of a one-component liquid into a simple-cubic (sc) lattice by bifurcation analysis and show that this transition is highly unfavorable, independent of interatomic potential choice. The bifurcation diagrams for the three lattices considered are compared and found to be strikingly different. Finally, a new stability analysis of the bifurcation diagrams is presented.« less

Underlying Mechanisms of Cooperativity, Input Specificity, and Associativity of Long-Term Potentiation Through a Positive Feedback of Local Protein Synthesis.

PubMed

Hao, Lijie; Yang, Zhuoqin; Lei, Jinzhi

2018-01-01

Long-term potentiation (LTP) is a specific form of activity-dependent synaptic plasticity that is a leading mechanism of learning and memory in mammals. The properties of cooperativity, input specificity, and associativity are essential for LTP; however, the underlying mechanisms are unclear. Here, based on experimentally observed phenomena, we introduce a computational model of synaptic plasticity in a pyramidal cell to explore the mechanisms responsible for the cooperativity, input specificity, and associativity of LTP. The model is based on molecular processes involved in synaptic plasticity and integrates gene expression involved in the regulation of neuronal activity. In the model, we introduce a local positive feedback loop of protein synthesis at each synapse, which is essential for bimodal response and synapse specificity. Bifurcation analysis of the local positive feedback loop of brain-derived neurotrophic factor (BDNF) signaling illustrates the existence of bistability, which is the basis of LTP induction. The local bifurcation diagram provides guidance for the realization of LTP, and the projection of whole system trajectories onto the two-parameter bifurcation diagram confirms the predictions obtained from bifurcation analysis. Moreover, model analysis shows that pre- and postsynaptic components are required to achieve the three properties of LTP. This study provides insights into the mechanisms underlying the cooperativity, input specificity, and associativity of LTP, and the further construction of neural networks for learning and memory.

Possibility of Atherosclerosis in an Arterial Bifurcation Model

PubMed Central

Arjmandi-Tash, Omid; Razavi, Seyed Esmail; Zanbouri, Ramin

2011-01-01

Introduction Arterial bifurcations are susceptible locations for formation of atherosclerotic plaques. In the present study, steady blood flow is investigated in a bifurcation model with a non-planar branch. Methods The influence of different bifurcation angles and non-planar branch is demonstrated on wall shear stress (WSS) distribution using three-dimensional Navier–Stokes equations. Results The WSS values are low in two locations at the top and bottom walls of the mother vessels just before the bifurcation, especially for higher bifurcation angles. These regions approach the apex of bifurcation with decreasing the bifurcation angle. The WSS magnitudes approach near to zero at the outer side of bifurcation plane and these locations are separation-prone. By increasing the bifurcation angle, the minimum WSS decreases at the outer side of bifurcation plane but low WSS region squeezes. WSS peaks exist on the inner side of bifurcation plane near the entry section of daughter vessels and these initial peaks drop as bifurcation angle is increased. Conclusion It is concluded that the non-planarity of the daughter vessel lowers the minimum WSS at the outer side of bifurcation and increases the maximum WSS at the inner side. So it seems that the formation of atherosclerotic plaques at bifurcation region in direction of non-planar daughter vessel is more risky. PMID:23678432

Effects of Maximal Sodium and Potassium Conductance on the Stability of Hodgkin-Huxley Model

PubMed Central

Wang, Kuanquan; Yuan, Yongfeng; Zhang, Henggui

2014-01-01

Hodgkin-Huxley (HH) equation is the first cell computing model in the world and pioneered the use of model to study electrophysiological problems. The model consists of four differential equations which are based on the experimental data of ion channels. Maximal conductance is an important characteristic of different channels. In this study, mathematical method is used to investigate the importance of maximal sodium conductance g-Na and maximal potassium conductance g-K. Applying stability theory, and taking g-Na and g-K as variables, we analyze the stability and bifurcations of the model. Bifurcations are found when the variables change, and bifurcation points and boundary are also calculated. There is only one bifurcation point when g-Na is the variable, while there are two points when g-K is variable. The (g-Na,  g-K) plane is partitioned into two regions and the upper bifurcation boundary is similar to a line when both g-Na and g-K are variables. Numerical simulations illustrate the validity of the analysis. The results obtained could be helpful in studying relevant diseases caused by maximal conductance anomaly. PMID:25104970

Deterministic and stochastic bifurcations in the Hindmarsh-Rose neuronal model

NASA Astrophysics Data System (ADS)

Dtchetgnia Djeundam, S. R.; Yamapi, R.; Kofane, T. C.; Aziz-Alaoui, M. A.

