Analysis, synchronisation and circuit design of a new highly nonlinear chaotic system

NASA Astrophysics Data System (ADS)

Mobayen, Saleh; Kingni, Sifeu Takougang; Pham, Viet-Thanh; Nazarimehr, Fahimeh; Jafari, Sajad

2018-02-01

This paper investigates a three-dimensional autonomous chaotic flow without linear terms. Dynamical behaviour of the proposed system is investigated through eigenvalue structures, phase portraits, bifurcation diagram, Lyapunov exponents and basin of attraction. For a suitable choice of the parameters, the proposed system can exhibit anti-monotonicity, periodic oscillations and double-scroll chaotic attractor. Basin of attraction of the proposed system shows that the chaotic attractor is self-excited. Furthermore, feasibility of double-scroll chaotic attractor in the real word is investigated by using the OrCAD-PSpice software via an electronic implementation of the proposed system. A good qualitative agreement is illustrated between the numerical simulations and the OrCAD-PSpice results. Finally, a finite-time control method based on dynamic sliding surface for the synchronisation of master and slave chaotic systems in the presence of external disturbances is performed. Using the suggested control technique, the superior master-slave synchronisation is attained. Illustrative simulation results on the studied chaotic system are presented to indicate the effectiveness of the suggested scheme.

Alternation of regular and chaotic dynamics in a simple two-degree-of-freedom system with nonlinear inertial coupling.

PubMed

Sigalov, G; Gendelman, O V; AL-Shudeifat, M A; Manevitch, L I; Vakakis, A F; Bergman, L A

2012-03-01

We show that nonlinear inertial coupling between a linear oscillator and an eccentric rotator can lead to very interesting interchanges between regular and chaotic dynamical behavior. Indeed, we show that this model demonstrates rather unusual behavior from the viewpoint of nonlinear dynamics. Specifically, at a discrete set of values of the total energy, the Hamiltonian system exhibits non-conventional nonlinear normal modes, whose shape is determined by phase locking of rotatory and oscillatory motions of the rotator at integer ratios of characteristic frequencies. Considering the weakly damped system, resonance capture of the dynamics into the vicinity of these modes brings about regular motion of the system. For energy levels far from these discrete values, the motion of the system is chaotic. Thus, the succession of resonance captures and escapes by a discrete set of the normal modes causes a sequence of transitions between regular and chaotic behavior, provided that the damping is sufficiently small. We begin from the Hamiltonian system and present a series of Poincaré sections manifesting the complex structure of the phase space of the considered system with inertial nonlinear coupling. Then an approximate analytical description is presented for the non-conventional nonlinear normal modes. We confirm the analytical results by numerical simulation and demonstrate the alternate transitions between regular and chaotic dynamics mentioned above. The origin of the chaotic behavior is also discussed.

Regular transport dynamics produce chaotic travel times.

PubMed

Villalobos, Jorge; Muñoz, Víctor; Rogan, José; Zarama, Roberto; Johnson, Neil F; Toledo, Benjamín; Valdivia, Juan Alejandro

2014-06-01

In the hope of making passenger travel times shorter and more reliable, many cities are introducing dedicated bus lanes (e.g., Bogota, London, Miami). Here we show that chaotic travel times are actually a natural consequence of individual bus function, and hence of public transport systems more generally, i.e., chaotic dynamics emerge even when the route is empty and straight, stops and lights are equidistant and regular, and loading times are negligible. More generally, our findings provide a novel example of chaotic dynamics emerging from a single object following Newton's laws of motion in a regularized one-dimensional system.

Regular transport dynamics produce chaotic travel times

NASA Astrophysics Data System (ADS)

Villalobos, Jorge; Muñoz, Víctor; Rogan, José; Zarama, Roberto; Johnson, Neil F.; Toledo, Benjamín; Valdivia, Juan Alejandro

2014-06-01

In the hope of making passenger travel times shorter and more reliable, many cities are introducing dedicated bus lanes (e.g., Bogota, London, Miami). Here we show that chaotic travel times are actually a natural consequence of individual bus function, and hence of public transport systems more generally, i.e., chaotic dynamics emerge even when the route is empty and straight, stops and lights are equidistant and regular, and loading times are negligible. More generally, our findings provide a novel example of chaotic dynamics emerging from a single object following Newton's laws of motion in a regularized one-dimensional system.

Improvement and empirical research on chaos control by theory of "chaos + chaos = order".

PubMed

Fulai, Wang

2012-12-01

This paper focuses on advancing the understanding of Parrondian effects and their paradoxical behavior in nonlinear dynamical systems. Some examples are given to show that a dynamics combined by more than two discrete chaotic dynamics in deterministic manners can give rise to order when combined. The chaotic maps in our study are more general than those in the current literatures as far as "chaos + chaos = order" is concerned. Some problems left over in the current literatures are solved. It is proved both theoretically and numerically that, given any m chaotic dynamics generated by the one-dimensional real Mandelbrot maps, it is no possible to get a periodic system when all the m chaotic dynamics are alternated in random manner, but for any integer m(m ≥ 2) a dynamics combined in deterministic manner by m Mandelbrot chaotic dynamics can be found to give rise to a periodic dynamics of m periods. Numerical and mathematical analysis prove that the paradoxical phenomenon of "chaos + chaos = order" also exist in the dynamics generated by non-Mandelbrot maps.

Synchronization transition in neuronal networks composed of chaotic or non-chaotic oscillators.

PubMed

Xu, Kesheng; Maidana, Jean Paul; Castro, Samy; Orio, Patricio

2018-05-30

Chaotic dynamics has been shown in the dynamics of neurons and neural networks, in experimental data and numerical simulations. Theoretical studies have proposed an underlying role of chaos in neural systems. Nevertheless, whether chaotic neural oscillators make a significant contribution to network behaviour and whether the dynamical richness of neural networks is sensitive to the dynamics of isolated neurons, still remain open questions. We investigated synchronization transitions in heterogeneous neural networks of neurons connected by electrical coupling in a small world topology. The nodes in our model are oscillatory neurons that - when isolated - can exhibit either chaotic or non-chaotic behaviour, depending on conductance parameters. We found that the heterogeneity of firing rates and firing patterns make a greater contribution than chaos to the steepness of the synchronization transition curve. We also show that chaotic dynamics of the isolated neurons do not always make a visible difference in the transition to full synchrony. Moreover, macroscopic chaos is observed regardless of the dynamics nature of the neurons. However, performing a Functional Connectivity Dynamics analysis, we show that chaotic nodes can promote what is known as multi-stable behaviour, where the network dynamically switches between a number of different semi-synchronized, metastable states.

Extreme multistability in a memristor-based multi-scroll hyper-chaotic system

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

Yuan, Fang, E-mail: yf210yf@163.com; Wang, Guangyi, E-mail: wanggyi@163.com; Wang, Xiaowei

In this paper, a new memristor-based multi-scroll hyper-chaotic system is designed. The proposed memristor-based system possesses multiple complex dynamic behaviors compared with other chaotic systems. Various coexisting attractors and hidden coexisting attractors are observed in this system, which means extreme multistability arises. Besides, by adjusting parameters of the system, this chaotic system can perform single-scroll attractors, double-scroll attractors, and four-scroll attractors. Basic dynamic characteristics of the system are investigated, including equilibrium points and stability, bifurcation diagrams, Lyapunov exponents, and so on. In addition, the presented system is also realized by an analog circuit to confirm the correction of the numericalmore » simulations.« less

Periodic or chaotic bursting dynamics via delayed pitchfork bifurcation in a slow-varying controlled system

NASA Astrophysics Data System (ADS)

Yu, Yue; Zhang, Zhengdi; Han, Xiujing

2018-03-01

In this work, we aim to demonstrate the novel routes to periodic and chaotic bursting, i.e., the different bursting dynamics via delayed pitchfork bifurcations around stable attractors, in the classical controlled Lü system. First, by computing the corresponding characteristic polynomial, we determine where some critical values about bifurcation behaviors appear in the Lü system. Moreover, the transition mechanism among different stable attractors has been introduced including homoclinic-type connections or chaotic attractors. Secondly, taking advantage of the above analytical results, we carry out a study of the mechanism for bursting dynamics in the Lü system with slowly periodic variation of certain control parameter. A distinct delayed supercritical pitchfork bifurcation behavior can be discussed when the control item passes through bifurcation points periodically. This delayed dynamical behavior may terminate at different parameter areas, which leads to different spiking modes around different stable attractors (equilibriums, limit cycles, or chaotic attractors). In particular, the chaotic attractor may appear by Shilnikov connections or chaos boundary crisis, which leads to the occurrence of impressive chaotic bursting oscillations. Our findings enrich the study of bursting dynamics and deepen the understanding of some similar sorts of delayed bursting phenomena. Finally, some numerical simulations are included to illustrate the validity of our study.