2013-09-01

We analyze the bifurcations occurring in the 3D Hindmarsh-Rose neuronal model with and without random signal. When under a sufficient stimulus, the neuron activity takes place; we observe various types of bifurcations that lead to chaotic transitions. Beside the equilibrium solutions and their stability, we also investigate the deterministic bifurcation. It appears that the neuronal activity consists of chaotic transitions between two periodic phases called bursting and spiking solutions. The stochastic bifurcation, defined as a sudden change in character of a stochastic attractor when the bifurcation parameter of the system passes through a critical value, or under certain condition as the collision of a stochastic attractor with a stochastic saddle, occurs when a random Gaussian signal is added. Our study reveals two kinds of stochastic bifurcation: the phenomenological bifurcation (P-bifurcations) and the dynamical bifurcation (D-bifurcations). The asymptotical method is used to analyze phenomenological bifurcation. We find that the neuronal activity of spiking and bursting chaos remains for finite values of the noise intensity.

True coronary bifurcation lesions: meta-analysis and review of literature.

PubMed

Athappan, Ganesh; Ponniah, Thirumalaikolundiusubramanian; Jeyaseelan, Lakshmanan

2010-02-01

Percutaneous intervention of true coronary bifurcation lesions is challenging. Based on the results of randomized trials and registry data, the approach of stenting of main vessel only with balloon dilatation of the side branch has become the default approach for false bifurcation lesions except when a complication occurs or in cases of suboptimal result. However, the optimal stenting strategy for true coronary bifurcation lesions - to stent or not to stent the side branch - is still a matter of debate. The purpose of this study was, therefore, to compare the clinical and angiographic outcomes of the double stent technique (stenting of the main branch and side branch) over the single stent technique (stenting of main vessel only with balloon dilatation of the side branch) for treatment of true coronary bifurcation lesions, with drug-eluting stents (DES). Comparative studies published between January 2000 and February 2009 of the double stent technique vs. single stent technique with DES for true coronary bifurcations were identified using an electronic search and reviewed using a random effects model. The primary endpoints of our study were side-branch and main-branch restenoses, all-cause mortality, myocardial infarction (MI) and target lesion revascularization (TLR) at longest available follow-up. The secondary endpoints of our analysis were postprocedural minimal luminal diameter (MLD) of the side branch and main branch, follow-up MLD of side branch and main branch and stent thrombosis. Heterogeneity was assessed and sensitivity analysis was performed to test the robustness of the overall summary odds ratios (ORs). Five studies comprising 1145 patients (616 single stent and 529 double stent) were included in the analysis. Three studies were randomized comparisons between the two techniques for true coronary bifurcation lesions. Incomplete reporting of data in the primary studies was common. The lengths of clinical and angiographic follow-up ranged between 6 and 12 months and 6 and 7 months, respectively. Postprocedural MLD of the side branch was significantly smaller in the single stent group [standardized mean difference (SMD) -0.71, 95% CI -0.88 to -0.54, P < 0.000, I2 = 0%]. The odds of side-branch restenosis (OR 1.11, 95% CI 0.47-2.67, P = 0.81, I2 = 76%), main-branch restenois (OR 0.88, 95% CI 0.56-1.39, P = 0.58, I = 0%), all-cause mortality (OR 0.52, 95% CI 0.11-2.45, P = 0.41, I2 = 0%), MI (OR 0.92, 95% CI 0.34-2.54, P = 0.87, I = 49%) and TLR (OR 0.87, 95% CI 0.46-1.65, P = 0.68, I2 = 0%) were similar between the two groups. Postprocedural MLD of the main branch [standardized mean difference (SMD) -0.08, 95% CI -0.42 to -0.26, P < 0.65, I2 = 67%], follow-up MLD of side branch (SMD -0.19, 95% CI -0.40 to 0.01, P < 0.31, I2 = 15%) and main branch MLD (SMD 0.17, 95% CI -0.18 to 0.542, P < 0.35, I2 = 65%) were also similar between the two groups. In patients undergoing percutaneous coronary intervention (PCI) for true coronary bifurcations, there is no added advantage of stenting both branches as compared with a conventional one-stent strategy. The results, however, need to be interpreted considering the poor study methods and/or poor quality of reporting in publications. We propose to move forward and consider the conduct of more systematic, well-designed and scientific trials to investigate the treatment of true coronary bifurcation lesions.