Dynamic analyses, FPGA implementation and engineering applications of multi-butterfly chaotic attractors generated from generalised Sprott C system

NASA Astrophysics Data System (ADS)

Lai, Qiang; Zhao, Xiao-Wen; Rajagopal, Karthikeyan; Xu, Guanghui; Akgul, Akif; Guleryuz, Emre

2018-01-01

This paper considers the generation of multi-butterfly chaotic attractors from a generalised Sprott C system with multiple non-hyperbolic equilibria. The system is constructed by introducing an additional variable whose derivative has a switching function to the Sprott C system. It is numerically found that the system creates two-, three-, four-, five-butterfly attractors and any other multi-butterfly attractors. First, the dynamic analyses of multi-butterfly chaotic attractors are presented. Secondly, the field programmable gate array implementation, electronic circuit realisation and random number generator are done with the multi-butterfly chaotic attractors.

Solar System Dynamics

NASA Technical Reports Server (NTRS)

Wisdom, Jack

2002-01-01

In these 18 years, the research has touched every major dynamical problem in the solar system, including: the effect of chaotic zones on the distribution of asteroids, the delivery of meteorites along chaotic pathways, the chaotic motion of Pluto, the chaotic motion of the outer planets and that of the whole solar system, the delivery of short period comets from the Kuiper belt, the tidal evolution of the Uranian arid Galilean satellites, the chaotic tumbling of Hyperion and other irregular satellites, the large chaotic variations of the obliquity of Mars, the evolution of the Earth-Moon system, and the resonant core- mantle dynamics of Earth and Venus. It has introduced new analytical and numerical tools that are in widespread use. Today, nearly every long-term integration of our solar system, its subsystems, and other solar systems uses algorithms that was invented. This research has all been primarily Supported by this sequence of PGG NASA grants. During this period published major investigations of tidal evolution of the Earth-Moon system and of the passage of the Earth and Venus through non-linear core-mantle resonances were completed. It has published a major innovation in symplectic algorithms: the symplectic corrector. A paper was completed on non-perturbative hydrostatic equilibrium.

Chaotic dynamics of large-scale double-diffusive convection in a porous medium

NASA Astrophysics Data System (ADS)

Kondo, Shutaro; Gotoda, Hiroshi; Miyano, Takaya; Tokuda, Isao T.

2018-02-01

We have studied chaotic dynamics of large-scale double-diffusive convection of a viscoelastic fluid in a porous medium from the viewpoint of dynamical systems theory. A fifth-order nonlinear dynamical system modeling the double-diffusive convection is theoretically obtained by incorporating the Darcy-Brinkman equation into transport equations through a physical dimensionless parameter representing porosity. We clearly show that the chaotic convective motion becomes much more complicated with increasing porosity. The degree of dynamic instability during chaotic convective motion is quantified by two important measures: the network entropy of the degree distribution in the horizontal visibility graph and the Kaplan-Yorke dimension in terms of Lyapunov exponents. We also present an interesting on-off intermittent phenomenon in the probability distribution of time intervals exhibiting nearly complete synchronization.

Entanglement as a signature of quantum chaos.

PubMed

Wang, Xiaoguang; Ghose, Shohini; Sanders, Barry C; Hu, Bambi

2004-01-01

We explore the dynamics of entanglement in classically chaotic systems by considering a multiqubit system that behaves collectively as a spin system obeying the dynamics of the quantum kicked top. In the classical limit, the kicked top exhibits both regular and chaotic dynamics depending on the strength of the chaoticity parameter kappa in the Hamiltonian. We show that the entanglement of the multiqubit system, considered for both the bipartite and the pairwise entanglement, yields a signature of quantum chaos. Whereas bipartite entanglement is enhanced in the chaotic region, pairwise entanglement is suppressed. Furthermore, we define a time-averaged entangling power and show that this entangling power changes markedly as kappa moves the system from being predominantly regular to being predominantly chaotic, thus sharply identifying the edge of chaos. When this entangling power is averaged over all states, it yields a signature of global chaos. The qualitative behavior of this global entangling power is similar to that of the classical Lyapunov exponent.

Urey Prize Lecture - Chaotic dynamics in the solar system

NASA Technical Reports Server (NTRS)

Wisdom, Jack

1987-01-01

Attention is given to solar system cases in which chaotic solutions of Newton's equations are important, as in chaotic rotation and orbital evolution. Hyperion is noted to be tumbling chaotically; chaotic orbital evolution is suggested to be of fundamental importance to an accounting for the Kirkwood gaps in asteroid distribution and for the phase space boundary of the chaotic zone at the 3/1 mean-motion commensurability with Jupiter. In addition, chaotic trajectories in the 2/1 chaotic zone reach very high eccentricities by a route that carries them to high inclinations temporarily.

Dynamic Long-Term Anticipation of Chaotic States

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

Voss, Henning U.

2001-07-02

Introducing a short time delay into the coupling of two synchronizing chaotic systems, it was shown recently that the driven system may anticipate the driving system in real time. Augmenting the phase space of the driven system, we accomplish anticipation times that are multiples of the coupling delay time and exceed characteristic time scales of the chaotic dynamics. The stability properties of the associated anticipatory synchronization manifold in certain cases turn out to be the same as for identically synchronizing oscillators.

Biologically inspired rate control of chaos.

PubMed

Olde Scheper, Tjeerd V

2017-10-01

The overall intention of chaotic control is to eliminate chaos and to force the system to become stable in the classical sense. In this paper, I demonstrate a more subtle method that does not eliminate all traces of chaotic behaviour; yet it consistently, and reliably, can provide control as intended. The Rate Control of Chaos (RCC) method is derived from metabolic control processes and has several remarkable properties. RCC can control complex systems continuously, and unsupervised, it can also maintain control across bifurcations, and in the presence of significant systemic noise. Specifically, I show that RCC can control a typical set of chaotic models, including the 3 and 4 dimensional chaotic Lorenz systems, in all modes. Furthermore, it is capable of controlling spatiotemporal chaos without supervision and maintains control of the system across bifurcations. This property of RCC allows a dynamic system to operate in parameter spaces that are difficult to control otherwise. This may be particularly interesting for the control of forced systems or dynamic systems that are chaotically perturbed. These control properties of RCC are applicable to a range of dynamic systems, thereby appearing to have far-reaching effects beyond just controlling chaos. RCC may also point to the existence of a biochemical control function of an enzyme, to stabilise the dynamics of the reaction cascade.

Temperature crossover of decoherence rates in chaotic and regular bath dynamics.

PubMed

Sanz, A S; Elran, Y; Brumer, P

2012-03-01

The effect of chaotic bath dynamics on the decoherence of a quantum system is examined for the vibrational degrees of freedom of a diatomic molecule in a realistic, constant temperature collisional bath. As an example, the specific case of I(2) in liquid xenon is examined as a function of temperature, and the results compared with an integrable xenon bath. A crossover in behavior is found: The integrable bath induces more decoherence at low bath temperatures than does the chaotic bath, whereas the opposite is the case at the higher bath temperatures. These results, verifying a conjecture due to Wilkie, shed light on the differing views of the effect of chaotic dynamics on system decoherence.

Characterizing chaotic melodies in automatic music composition

NASA Astrophysics Data System (ADS)

Coca, Andrés E.; Tost, Gerard O.; Zhao, Liang

2010-09-01

In this paper, we initially present an algorithm for automatic composition of melodies using chaotic dynamical systems. Afterward, we characterize chaotic music in a comprehensive way as comprising three perspectives: musical discrimination, dynamical influence on musical features, and musical perception. With respect to the first perspective, the coherence between generated chaotic melodies (continuous as well as discrete chaotic melodies) and a set of classical reference melodies is characterized by statistical descriptors and melodic measures. The significant differences among the three types of melodies are determined by discriminant analysis. Regarding the second perspective, the influence of dynamical features of chaotic attractors, e.g., Lyapunov exponent, Hurst coefficient, and correlation dimension, on melodic features is determined by canonical correlation analysis. The last perspective is related to perception of originality, complexity, and degree of melodiousness (Euler's gradus suavitatis) of chaotic and classical melodies by nonparametric statistical tests.

Complex dynamics of a new 3D Lorenz-type autonomous chaotic system

NASA Astrophysics Data System (ADS)

Zhang, Fuchen; Liao, Xiaofeng; Zhang, Guangyun; Mu, Chunlai

2017-12-01

This paper investigates a new three-dimensional continuous quadratic autonomous chaotic system which is not topologically equivalent to the Lorenz system. The dynamical behaviours of this system are further investigated in detail, including the ultimate boundedness, the invariant sets and the global attraction domain according to Lyapunov stability theory of dynamical systems. The innovation of the paper lies in the fact that this paper not only proves this chaotic system is globally bounded for the parameters of this system but also gives a family of mathematical expressions of global exponential attractive sets with respect to the parameters of this system. To validate the ultimate bound estimation, numerical simulations are also investigated. Numerical simulations verify the effectiveness and feasibility of the theoretical scheme.