Diversity of Knot Solitons in Liquid Crystals Manifested by Linking of Preimages in Torons and Hopfions

NASA Astrophysics Data System (ADS)

Ackerman, Paul J.; Smalyukh, Ivan I.

2017-01-01

Topological solitons are knots in continuous physical fields classified by nonzero Hopf index values. Despite arising in theories that span many branches of physics, from elementary particles to condensed matter and cosmology, they remain experimentally elusive and poorly understood. We introduce a method of experimental and numerical analysis of such localized structures in liquid crystals that, similar to the mathematical Hopf maps, relates all points of the medium's order parameter space to their closed-loop preimages within the three-dimensional solitons. We uncover a surprisingly large diversity of naturally occurring and laser-generated topologically nontrivial solitons with differently knotted nematic fields, which previously have not been realized in theories and experiments alike. We discuss the implications of the liquid crystal's nonpolar nature on the knot soliton topology and how the medium's chirality, confinement, and elastic anisotropy help to overcome the constraints of the Hobart-Derrick theorem, yielding static three-dimensional solitons without or with additional defects. Our findings will establish chiral nematics as a model system for experimental exploration of topological solitons and may impinge on understanding of such nonsingular field configurations in other branches of physics, as well as may lead to technological applications.

Equilibrium problems for Raney densities

NASA Astrophysics Data System (ADS)

Forrester, Peter J.; Liu, Dang-Zheng; Zinn-Justin, Paul

2015-07-01

The Raney numbers are a class of combinatorial numbers generalising the Fuss-Catalan numbers. They are indexed by a pair of positive real numbers (p, r) with p > 1 and 0 < r ⩽ p, and form the moments of a probability density function. For certain (p, r) the latter has the interpretation as the density of squared singular values for certain random matrix ensembles, and in this context equilibrium problems characterising the Raney densities for (p, r) = (θ + 1, 1) and (θ/2 + 1, 1/2) have recently been proposed. Using two different techniques—one based on the Wiener-Hopf method for the solution of integral equations and the other on an analysis of the algebraic equation satisfied by the Green's function—we establish the validity of the equilibrium problems for general θ > 0 and similarly use both methods to identify the equilibrium problem for (p, r) = (θ/q + 1, 1/q), θ > 0 and q \\in Z+ . The Wiener-Hopf method is used to extend the latter to parameters (p, r) = (θ/q + 1, m + 1/q) for m a non-negative integer, and also to identify the equilibrium problem for a family of densities with moments given by certain binomial coefficients.

Non-geometric fluxes, quasi-Hopf twist deformations, and nonassociative quantum mechanics

NASA Astrophysics Data System (ADS)

Mylonas, Dionysios; Schupp, Peter; Szabo, Richard J.

2014-12-01

We analyse the symmetries underlying nonassociative deformations of geometry in non-geometric R-flux compactifications which arise via T-duality from closed strings with constant geometric fluxes. Starting from the non-abelian Lie algebra of translations and Bopp shifts in phase space, together with a suitable cochain twist, we construct the quasi-Hopf algebra of symmetries that deforms the algebra of functions and the exterior differential calculus in the phase space description of nonassociative R-space. In this setting, nonassociativity is characterised by the associator 3-cocycle which controls non-coassociativity of the quasi-Hopf algebra. We use abelian 2-cocycle twists to construct maps between the dynamical nonassociative star product and a family of associative star products parametrized by constant momentum surfaces in phase space. We define a suitable integration on these nonassociative spaces and find that the usual cyclicity of associative noncommutative deformations is replaced by weaker notions of 2-cyclicity and 3-cyclicity. Using this star product quantization on phase space together with 3-cyclicity, we formulate a consistent version of nonassociative quantum mechanics, in which we calculate the expectation values of area and volume operators, and find coarse-graining of the string background due to the R-flux.