Chaos in driven Alfvén systems: unstable periodic orbits and chaotic saddles

NASA Astrophysics Data System (ADS)

Chian, A. C.-L.; Santana, W. M.; Rempel, E. L.; Borotto, F. A.; Hada, T.; Kamide, Y.

2007-01-01

The chaotic dynamics of Alfvén waves in space plasmas governed by the derivative nonlinear Schrödinger equation, in the low-dimensional limit described by stationary spatial solutions, is studied. A bifurcation diagram is constructed, by varying the driver amplitude, to identify a number of nonlinear dynamical processes including saddle-node bifurcation, boundary crisis, and interior crisis. The roles played by unstable periodic orbits and chaotic saddles in these transitions are analyzed, and the conversion from a chaotic saddle to a chaotic attractor in these dynamical processes is demonstrated. In particular, the phenomenon of gap-filling in the chaotic transition from weak chaos to strong chaos via an interior crisis is investigated. A coupling unstable periodic orbit created by an explosion, within the gaps of the chaotic saddles embedded in a chaotic attractor following an interior crisis, is found numerically. The gap-filling unstable periodic orbits are responsible for coupling the banded chaotic saddle (BCS) to the surrounding chaotic saddle (SCS), leading to crisis-induced intermittency. The physical relevance of chaos for Alfvén intermittent turbulence observed in the solar wind is discussed.

Recent developments in chaotic dynamics

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

Ott, E.

1994-02-01

Before the relatively recent wide acceptance of the existence of chaotic dynamics, many physicists and engineers were under the impression that simple systems could necessarily only display simple solutions. This feeling had been unintentionally reinforced by conventional college courses which emphasize linear dynamics (partly because that is the only case with nice general solutions). More recently, physical experiments and numerical examples have abundantly demonstrated how wrong this feeling is. A brief review of chaotic dynamics is presented. Topics discussed include basic concepts, recent developments, and applications.

Space-Group Symmetries Generate Chaotic Fluid Advection in Crystalline Granular Media

NASA Astrophysics Data System (ADS)

Turuban, R.; Lester, D. R.; Le Borgne, T.; Méheust, Y.

2018-01-01

The classical connection between symmetry breaking and the onset of chaos in dynamical systems harks back to the seminal theory of Noether [Transp. Theory Statist. Phys. 1, 186 (1918), 10.1080/00411457108231446]. We study the Lagrangian kinematics of steady 3D Stokes flow through simple cubic and body-centered cubic (bcc) crystalline lattices of close-packed spheres, and uncover an important exception. While breaking of point-group symmetries is a necessary condition for chaotic mixing in both lattices, a further space-group (glide) symmetry of the bcc lattice generates a transition from globally regular to globally chaotic dynamics. This finding provides new insights into chaotic mixing in porous media and has significant implications for understanding the impact of symmetries upon generic dynamical systems.

A novel double-convection chaotic attractor, its adaptive control and circuit simulation

NASA Astrophysics Data System (ADS)

Mamat, M.; Vaidyanathan, S.; Sambas, A.; Mujiarto; Sanjaya, W. S. M.; Subiyanto

2018-03-01

A 3-D novel double-convection chaotic system with three nonlinearities is proposed in this research work. The dynamical properties of the new chaotic system are described in terms of phase portraits, Lyapunov exponents, Kaplan-Yorke dimension, dissipativity, stability analysis of equilibria, etc. Adaptive control and synchronization of the new chaotic system with unknown parameters are achieved via nonlinear controllers and the results are established using Lyapunov stability theory. Furthermore, an electronic circuit realization of the new 3-D novel chaotic system is presented in detail. Finally, the circuit experimental results of the 3-D novel chaotic attractor show agreement with the numerical simulations.

Randomly chosen chaotic maps can give rise to nearly ordered behavior

NASA Astrophysics Data System (ADS)

Boyarsky, Abraham; Góra, Paweł; Islam, Md. Shafiqul

2005-10-01

Parrondo’s paradox [J.M.R. Parrondo, G.P. Harmer, D. Abbott, New paradoxical games based on Brownian ratchets, Phys. Rev. Lett. 85 (2000), 5226-5229] (see also [O.E. Percus, J.K. Percus, Can two wrongs make a right? Coin-tossing games and Parrondo’s paradox, Math. Intelligencer 24 (3) (2002) 68-72]) states that two losing gambling games when combined one after the other (either deterministically or randomly) can result in a winning game: that is, a losing game followed by a losing game = a winning game. Inspired by this paradox, a recent study [J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order? Physica D 200 (2005) 124-132] asked an analogous question in discrete time dynamical system: can two chaotic systems give rise to order, namely can they be combined into another dynamical system which does not behave chaotically? Numerical evidence is provided in [J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order? Physica D 200 (2005) 124-132] that two chaotic quadratic maps, when composed with each other, create a new dynamical system which has a stable period orbit. The question of what happens in the case of random composition of maps is posed in [J. Almeida, D. Peralta-Salas, M. Romera, Can two chaotic systems give rise to order? Physica D 200 (2005) 124-132] but left unanswered. In this note we present an example of a dynamical system where, at each iteration, a map is chosen in a probabilistic manner from a collection of chaotic maps. The resulting random map is proved to have an infinite absolutely continuous invariant measure (acim) with spikes at two points. From this we show that the dynamics behaves in a nearly ordered manner. When the foregoing maps are applied one after the other, deterministically as in [O.E. Percus, J.K. Percus, Can two wrongs make a right? Coin-tossing games and Parrondo’s paradox, Math. Intelligencer 24 (3) (2002) 68-72], the resulting composed map has a periodic orbit which is stable.

Proceedings of the 2nd Experimental Chaos Conference

NASA Astrophysics Data System (ADS)

Ditto, William; Pecora, Lou; Shlesinger, Michael; Spano, Mark; Vohra, Sandeep

1995-02-01

The Table of Contents for the full book PDF is as follows: * Introduction * Spatiotemporal Phenomena * Experimental Studies of Chaotic Mixing * Using Random Maps in the Analysis of Experimental Fluid Flows * Transition to Spatiotemporal Chaos in a Reaction-Diffusion System * Ion-Dynamical Chaos in Plasmas * Optics * Chaos in a Synchronously Driven Optical Resonator * Chaos, Patterns and Defects in Stimulated Scattering Phenomena * Test of the Normal Form for a Subcritical Bifurcation * Observation of Bifurcations and Chaos in a Driven Fiber Optic Coil * Applications -- Communications * Robustness and Signal Recovery in a Synchronized Chaotic System * Synchronizing Nonautonomous Chaotic Circuits * Synchronization of Pulse-Coupled Chaotic Oscillators * Ocean Transmission Effects on Chaotic Signals * Controlling Symbolic Dynamics for Communication * Applications -- Control * Analysis of Nonlinear Actuators Using Chaotic Waveforms * Controlling Chaos in a Quasiperiodic Electronic System * Control of Chaos in a CO2 Laser * General Research * Video-Based Analysis of Bifurcation Phenomena in Radio-Frequency-Excited Inert Gas Plasmas * Transition from Soliton to Chaotic Motion During the Impact of a Nonlinear Structure * Sonoluminescence in a Single Bubble: Periodic, Quasiperiodic and Chaotic Light Source * Quantum Chaos Experiments Using Microwave Cavities * Experiments on Quantum Chaos With and Without Time Reversibility * When Small Noise Imposed on Deterministic Dynamics Becomes Important * Biology * Chaos Control for Cardiac Arrhythmias * Irregularities in Spike Trains of Cat Retinal Ganglion Cells * Broad-Band Synchronization in Monkey Neocortex * Applicability of Correlation Dimension Calculations to Blood Pressure Signal in Rats * Tests for Deterministic Chaos in Noisy Time Series * The Crayfish Mechanoreceptor Cell: A Biological Example of Stochastic Resonance * Chemistry * Chaos During Heterogeneous Chemical Reactions * Stabilizing and Tracking Unstable Periodic Orbits and Stationary States in Chemical Systems * Recursive Proportional-Feedback and Its Use to Control Chaos in an Electrochemical System * Temperature Patterns on Catalytic Surfaces * Meteorology/Oceanography * Nonlinear Evolution of Water Waves: Hilbert's View * Fractal Properties of Isoconcentration Surfaces in a Smoke Plume * Fractal Dimensions of Remotely Sensed Atmospheric Signals * Are Ocean Surface Waves Chaotic? * Dynamical Attractor Reconstruction for a Marine Stratocumulus Cloud

Active synchronization between two different chaotic dynamical system

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

Maheri, M.; Arifin, N. Md; Ismail, F.

2015-05-15

In this paper we investigate on the synchronization problem between two different chaotic dynamical system based on the Lyapunov stability theorem by using nonlinear control functions. Active control schemes are used for synchronization Liu system as drive and Rossler system as response. Numerical simulation by using Maple software are used to show effectiveness of the proposed schemes.

Dynamic Analysis and Adaptive Sliding Mode Controller for a Chaotic Fractional Incommensurate Order Financial System

NASA Astrophysics Data System (ADS)

Hajipour, Ahmad; Tavakoli, Hamidreza

2017-12-01

In this study, the dynamic behavior and chaos control of a chaotic fractional incommensurate-order financial system are investigated. Using well-known tools of nonlinear theory, i.e. Lyapunov exponents, phase diagrams and bifurcation diagrams, we observe some interesting phenomena, e.g. antimonotonicity, crisis phenomena and route to chaos through a period doubling sequence. Adopting largest Lyapunov exponent criteria, we find that the system yields chaos at the lowest order of 2.15. Next, in order to globally stabilize the chaotic fractional incommensurate order financial system with uncertain dynamics, an adaptive fractional sliding mode controller is designed. Numerical simulations are used to demonstrate the effectiveness of the proposed control method.

A new chaotic attractor with two quadratic nonlinearities, its synchronization and circuit implementation

NASA Astrophysics Data System (ADS)

Vaidyanathan, S.; Sambas, A.; Sukono; Mamat, M.; Gundara, G.; Mada Sanjaya, W. S.; Subiyanto

2018-03-01

A 3-D new chaotic attractor with two quadratic nonlinearities is proposed in this paper. The dynamical properties of the new chaotic system are described in terms of phase portraits, equilibrium points, Lyapunov exponents, Kaplan-Yorke dimension, dissipativity, etc. We show that the new chaotic system has three unstable equilibrium points. The new chaotic attractor is dissipative in nature. As an engineering application, adaptive synchronization of identical new chaotic attractors is designed via nonlinear control and Lyapunov stability theory. Furthermore, an electronic circuit realization of the new chaotic attractor is presented in detail to confirm the feasibility of the theoretical chaotic attractor model.

Synchronizability of nonidentical weakly dissipative systems

NASA Astrophysics Data System (ADS)

Sendiña-Nadal, Irene; Letellier, Christophe

2017-10-01

Synchronization is a very generic process commonly observed in a large variety of dynamical systems which, however, has been rarely addressed in systems with low dissipation. Using the Rössler, the Lorenz 84, and the Sprott A systems as paradigmatic examples of strongly, weakly, and non-dissipative chaotic systems, respectively, we show that a parameter or frequency mismatch between two coupled such systems does not affect the synchronizability and the underlying structure of the joint attractor in the same way. By computing the Shannon entropy associated with the corresponding recurrence plots, we were able to characterize how two coupled nonidentical chaotic oscillators organize their dynamics in different dissipation regimes. While for strongly dissipative systems, the resulting dynamics exhibits a Shannon entropy value compatible with the one having an average parameter mismatch, for weak dissipation synchronization dynamics corresponds to a more complex behavior with higher values of the Shannon entropy. In comparison, conservative dynamics leads to a less rich picture, providing either similar chaotic dynamics or oversimplified periodic ones.

A new two-scroll chaotic attractor with three quadratic nonlinearities, its adaptive control and circuit design

NASA Astrophysics Data System (ADS)

Lien, C.-H.; Vaidyanathan, S.; Sambas, A.; Sukono; Mamat, M.; Sanjaya, W. S. M.; Subiyanto

2018-03-01

A 3-D new two-scroll chaotic attractor with three quadratic nonlinearities is investigated in this paper. First, the qualitative and dynamical properties of the new two-scroll chaotic system are described in terms of phase portraits, equilibrium points, Lyapunov exponents, Kaplan-Yorke dimension, dissipativity, etc. We show that the new two-scroll dissipative chaotic system has three unstable equilibrium points. As an engineering application, global chaos control of the new two-scroll chaotic system with unknown system parameters is designed via adaptive feedback control and Lyapunov stability theory. Furthermore, an electronic circuit realization of the new chaotic attractor is presented in detail to confirm the feasibility of the theoretical chaotic two-scroll attractor model.

Time Series Analysis of the Bacillus subtilis Sporulation Network Reveals Low Dimensional Chaotic Dynamics.

PubMed

Lecca, Paola; Mura, Ivan; Re, Angela; Barker, Gary C; Ihekwaba, Adaoha E C

2016-01-01

Chaotic behavior refers to a behavior which, albeit irregular, is generated by an underlying deterministic process. Therefore, a chaotic behavior is potentially controllable. This possibility becomes practically amenable especially when chaos is shown to be low-dimensional, i.e., to be attributable to a small fraction of the total systems components. In this case, indeed, including the major drivers of chaos in a system into the modeling approach allows us to improve predictability of the systems dynamics. Here, we analyzed the numerical simulations of an accurate ordinary differential equation model of the gene network regulating sporulation initiation in Bacillus subtilis to explore whether the non-linearity underlying time series data is due to low-dimensional chaos. Low-dimensional chaos is expectedly common in systems with few degrees of freedom, but rare in systems with many degrees of freedom such as the B. subtilis sporulation network. The estimation of a number of indices, which reflect the chaotic nature of a system, indicates that the dynamics of this network is affected by deterministic chaos. The neat separation between the indices obtained from the time series simulated from the model and those obtained from time series generated by Gaussian white and colored noise confirmed that the B. subtilis sporulation network dynamics is affected by low dimensional chaos rather than by noise. Furthermore, our analysis identifies the principal driver of the networks chaotic dynamics to be sporulation initiation phosphotransferase B (Spo0B). We then analyzed the parameters and the phase space of the system to characterize the instability points of the network dynamics, and, in turn, to identify the ranges of values of Spo0B and of the other drivers of the chaotic dynamics, for which the whole system is highly sensitive to minimal perturbation. In summary, we described an unappreciated source of complexity in the B. subtilis sporulation network by gathering evidence for the chaotic behavior of the system, and by suggesting candidate molecules driving chaos in the system. The results of our chaos analysis can increase our understanding of the intricacies of the regulatory network under analysis, and suggest experimental work to refine our behavior of the mechanisms underlying B. subtilis sporulation initiation control.

Chaotic Stochasticity: A Ubiquitous Source of Unpredictability in Epidemics

NASA Astrophysics Data System (ADS)

Rand, D. A.; Wilson, H. B.

1991-11-01

We address the question of whether or not childhood epidemics such as measles and chickenpox are chaotic, and argue that the best explanation of the observed unpredictability is that it is a manifestation of what we call chaotic stochasticity. Such chaos is driven and made permanent by the fluctuations from the mean field encountered in epidemics, or by extrinsic stochastic noise, and is dependent upon the existence of chaotic repellors in the mean field dynamics. Its existence is also a consequence of the near extinctions in the epidemic. For such systems, chaotic stochasticity is likely to be far more ubiquitous than the presence of deterministic chaotic attractors. It is likely to be a common phenomenon in biological dynamics.

Chimera states in coupled Kuramoto oscillators with inertia.

PubMed

Olmi, Simona

2015-12-01

The dynamics of two symmetrically coupled populations of rotators is studied for different values of the inertia. The system is characterized by different types of solutions, which all coexist with the fully synchronized state. At small inertia, the system is no more chaotic and one observes mainly quasi-periodic chimeras, while the usual (stationary) chimera state is not anymore observable. At large inertia, one observes two different kind of chaotic solutions with broken symmetry: the intermittent chaotic chimera, characterized by a synchronized population and a population displaying a turbulent behaviour, and a second state where the two populations are both chaotic but whose dynamics adhere to two different macroscopic attractors. The intermittent chaotic chimeras are characterized by a finite life-time, whose duration increases as a power-law with the system size and the inertia value. Moreover, the chaotic population exhibits clear intermittent behavior, displaying a laminar phase where the two populations tend to synchronize, and a turbulent phase where the macroscopic motion of one population is definitely erratic. In the thermodynamic limit, these states survive for infinite time and the laminar regimes tends to disappear, thus giving rise to stationary chaotic solutions with broken symmetry contrary to what observed for chaotic chimeras on a ring geometry.

Effect of Parametric Dichotomic Markov Noise on the Properties of Chaotic Transitions in Dynamical Systems

NASA Astrophysics Data System (ADS)

Gac, J. M.; Żebrowski, J. J.

A chaotic transition occurs when a continuous change of one of the parameters of the system causes a discontinuous change in the properties of the chaotic attractor of the system. Such phenomena are present in many dynamical systems, in which a chaotic behavior occurs. The best known of these transitions are: the period-doubling bifurcation cascade, intermittency and crises. The effect of dichotomous Markov noise (DMN) on the properties of systems with chaotic transitions is discussed. DMN is a very simple two-valued stochastic process, with constant transition rates between the two states. In spite of its simplicity, this kind of noise is a very powerful tool to describe various phenomena present in many physical, chemical or biological systems. Many interesting phenomena induced by DMN are known. However, there is no research on the effect of this kind of noise on intermittency or crises. We present the change of the mean laminar phase length and of laminar phase length distribution caused by DMN modulating the parameters of a system with intermittency and the modification of the mean life time on the pre-crisis attractor in the case of a boundary crisis. The results obtained analytically are compared with numerical simulations for several simple dynamical systems.