Buckling and Damage Resistance of Transversely-Loaded Composite Shells

NASA Technical Reports Server (NTRS)

Wardle, Brian L.

1998-01-01

Experimental and numerical work was conducted to better understand composite shell response to transverse loadings which simulate damage-causing impact events. The quasi-static, centered, transverse loading response of laminated graphite/epoxy shells in a [+/-45(sub n)/O(sub n)](sub s) layup having geometric characteristics of a commercial fuselage are studied. The singly-curved composite shell structures are hinged along the straight circumferential edges and are either free or simply supported along the curved axial edges. Key components of the shell response are response instabilities due to limit-point and/or bifurcation buckling. Experimentally, deflection-controlled shell response is characterized via load-deflection data, deformation-shape evolutions, and the resulting damage state. Finite element models are used to study the kinematically nonlinear shell response, including bifurcation, limit-points, and postbuckling. A novel technique is developed for evaluating bifurcation from nonlinear prebuckling states utilizing asymmetric spatial discretization to introduce numerical perturbations. Advantages of the asymmetric meshing technique (AMT) over traditional techniques include efficiency, robustness, ease of application, and solution of the actual (not modified) problems. The AMT is validated by comparison to traditional numerical analysis of a benchmark problem and verified by comparison to experimental data. Applying the technique, bifurcation in a benchmark shell-buckling problem is correctly identified. Excellent agreement between the numerical and experimental results are obtained for a number of composite shells although predictive capability decreases for stiffer (thicker) specimens which is attributed to compliance of the test fixture. Restraining the axial edge (simple support) has the effect of creating a more complex response which involves unstable bifurcation, limit-point buckling, and dynamic collapse. Such shells were noted to bifurcate into asymmetric deformation modes but were undamaged during testing. Shells in this study which were damaged were not observed to bifurcate. Thus, a direct link between bifurcation and atypical damage could not be established although the mechanism (bifurcation) was identified. Recommendations for further work in these related areas are provided and include extensions of the AMT to other shell geometries and structural problems.

Coarse analysis of collective behaviors: Bifurcation analysis of the optimal velocity model for traffic jam formation

NASA Astrophysics Data System (ADS)

Miura, Yasunari; Sugiyama, Yuki

2017-12-01

We present a general method for analyzing macroscopic collective phenomena observed in many-body systems. For this purpose, we employ diffusion maps, which are one of the dimensionality-reduction techniques, and systematically define a few relevant coarse-grained variables for describing macroscopic phenomena. The time evolution of macroscopic behavior is described as a trajectory in the low-dimensional space constructed by these coarse variables. We apply this method to the analysis of the traffic model, called the optimal velocity model, and reveal a bifurcation structure, which features a transition to the emergence of a moving cluster as a traffic jam.

Guess-Work and Reasonings on Centennial Evolution of Surface Air Temperature in Russia. Part III: Where is the Joint Between Norms and Hazards from a Bifurcation Analysis Viewpoint?

NASA Astrophysics Data System (ADS)