Transversal homoclinic orbits in a transiently chaotic neural network.

PubMed

Chen, Shyan-Shiou; Shih, Chih-Wen

2002-09-01

We study the existence of snap-back repellers, hence the existence of transversal homoclinic orbits in a discrete-time neural network. Chaotic behaviors for the network system in the sense of Li and Yorke or Marotto can then be concluded. The result is established by analyzing the structures of the system and allocating suitable parameters in constructing the fixed points and their pre-images for the system. The investigation provides a theoretical confirmation on the scenario of transient chaos for the system. All the parameter conditions for the theory can be examined numerically. The numerical ranges for the parameters which yield chaotic dynamics and convergent dynamics provide significant information in the annealing process in solving combinatorial optimization problems using this transiently chaotic neural network. (c) 2002 American Institute of Physics.

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.

Qualitative models and experimental investigation of chaotic NOR gates and set/reset flip-flops

NASA Astrophysics Data System (ADS)

Rahman, Aminur; Jordan, Ian; Blackmore, Denis

2018-01-01

It has been observed through experiments and SPICE simulations that logical circuits based upon Chua's circuit exhibit complex dynamical behaviour. This behaviour can be used to design analogues of more complex logic families and some properties can be exploited for electronics applications. Some of these circuits have been modelled as systems of ordinary differential equations. However, as the number of components in newer circuits increases so does the complexity. This renders continuous dynamical systems models impractical and necessitates new modelling techniques. In recent years, some discrete dynamical models have been developed using various simplifying assumptions. To create a robust modelling framework for chaotic logical circuits, we developed both deterministic and stochastic discrete dynamical models, which exploit the natural recurrence behaviour, for two chaotic NOR gates and a chaotic set/reset flip-flop. This work presents a complete applied mathematical investigation of logical circuits. Experiments on our own designs of the above circuits are modelled and the models are rigorously analysed and simulated showing surprisingly close qualitative agreement with the experiments. Furthermore, the models are designed to accommodate dynamics of similarly designed circuits. This will allow researchers to develop ever more complex chaotic logical circuits with a simple modelling framework.

Qualitative models and experimental investigation of chaotic NOR gates and set/reset flip-flops.

PubMed

Rahman, Aminur; Jordan, Ian; Blackmore, Denis

2018-01-01

It has been observed through experiments and SPICE simulations that logical circuits based upon Chua's circuit exhibit complex dynamical behaviour. This behaviour can be used to design analogues of more complex logic families and some properties can be exploited for electronics applications. Some of these circuits have been modelled as systems of ordinary differential equations. However, as the number of components in newer circuits increases so does the complexity. This renders continuous dynamical systems models impractical and necessitates new modelling techniques. In recent years, some discrete dynamical models have been developed using various simplifying assumptions. To create a robust modelling framework for chaotic logical circuits, we developed both deterministic and stochastic discrete dynamical models, which exploit the natural recurrence behaviour, for two chaotic NOR gates and a chaotic set/reset flip-flop. This work presents a complete applied mathematical investigation of logical circuits. Experiments on our own designs of the above circuits are modelled and the models are rigorously analysed and simulated showing surprisingly close qualitative agreement with the experiments. Furthermore, the models are designed to accommodate dynamics of similarly designed circuits. This will allow researchers to develop ever more complex chaotic logical circuits with a simple modelling framework.

Hybrid forecasting of chaotic processes: Using machine learning in conjunction with a knowledge-based model

NASA Astrophysics Data System (ADS)

Pathak, Jaideep; Wikner, Alexander; Fussell, Rebeckah; Chandra, Sarthak; Hunt, Brian R.; Girvan, Michelle; Ott, Edward

2018-04-01

A model-based approach to forecasting chaotic dynamical systems utilizes knowledge of the mechanistic processes governing the dynamics to build an approximate mathematical model of the system. In contrast, machine learning techniques have demonstrated promising results for forecasting chaotic systems purely from past time series measurements of system state variables (training data), without prior knowledge of the system dynamics. The motivation for this paper is the potential of machine learning for filling in the gaps in our underlying mechanistic knowledge that cause widely-used knowledge-based models to be inaccurate. Thus, we here propose a general method that leverages the advantages of these two approaches by combining a knowledge-based model and a machine learning technique to build a hybrid forecasting scheme. Potential applications for such an approach are numerous (e.g., improving weather forecasting). We demonstrate and test the utility of this approach using a particular illustrative version of a machine learning known as reservoir computing, and we apply the resulting hybrid forecaster to a low-dimensional chaotic system, as well as to a high-dimensional spatiotemporal chaotic system. These tests yield extremely promising results in that our hybrid technique is able to accurately predict for a much longer period of time than either its machine-learning component or its model-based component alone.

Generating a Double-Scroll Attractor by Connecting a Pair of Mutual Mirror-Image Attractors via Planar Switching Control

NASA Astrophysics Data System (ADS)

Sun, Changchun; Chen, Zhongtang; Xu, Qicheng

2017-12-01

An original three-dimensional (3D) smooth continuous chaotic system and its mirror-image system with eight common parameters are constructed and a pair of symmetric chaotic attractors can be generated simultaneously. Basic dynamical behaviors of two 3D chaotic systems are investigated respectively. A double-scroll chaotic attractor by connecting the pair of mutual mirror-image attractors is generated via a novel planar switching control approach. Chaos can also be controlled to a fixed point, a periodic orbit and a divergent orbit respectively by switching between two chaotic systems. Finally, an equivalent 3D chaotic system by combining two 3D chaotic systems with a switching law is designed by utilizing a sign function. Two circuit diagrams for realizing the double-scroll attractor are depicted by employing an improved module-based design approach.

A Simple Snap Oscillator with Coexisting Attractors, Its Time-Delayed Form, Physical Realization, and Communication Designs

NASA Astrophysics Data System (ADS)

Rajagopal, Karthikeyan; Jafari, Sajad; Akgul, Akif; Karthikeyan, Anitha; Çiçek, Serdar; Shekofteh, Yasser

2018-05-01

In this paper, we report a novel chaotic snap oscillator with one nonlinear function. Dynamic analysis of the system shows the existence of bistability. To study the time delay effects on the proposed snap oscillator, we introduce multiple time delay in the fourth state equation. Investigation of dynamical properties of the time-delayed system shows that the snap oscillator exhibits the same multistable properties as the nondelayed system. The new multistable hyperjerk chaotic system has been tested in chaos shift keying and symmetric choc shift keying modulated communication designs for engineering applications. It has been determined that the symmetric chaos shift keying modulated communication system implemented with the new chaotic system is more successful than the chaos shift keying modulation for secure communication. Also, circuit implementation of the chaotic snap oscillator with tangent function is carried out showing its feasibility.

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.

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

Munoz, Pablo R.; Rempel, Erico L.; Barroso, Joaquim J.

We study the chaotic dynamics of the Pierce diode, a simple spatially extended system for collisionless bounded plasmas, focusing on the concept of edge of chaos, the boundary that separates transient from asymptotic dynamics. We fully characterize an interior crisis at the end of a periodic window, thereby showing direct evidence of the collision between a chaotic attractor, a chaotic saddle, and the edge of chaos, formed by a period-3 unstable periodic orbit and its stable manifold. The edge of chaos persists after the interior crisis, when the global attractor of the system increases its size in the phase space.

A new approach of optimal control for a class of continuous-time chaotic systems by an online ADP algorithm

NASA Astrophysics Data System (ADS)

Song, Rui-Zhuo; Xiao, Wen-Dong; Wei, Qing-Lai

2014-05-01

We develop an online adaptive dynamic programming (ADP) based optimal control scheme for continuous-time chaotic systems. The idea is to use the ADP algorithm to obtain the optimal control input that makes the performance index function reach an optimum. The expression of the performance index function for the chaotic system is first presented. The online ADP algorithm is presented to achieve optimal control. In the ADP structure, neural networks are used to construct a critic network and an action network, which can obtain an approximate performance index function and the control input, respectively. It is proven that the critic parameter error dynamics and the closed-loop chaotic systems are uniformly ultimately bounded exponentially. Our simulation results illustrate the performance of the established optimal control method.