Kolokolov, Yury; Monovskaya, Anna

2016-06-01

The paper continues the application of the bifurcation analysis in the research on local climate dynamics based on processing the historically observed data on the daily average land surface air temperature. Since the analyzed data are from instrumental measurements, we are doing the experimental bifurcation analysis. In particular, we focus on the discussion where is the joint between the normal dynamics of local climate systems (norms) and situations with the potential to create damages (hazards)? We illustrate that, perhaps, the criteria for hazards (or violent and unfavorable weather factors) relate mainly to empirical considerations from human opinion, but not to the natural qualitative changes of climate dynamics. To build the bifurcation diagrams, we base on the unconventional conceptual model (HDS-model) which originates from the hysteresis regulator with double synchronization. The HDS-model is characterized by a variable structure with the competition between the amplitude quantization and the time quantization. Then the intermittency between three periodical processes is considered as the typical behavior of local climate systems instead of both chaos and quasi-periodicity in order to excuse the variety of local climate dynamics. From the known specific regularities of the HDS-model dynamics, we try to find a way to decompose the local behaviors into homogeneous units within the time sections with homogeneous dynamics. Here, we present the first results of such decomposition, where the quasi-homogeneous sections (QHS) are determined on the basis of the modified bifurcation diagrams, and the units are reconstructed within the limits connected with the problem of shape defects. Nevertheless, the proposed analysis of the local climate dynamics (QHS-analysis) allows to exhibit how the comparatively modest temperature differences between the mentioned units in an annual scale can step-by-step expand into the great temperature differences of the daily variability at a centennial scale. Then the norms and the hazards relate to the fundamentally different viewpoints, where the time sections of months and, especially, seasons distort the causal effects of natural dynamical processes. The specific circumstances to realize the qualitative changes of the local climate dynamics are summarized by the notion of a likely periodicity. That, in particular, allows to explain why 30-year averaging remains the most common rule so far, but the decadal averaging begins to substitute that rule. We believe that the QHS-analysis can be considered as the joint between the norms and the hazards from a bifurcation analysis viewpoint, where the causal effects of the local climate dynamics are projected into the customary timescale only at the last step. We believe that the results could be interesting to develop the fields connected with climatic change and risk assessment.

Gender, race, and electrophysiologic characteristics of the branched recurrent laryngeal nerve.

PubMed

Fontenot, Tatyana E; Randolph, Gregory W; Friedlander, Paul L; Masoodi, Hammad; Yola, Ibrahim M; Kandil, Emad

2014-10-01

The extralaryngeal branching of recurrent laryngeal nerves (RLN) conveys an increased risk of nerve injury during thyroid surgery. We hypothesized that racial and gender variations in prevalence of branched RLN exist. A retrospective review of all patients who underwent thyroid surgery in a 4-year period in a single surgeon practice. The RLN was routinely identified during thyroid surgery. Presence of RLN branching, its distance from the laryngeal nerve entry point (NEP), and functionality of the branches were ascertained. Patient demographics, rates of neural branching, and distance of bifurcation from the NEP were evaluated using statistical analysis. We identified 719 RLNs at risk in 491 patients who underwent central neck surgery. Four hundred and five (82.5%) patients were female and 86 (17.5%) patients were male. There were 218 (44.4%) African American patients and 251 (51.1 %) Caucasian patients. In African American patients, 42.1% RLNs bifurcated compared to 33.2% RLNs in Caucasian (P = 0.017) patients. The RLNs of African American and Caucasian patients bifurcated at comparable distances (P = 0.30). In male patients, 39.1% RLNs bifurcated; whereas in female patients 36.2% RLNs bifurcated (P = 0.53). On average, RLN bifurcation in female patients was at a longer distance from NEP compared to that of male patients (P = 0.012). Electrophysiologic testing found motor fibers in all anterior branches and three posterior extralaryngeal RLN branches. African American patients have a higher rate of RLN bifurcation compared to Caucasian patients but no statistically significant difference in distance from NEP. Female patients tend to have longer branching variants of bifid RLNs. RLN motor fibers reside primarily in the anterior branch but may occur in the posterior branch. © 2014 The American Laryngological, Rhinological and Otological Society, Inc.

Molecular Analysis of Sensory Axon Branching Unraveled a cGMP-Dependent Signaling Cascade.