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

Olmi, Simona, E-mail: simona.olmi@fi.isc.cnr.it; INFN Sez. Firenze, via Sansone, 1 - I-50019 Sesto Fiorentino

The dynamics of two symmetrically coupled populations of rotators is studied for different values of the inertia. The system is characterized by different types of solutions, which all coexist with the fully synchronized state. At small inertia, the system is no more chaotic and one observes mainly quasi-periodic chimeras, while the usual (stationary) chimera state is not anymore observable. At large inertia, one observes two different kind of chaotic solutions with broken symmetry: the intermittent chaotic chimera, characterized by a synchronized population and a population displaying a turbulent behaviour, and a second state where the two populations are both chaoticmore » but whose dynamics adhere to two different macroscopic attractors. The intermittent chaotic chimeras are characterized by a finite life-time, whose duration increases as a power-law with the system size and the inertia value. Moreover, the chaotic population exhibits clear intermittent behavior, displaying a laminar phase where the two populations tend to synchronize, and a turbulent phase where the macroscopic motion of one population is definitely erratic. In the thermodynamic limit, these states survive for infinite time and the laminar regimes tends to disappear, thus giving rise to stationary chaotic solutions with broken symmetry contrary to what observed for chaotic chimeras on a ring geometry.« less

A full computation-relevant topological dynamics classification of elementary cellular automata.

PubMed

Schüle, Martin; Stoop, Ruedi

2012-12-01

Cellular automata are both computational and dynamical systems. We give a complete classification of the dynamic behaviour of elementary cellular automata (ECA) in terms of fundamental dynamic system notions such as sensitivity and chaoticity. The "complex" ECA emerge to be sensitive, but not chaotic and not eventually weakly periodic. Based on this classification, we conjecture that elementary cellular automata capable of carrying out complex computations, such as needed for Turing-universality, are at the "edge of chaos."

Analytically solvable chaotic oscillator based on a first-order filter.

PubMed

Corron, Ned J; Cooper, Roy M; Blakely, Jonathan N

2016-02-01

A chaotic hybrid dynamical system is introduced and its analytic solution is derived. The system is described as an unstable first order filter subject to occasional switching of a set point according to a feedback rule. The system qualitatively differs from other recently studied solvable chaotic hybrid systems in that the timing of the switching is regulated by an external clock. The chaotic analytic solution is an optimal waveform for communications in noise when a resistor-capacitor-integrate-and-dump filter is used as a receiver. As such, these results provide evidence in support of a recent conjecture that the optimal communication waveform for any stable infinite-impulse response filter is chaotic.

Analytically solvable chaotic oscillator based on a first-order filter

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

Corron, Ned J.; Cooper, Roy M.; Blakely, Jonathan N.

2016-02-15

A chaotic hybrid dynamical system is introduced and its analytic solution is derived. The system is described as an unstable first order filter subject to occasional switching of a set point according to a feedback rule. The system qualitatively differs from other recently studied solvable chaotic hybrid systems in that the timing of the switching is regulated by an external clock. The chaotic analytic solution is an optimal waveform for communications in noise when a resistor-capacitor-integrate-and-dump filter is used as a receiver. As such, these results provide evidence in support of a recent conjecture that the optimal communication waveform formore » any stable infinite-impulse response filter is chaotic.« less

Stability enhancement of high Prandtl number chaotic convection in an anisotropic porous layer with feedback control

NASA Astrophysics Data System (ADS)

Mahmud, M. N.

2018-04-01

The chaotic dynamical behaviour of thermal convection in an anisotropic porous layer subject to gravity, heated from below and cooled from above, is studied based on theory of dynamical system in the presence of feedback control. The extended Darcy model, which includes the time derivative has been employed in the momentum equation to derive a low dimensional Lorenz-like equation by using Galerkin-truncated approximation. The classical fourth-order Runge-Kutta method is used to obtain the numerical solution in order to exemplify the dynamics of the nonlinear autonomous system. The results show that stability enhancement of chaotic convection is feasible via feedback control.

Gross-Pitaevski map as a chaotic dynamical system.

PubMed

Guarneri, Italo

2017-03-01

The Gross-Pitaevski map is a discrete time, split-operator version of the Gross-Pitaevski dynamics in the circle, for which exponential instability has been recently reported. Here it is studied as a classical dynamical system in its own right. A systematic analysis of Lyapunov exponents exposes strongly chaotic behavior. Exponential growth of energy is then shown to be a direct consequence of rotational invariance and for stationary solutions the full spectrum of Lyapunov exponents is analytically computed. The present analysis includes the "resonant" case, when the free rotation period is commensurate to 2π, and the map has countably many constants of the motion. Except for lowest-order resonances, this case exhibits an integrable-chaotic transition.

Chaos in the sunspot cycle - Analysis and prediction

NASA Technical Reports Server (NTRS)

Mundt, Michael D.; Maguire, W. Bruce, II; Chase, Robert R. P.

1991-01-01

The variability of solar activity over long time scales, given semiquantitatively by measurements of sunspot numbers, is examined as a nonlinear dynamical system. First, a discussion of the data set used and the techniques utilized to reduce the noise and capture the long-term dynamics inherent in the data is presented. Subsequently, an attractor is reconstructed from the data set using the method of time delays. The reconstructed attractor is then used to determine both the dimension of the underlying system and also the largest Lyapunov exponent, which together indicate that the sunspot cycle is indeed chaotic and also low dimensional. In addition, recent techniques of exploiting chaotic dynamics to provide accurate, short-term predictions are utilized in order to improve upon current forecasting methods and also to place theoretical limits on predictability extent. The results are compared to chaotic solar-dynamo models as a possible physically motivated source of this chaotic behavior.

Quantum-chaotic cryptography

NASA Astrophysics Data System (ADS)

de Oliveira, G. L.; Ramos, R. V.

2018-03-01

In this work, it is presented an optical scheme for quantum key distribution employing two synchronized optoelectronic oscillators (OEO) working in the chaotic regime. The produced key depends on the chaotic dynamic, and the synchronization between Alice's and Bob's OEOs uses quantum states. An attack on the synchronization signals will disturb the synchronization of the chaotic systems increasing the error rate in the final key.

A chaotic secure communication scheme using fractional chaotic systems based on an extended fractional Kalman filter

NASA Astrophysics Data System (ADS)

Kiani-B, Arman; Fallahi, Kia; Pariz, Naser; Leung, Henry

2009-03-01

In recent years chaotic secure communication and chaos synchronization have received ever increasing attention. In this paper, for the first time, a fractional chaotic communication method using an extended fractional Kalman filter is presented. The chaotic synchronization is implemented by the EFKF design in the presence of channel additive noise and processing noise. Encoding chaotic communication achieves a satisfactory, typical secure communication scheme. In the proposed system, security is enhanced based on spreading the signal in frequency and encrypting it in time domain. In this paper, the main advantages of using fractional order systems, increasing nonlinearity and spreading the power spectrum are highlighted. To illustrate the effectiveness of the proposed scheme, a numerical example based on the fractional Lorenz dynamical system is presented and the results are compared to the integer Lorenz system.

A reducing of a chaotic movement to a periodic orbit, of a micro-electro-mechanical system, by using an optimal linear control design

NASA Astrophysics Data System (ADS)

Chavarette, Fábio Roberto; Balthazar, José Manoel; Felix, Jorge L. P.; Rafikov, Marat

2009-05-01

This paper analyzes the non-linear dynamics, with a chaotic behavior of a particular micro-electro-mechanical system. We used a technique of the optimal linear control for reducing the irregular (chaotic) oscillatory movement of the non-linear systems to a periodic orbit. We use the mathematical model of a (MEMS) proposed by Luo and Wang.

Chaotic behavior in Malaysian stock market: A study with recurrence quantification analysis

NASA Astrophysics Data System (ADS)

Niu, Betty Voon Wan; Noorani, Mohd Salmi Md; Jaaman, Saiful Hafizah

2016-11-01

The dynamics of stock market has been questioned for decades. Its behavior appeared random yet some found it behaves as chaos. Up to 5000 daily adjusted closing data of FTSE Bursa Malaysia Kuala Lumpur Composite Index (KLSE) was investigated through recurrence plot and recurrence quantification analysis. Results were compared between stochastic system, chaotic system and deterministic system. Results show that KLSE daily adjusted closing data behaves chaotically.

Chaotic Motions in the Real Fuzzy Electronic Circuits

DTIC Science & Technology

2012-12-30

field of secure communications, the original source should be blended with other complex signals. Chaotic signals are one of the good sources to be...Takagi-Sugeno (T-S) fuzzy chaotic systems on electronic circuit. In the research field of secure communications, the original source should be blended ...model. The overall fuzzy model of the system is achieved by fuzzy blending of the linear system models. Consider a continuous-time nonlinear dynamic

Bluetooth based chaos synchronization using particle swarm optimization and its applications to image encryption.

PubMed

Yau, Her-Terng; Hung, Tzu-Hsiang; Hsieh, Chia-Chun

2012-01-01

This study used the complex dynamic characteristics of chaotic systems and Bluetooth to explore the topic of wireless chaotic communication secrecy and develop a communication security system. The PID controller for chaos synchronization control was applied, and the optimum parameters of this PID controller were obtained using a Particle Swarm Optimization (PSO) algorithm. Bluetooth was used to realize wireless transmissions, and a chaotic wireless communication security system was developed in the design concept of a chaotic communication security system. The experimental results show that this scheme can be used successfully in image encryption.