PubMed

Dumoulin, Alexandre; Ter-Avetisyan, Gohar; Schmidt, Hannes; Rathjen, Fritz G

2018-04-24

Axonal branching is a key process in the establishment of circuit connectivity within the nervous system. Molecular-genetic studies have shown that a specific form of axonal branching—the bifurcation of sensory neurons at the transition zone between the peripheral and the central nervous system—is regulated by a cyclic guanosine monophosphate (cGMP)-dependent signaling cascade which is composed of C-type natriuretic peptide (CNP), the receptor guanylyl cyclase Npr2, and cGMP-dependent protein kinase Iα (cGKIα). In the absence of any one of these components, neurons in dorsal root ganglia (DRG) and cranial sensory ganglia no longer bifurcate, and instead turn in either an ascending or a descending direction. In contrast, collateral axonal branch formation which represents a second type of axonal branch formation is not affected by inactivation of CNP, Npr2, or cGKI. Whereas axon bifurcation was lost in mouse mutants deficient for components of CNP-induced cGMP formation; the absence of the cGMP-degrading enzyme phosphodiesterase 2A had no effect on axon bifurcation. Adult mice that lack sensory axon bifurcation due to the conditional inactivation of Npr2-mediated cGMP signaling in DRG neurons demonstrated an altered shape of sensory axon terminal fields in the spinal cord, indicating that elaborate compensatory mechanisms reorganize neuronal circuits in the absence of bifurcation. On a functional level, these mice showed impaired heat sensation and nociception induced by chemical irritants, whereas responses to cold sensation, mechanical stimulation, and motor coordination are normal. These data point to a critical role of axon bifurcation for the processing of acute pain perception.

Bifurcation analysis of a discrete-time ratio-dependent predator-prey model with Allee Effect

NASA Astrophysics Data System (ADS)

Cheng, Lifang; Cao, Hongjun

2016-09-01

A discrete-time predator-prey model with Allee effect is investigated in this paper. We consider the strong and the weak Allee effect (the population growth rate is negative and positive at low population density, respectively). From the stability analysis and the bifurcation diagrams, we get that the model with Allee effect (strong or weak) growth function and the model with logistic growth function have somewhat similar bifurcation structures. If the predator growth rate is smaller than its death rate, two species cannot coexist due to having no interior fixed points. When the predator growth rate is greater than its death rate and other parameters are fixed, the model can have two interior fixed points. One is always unstable, and the stability of the other is determined by the integral step size, which decides the species coexistence or not in some extent. If we increase the value of the integral step size, then the bifurcated period doubled orbits or invariant circle orbits may arise. So the numbers of the prey and the predator deviate from one stable state and then circulate along the period orbits or quasi-period orbits. When the integral step size is increased to a critical value, chaotic orbits may appear with many uncertain period-windows, which means that the numbers of prey and predator will be chaotic. In terms of bifurcation diagrams and phase portraits, we know that the complexity degree of the model with strong Allee effect decreases, which is related to the fact that the persistence of species can be determined by the initial species densities.

Turing mechanism underlying a branching model for lung morphogenesis.

PubMed

Xu, Hui; Sun, Mingzhu; Zhao, Xin

2017-01-01

The mammalian lung develops through branching morphogenesis. Two primary forms of branching, which occur in order, in the lung have been identified: tip bifurcation and side branching. However, the mechanisms of lung branching morphogenesis remain to be explored. In our previous study, a biological mechanism was presented for lung branching pattern formation through a branching model. Here, we provide a mathematical mechanism underlying the branching patterns. By decoupling the branching model, we demonstrated the existence of Turing instability. We performed Turing instability analysis to reveal the mathematical mechanism of the branching patterns. Our simulation results show that the Turing patterns underlying the branching patterns are spot patterns that exhibit high local morphogen concentration. The high local morphogen concentration induces the growth of branching. Furthermore, we found that the sparse spot patterns underlie the tip bifurcation patterns, while the dense spot patterns underlies the side branching patterns. The dispersion relation analysis shows that the Turing wavelength affects the branching structure. As the wavelength decreases, the spot patterns change from sparse to dense, the rate of tip bifurcation decreases and side branching eventually occurs instead. In the process of transformation, there may exists hybrid branching that mixes tip bifurcation and side branching. Since experimental studies have reported that branching mode switching from side branching to tip bifurcation in the lung is under genetic control, our simulation results suggest that genes control the switch of the branching mode by regulating the Turing wavelength. Our results provide a novel insight into and understanding of the formation of branching patterns in the lung and other biological systems.

Classification of bifurcations regions in IVOCT images using support vector machine and artificial neural network models

NASA Astrophysics Data System (ADS)

Porto, C. D. N.; Costa Filho, C. F. F.; Macedo, M. M. G.; Gutierrez, M. A.; Costa, M. G. F.