Uncovering low dimensional macroscopic chaotic dynamics of large finite size complex systems

NASA Astrophysics Data System (ADS)

Skardal, Per Sebastian; Restrepo, Juan G.; Ott, Edward

2017-08-01

In the last decade, it has been shown that a large class of phase oscillator models admit low dimensional descriptions for the macroscopic system dynamics in the limit of an infinite number N of oscillators. The question of whether the macroscopic dynamics of other similar systems also have a low dimensional description in the infinite N limit has, however, remained elusive. In this paper, we show how techniques originally designed to analyze noisy experimental chaotic time series can be used to identify effective low dimensional macroscopic descriptions from simulations with a finite number of elements. We illustrate and verify the effectiveness of our approach by applying it to the dynamics of an ensemble of globally coupled Landau-Stuart oscillators for which we demonstrate low dimensional macroscopic chaotic behavior with an effective 4-dimensional description. By using this description, we show that one can calculate dynamical invariants such as Lyapunov exponents and attractor dimensions. One could also use the reconstruction to generate short-term predictions of the macroscopic dynamics.

From globally coupled maps to complex-systems biology

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

Kaneko, Kunihiko, E-mail: kaneko@complex.c.u-tokyo.ac.jp

Studies of globally coupled maps, introduced as a network of chaotic dynamics, are briefly reviewed with an emphasis on novel concepts therein, which are universal in high-dimensional dynamical systems. They include clustering of synchronized oscillations, hierarchical clustering, chimera of synchronization and desynchronization, partition complexity, prevalence of Milnor attractors, chaotic itinerancy, and collective chaos. The degrees of freedom necessary for high dimensionality are proposed to equal the number in which the combinatorial exceeds the exponential. Future analysis of high-dimensional dynamical systems with regard to complex-systems biology is briefly discussed.

Chaotic behaviour of the Rossler model and its analysis by using bifurcations of limit cycles and chaotic attractors

NASA Astrophysics Data System (ADS)

Ibrahim, K. M.; Jamal, R. K.; Ali, F. H.

2018-05-01

The behaviour of certain dynamical nonlinear systems are described in term as chaos, i.e., systems’ variables change with the time, displaying very sensitivity to initial conditions of chaotic dynamics. In this paper, we study archetype systems of ordinary differential equations in two-dimensional phase spaces of the Rössler model. A system displays continuous time chaos and is explained by three coupled nonlinear differential equations. We study its characteristics and determine the control parameters that lead to different behavior of the system output, periodic, quasi-periodic and chaos. The time series, attractor, Fast Fourier Transformation and bifurcation diagram for different values have been described.

Chimeras and clusters in networks of hyperbolic chaotic oscillators

NASA Astrophysics Data System (ADS)

Cano, A. V.; Cosenza, M. G.

2017-03-01

We show that chimera states, where differentiated subsets of synchronized and desynchronized dynamical elements coexist, can emerge in networks of hyperbolic chaotic oscillators subject to global interactions. As local dynamics we employ Lozi maps, which possess hyperbolic chaotic attractors. We consider a globally coupled system of these maps and use two statistical quantities to describe its collective behavior: the average fraction of elements belonging to clusters and the average standard deviation of state variables. Chimera states, clusters, complete synchronization, and incoherence are thus characterized on the space of parameters of the system. We find that chimera states are related to the formation of clusters in the system. In addition, we show that chimera states arise for a sufficiently long range of interactions in nonlocally coupled networks of these maps. Our results reveal that, under some circumstances, hyperbolicity does not impede the formation of chimera states in networks of coupled chaotic systems, as it had been previously hypothesized.

Statistical inference for noisy nonlinear ecological dynamic systems.

PubMed

Wood, Simon N

2010-08-26

Chaotic ecological dynamic systems defy conventional statistical analysis. Systems with near-chaotic dynamics are little better. Such systems are almost invariably driven by endogenous dynamic processes plus demographic and environmental process noise, and are only observable with error. Their sensitivity to history means that minute changes in the driving noise realization, or the system parameters, will cause drastic changes in the system trajectory. This sensitivity is inherited and amplified by the joint probability density of the observable data and the process noise, rendering it useless as the basis for obtaining measures of statistical fit. Because the joint density is the basis for the fit measures used by all conventional statistical methods, this is a major theoretical shortcoming. The inability to make well-founded statistical inferences about biological dynamic models in the chaotic and near-chaotic regimes, other than on an ad hoc basis, leaves dynamic theory without the methods of quantitative validation that are essential tools in the rest of biological science. Here I show that this impasse can be resolved in a simple and general manner, using a method that requires only the ability to simulate the observed data on a system from the dynamic model about which inferences are required. The raw data series are reduced to phase-insensitive summary statistics, quantifying local dynamic structure and the distribution of observations. Simulation is used to obtain the mean and the covariance matrix of the statistics, given model parameters, allowing the construction of a 'synthetic likelihood' that assesses model fit. This likelihood can be explored using a straightforward Markov chain Monte Carlo sampler, but one further post-processing step returns pure likelihood-based inference. I apply the method to establish the dynamic nature of the fluctuations in Nicholson's classic blowfly experiments.

Experimental study of firing death in a network of chaotic FitzHugh-Nagumo neurons

NASA Astrophysics Data System (ADS)

Ciszak, Marzena; Euzzor, Stefano; Arecchi, F. Tito; Meucci, Riccardo

2013-02-01

The FitzHugh-Nagumo neurons driven by a periodic forcing undergo a period-doubling route to chaos and a transition to mixed-mode oscillations. When coupled, their dynamics tend to be synchronized. We show that the chaotically spiking neurons change their internal dynamics to subthreshold oscillations, the phenomenon referred to as firing death. These dynamical changes are observed below the critical coupling strength at which the transition to full chaotic synchronization occurs. Moreover, we find various dynamical regimes in the subthreshold oscillations, namely, regular, quasiperiodic, and chaotic states. We show numerically that these dynamical states may coexist with large-amplitude spiking regimes and that this coexistence is characterized by riddled basins of attraction. The reported results are obtained for neurons implemented in the electronic circuits as well as for the model equations. Finally, we comment on the possible scenarios where the coupling-induced firing death could play an important role in biological systems.

Generation of 2N + 1-scroll existence in new three-dimensional chaos systems.

PubMed

Liu, Yue; Guan, Jian; Ma, Chunyang; Guo, Shuxu

2016-08-01

We propose a systematic methodology for creating 2N + 1-scroll chaotic attractors from a simple three-dimensional system, which is named as the translation chaotic system. It satisfies the condition a12a21 = 0, while the Chua system satisfies a12a21 > 0. In this paper, we also propose a successful (an effective) design and an analytical approach for constructing 2N + 1-scrolls, the translation transformation principle. Also, the dynamics properties of the system are studied in detail. MATLAB simulation results show very sophisticated dynamical behaviors and unique chaotic behaviors of the system. It provides a new approach for 2N + 1-scroll attractors. Finally, to explore the potential use in technological applications, a novel block circuit diagram is also designed for the hardware implementation of 1-, 3-, 5-, and 7-scroll attractors via switching the switches. Translation chaotic system has the merit of convenience and high sensitivity to initial values, emerging potentials in future engineering chaos design.

Modelling chaotic vibrations using NASTRAN

NASA Technical Reports Server (NTRS)

Sheerer, T. J.

1993-01-01

Due to the unavailability and, later, prohibitive cost of the computational power required, many phenomena in nonlinear dynamic systems have in the past been addressed in terms of linear systems. Linear systems respond to periodic inputs with periodic outputs, and may be characterized in the time domain or in the frequency domain as convenient. Reduction to the frequency domain is frequently desireable to reduce the amount of computation required for solution. Nonlinear systems are only soluble in the time domain, and may exhibit a time history which is extremely sensitive to initial conditions. Such systems are termed chaotic. Dynamic buckling, aeroelasticity, fatigue analysis, control systems and electromechanical actuators are among the areas where chaotic vibrations have been observed. Direct transient analysis over a long time period presents a ready means of simulating the behavior of self-excited or externally excited nonlinear systems for a range of experimental parameters, either to characterize chaotic behavior for development of load spectra, or to define its envelope and preclude its occurrence.

A Novel Type of Chaotic Attractor for Quadratic Systems Without Equilibriums

NASA Astrophysics Data System (ADS)

Dantsev, Danylo

In this paper, a new chaotic dynamic system without equilibriums is presented. A conducted research of the qualitative properties of the discovered system reveals a noncompliance between the bifurcation behavior of the system and the Feigenbaum-Sharkovskii-Magnitsky theory. Additional research of known systems confirms the discrepancy.

Exact folded-band chaotic oscillator.