2017-03-01

Studies in intravascular optical coherence tomography (IV-OCT) have demonstrated the importance of coronary bifurcation regions in intravascular medical imaging analysis, as plaques are more likely to accumulate in this region leading to coronary disease. A typical IV-OCT pullback acquires hundreds of frames, thus developing an automated tool to classify the OCT frames as bifurcation or non-bifurcation can be an important step to speed up OCT pullbacks analysis and assist automated methods for atherosclerotic plaque quantification. In this work, we evaluate the performance of two state-of-the-art classifiers, SVM and Neural Networks in the bifurcation classification task. The study included IV-OCT frames from 9 patients. In order to improve classification performance, we trained and tested the SVM with different parameters by means of a grid search and different stop criteria were applied to the Neural Network classifier: mean square error, early stop and regularization. Different sets of features were tested, using feature selection techniques: PCA, LDA and scalar feature selection with correlation. Training and test were performed in sets with a maximum of 1460 OCT frames. We quantified our results in terms of false positive rate, true positive rate, accuracy, specificity, precision, false alarm, f-measure and area under ROC curve. Neural networks obtained the best classification accuracy, 98.83%, overcoming the results found in literature. Our methods appear to offer a robust and reliable automated classification of OCT frames that might assist physicians indicating potential frames to analyze. Methods for improving neural networks generalization have increased the classification performance.

Bifurcations of periodic motion in a three-degree-of-freedom vibro-impact system with clearance

NASA Astrophysics Data System (ADS)

Liu, Yongbao; Wang, Qiang; Xu, Huidong

2017-07-01

The smooth bifurcation and grazing non-smooth bifurcation of periodic motion of a three-degree-of-freedom vibro-impact system with clearance are studied in this paper. Firstly, a periodic solution of vibro-impact system is solved and a six-dimensional Poincaré map is established. Then, for the six-dimensional Poincaré map, the analytic expressions of all eigenvalues of Jacobi matrix with respect to parameters are unavailable. This implies that with application of the classical critical criterion described by the properties of eigenvalues, we have to numerically compute eigenvalues point by point and check their properties to search for the bifurcation points. Such the numerical calculation is a laborious job in the process of determining bifurcation points. To overcome the difficulty that originates from the classical bifurcation criteria, the explicit critical criteria without using eigenvalues calculation of high-dimensional map are applied to determine bifurcation points of Co-dimension-one period doubling bifurcation and Co-dimension-one Neimark-Sacker bifurcation and Co-dimension-two Flip-Neimark-Sacker bifurcation, and then local dynamical behaviors of these bifurcations are analyzed. Moreover, the directions of period doubling bifurcation and Neimark-Sacker bifurcation are analyzed by center manifold reduction theory and normal form approach. Finally, the existence of the grazing periodic motion of the vibro-impact system is analyzed and the grazing bifurcation point is obtained, the discontinuous grazing bifurcation behavior is studied based on the compound normal form map near the grazing point, the discontinuous jumping phenomenon and co-existing multiple solutions near the grazing bifurcation point are revealed.

Closed set of the uniqueness conditions and bifurcation criteria in generalized coupled thermoplasticity for small deformations

NASA Astrophysics Data System (ADS)

Śloderbach, Zdzisław

2016-05-01

This paper reports the results of a study into global and local conditions of uniqueness and the criteria excluding the possibility of bifurcation of the equilibrium state for small strains. The conditions and criteria are derived on the basis of an analysis of the problem of uniqueness of a solution involving the basic incremental boundary problem of coupled generalized thermo-elasto-plasticity. This work forms a follow-up of previous research (Śloderbach in Bifurcations criteria for equilibrium states in generalized thermoplasticity, IFTR Reports, 1980, Arch Mech 3(35):337-349, 351-367, 1983), but contains a new derivation of global and local criteria excluding a possibility of bifurcation of an equilibrium state regarding a comparison body dependent on the admissible fields of stress rate. The thermal elasto-plastic coupling effects, non-associated laws of plastic flow and influence of plastic strains on thermoplastic properties of a body were taken into account in this work. Thus, the mathematical problem considered here is not a self-conjugated problem.