PubMed

Corron, Ned J; Blakely, Jonathan N

2012-06-01

An exactly solvable chaotic oscillator with folded-band dynamics is shown. The oscillator is a hybrid dynamical system containing a linear ordinary differential equation and a nonlinear switching condition. Bounded oscillations are provably chaotic, and successive waveform maxima yield a one-dimensional piecewise-linear return map with segments of both positive and negative slopes. Continuous-time dynamics exhibit a folded-band topology similar to Rössler's oscillator. An exact solution is written as a linear convolution of a fixed basis pulse and a discrete binary sequence, from which an equivalent symbolic dynamics is obtained. The folded-band topology is shown to be dependent on the symbol grammar.

A mixed analog/digital chaotic neuro-computer system for quadratic assignment problems.

PubMed

Horio, Yoshihiko; Ikeguchi, Tohru; Aihara, Kazuyuki

2005-01-01

We construct a mixed analog/digital chaotic neuro-computer prototype system for quadratic assignment problems (QAPs). The QAP is one of the difficult NP-hard problems, and includes several real-world applications. Chaotic neural networks have been used to solve combinatorial optimization problems through chaotic search dynamics, which efficiently searches optimal or near optimal solutions. However, preliminary experiments have shown that, although it obtained good feasible solutions, the Hopfield-type chaotic neuro-computer hardware system could not obtain the optimal solution of the QAP. Therefore, in the present study, we improve the system performance by adopting a solution construction method, which constructs a feasible solution using the analog internal state values of the chaotic neurons at each iteration. In order to include the construction method into our hardware, we install a multi-channel analog-to-digital conversion system to observe the internal states of the chaotic neurons. We show experimentally that a great improvement in the system performance over the original Hopfield-type chaotic neuro-computer is obtained. That is, we obtain the optimal solution for the size-10 QAP in less than 1000 iterations. In addition, we propose a guideline for parameter tuning of the chaotic neuro-computer system according to the observation of the internal states of several chaotic neurons in the network.

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

Grebogi, C.; Yorke, J.A.

This report discusses the following topics: controlling chaotic dynamical systems; embedding of experimental data; effect of noise on critical exponents of crises; transition to chaotic scattering; and distribution of floaters on a fluid surface. (LSP)

Least Squares Shadowing Sensitivity Analysis of Chaotic Flow Around a Two-Dimensional Airfoil

NASA Technical Reports Server (NTRS)

Blonigan, Patrick J.; Wang, Qiqi; Nielsen, Eric J.; Diskin, Boris

2016-01-01

Gradient-based sensitivity analysis has proven to be an enabling technology for many applications, including design of aerospace vehicles. However, conventional sensitivity analysis methods break down when applied to long-time averages of chaotic systems. This breakdown is a serious limitation because many aerospace applications involve physical phenomena that exhibit chaotic dynamics, most notably high-resolution large-eddy and direct numerical simulations of turbulent aerodynamic flows. A recently proposed methodology, Least Squares Shadowing (LSS), avoids this breakdown and advances the state of the art in sensitivity analysis for chaotic flows. The first application of LSS to a chaotic flow simulated with a large-scale computational fluid dynamics solver is presented. The LSS sensitivity computed for this chaotic flow is verified and shown to be accurate, but the computational cost of the current LSS implementation is high.

Chaotic dynamics of Heisenberg ferromagnetic spin chain with bilinear and biquadratic interactions

NASA Astrophysics Data System (ADS)

Blessy, B. S. Gnana; Latha, M. M.

2017-10-01

We investigate the chaotic dynamics of one dimensional Heisenberg ferromagnetic spin chain by constructing the Hamiltonian equations of motion. We present the trajectory and phase plots of the system with bilinear and also biquadratic interactions. The stability of the system is analysed in both cases by constructing the Jacobian matrix and by measuring the Lyapunov exponents. The results are illustrated graphically.

Extreme multistability analysis of memristor-based chaotic system and its application in image decryption

NASA Astrophysics Data System (ADS)

Li, Chuang; Min, Fuhong; Jin, Qiusen; Ma, Hanyuan

2017-12-01

An active charge-controlled memristive Chua's circuit is implemented, and its basic properties are analyzed. Firstly, with the system trajectory starting from an equilibrium point, the dynamic behavior of multiple coexisting attractors depending on the memristor initial value and the system parameter is studied, which shows the coexisting behaviors of point, period, chaos, and quasic-period. Secondly, with the system motion starting from a non-equilibrium point, the dynamics of extreme multistability in a wide initial value domain are easily conformed by new analytical methods. Furthermore, the simulation results indicate that some strange chaotic attractors like multi-wing type and multi-scroll type are observed when the observed signals are extended from voltage and current to power and energy, respectively. Specially, when different initial conditions are taken, the coexisting strange chaotic attractors between the power and energy signals are exhibited. Finally, the chaotic sequences of the new system are used for encrypting color image to protect image information security. The encryption performance is analyzed by statistic histogram, correlation, key spaces and key sensitivity. Simulation results show that the new memristive chaotic system has high security in color image encryption.

On Chaotic and Hyperchaotic Complex Nonlinear Dynamical Systems

NASA Astrophysics Data System (ADS)

Mahmoud, Gamal M.

Dynamical systems described by real and complex variables are currently one of the most popular areas of scientific research. These systems play an important role in several fields of physics, engineering, and computer sciences, for example, laser systems, control (or chaos suppression), secure communications, and information science. Dynamical basic properties, chaos (hyperchaos) synchronization, chaos control, and generating hyperchaotic behavior of these systems are briefly summarized. The main advantage of introducing complex variables is the reduction of phase space dimensions by a half. They are also used to describe and simulate the physics of detuned laser and thermal convection of liquid flows, where the electric field and the atomic polarization amplitudes are both complex. Clearly, if the variables of the system are complex the equations involve twice as many variables and control parameters, thus making it that much harder for a hostile agent to intercept and decipher the coded message. Chaotic and hyperchaotic complex systems are stated as examples. Finally there are many open problems in the study of chaotic and hyperchaotic complex nonlinear dynamical systems, which need further investigations. Some of these open problems are given.

Devaney chaos plus shadowing implies distributional chaos.

PubMed

Li, Jian; Li, Jie; Tu, Siming

2016-09-01

We explore connections among the regional proximal relation, the asymptotic relation, and the distal relation for a topological dynamical system with the shadowing property and show that if a Devaney chaotic system has the shadowing property then it is distributionally chaotic.

On the robustness of complex heterogeneous gene expression networks.

PubMed

Gómez-Gardeñes, Jesús; Moreno, Yamir; Floría, Luis M

2005-04-01

We analyze a continuous gene expression model on the underlying topology of a complex heterogeneous network. Numerical simulations aimed at studying the chaotic and periodic dynamics of the model are performed. The results clearly indicate that there is a region in which the dynamical and structural complexity of the system avoid chaotic attractors. However, contrary to what has been reported for Random Boolean Networks, the chaotic phase cannot be completely suppressed, which has important bearings on network robustness and gene expression modeling.

Impact of hyperbolicity on chimera states in ensembles of nonlocally coupled chaotic oscillators

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

Semenova, N.; Anishchenko, V.; Zakharova, A.

2016-06-08

In this work we analyse nonlocally coupled networks of identical chaotic oscillators. We study both time-discrete and time-continuous systems (Henon map, Lozi map, Lorenz system). We hypothesize that chimera states, in which spatial domains of coherent (synchronous) and incoherent (desynchronized) dynamics coexist, can be obtained only in networks of chaotic non-hyperbolic systems and cannot be found in networks of hyperbolic systems. This hypothesis is supported by numerical simulations for hyperbolic and non-hyperbolic cases.

Synchronization of chaotic systems

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

Pecora, Louis M.; Carroll, Thomas L.

2015-09-15

We review some of the history and early work in the area of synchronization in chaotic systems. We start with our own discovery of the phenomenon, but go on to establish the historical timeline of this topic back to the earliest known paper. The topic of synchronization of chaotic systems has always been intriguing, since chaotic systems are known to resist synchronization because of their positive Lyapunov exponents. The convergence of the two systems to identical trajectories is a surprise. We show how people originally thought about this process and how the concept of synchronization changed over the years tomore » a more geometric view using synchronization manifolds. We also show that building synchronizing systems leads naturally to engineering more complex systems whose constituents are chaotic, but which can be tuned to output various chaotic signals. We finally end up at a topic that is still in very active exploration today and that is synchronization of dynamical systems in networks of oscillators.« less

Geometric and dynamic perspectives on phase-coherent and noncoherent chaos.

PubMed

Zou, Yong; Donner, Reik V; Kurths, Jürgen

2012-03-01

Statistically distinguishing between phase-coherent and noncoherent chaotic dynamics from time series is a contemporary problem in nonlinear sciences. In this work, we propose different measures based on recurrence properties of recorded trajectories, which characterize the underlying systems from both geometric and dynamic viewpoints. The potentials of the individual measures for discriminating phase-coherent and noncoherent chaotic oscillations are discussed. A detailed numerical analysis is performed for the chaotic Rössler system, which displays both types of chaos as one control parameter is varied, and the Mackey-Glass system as an example of a time-delay system with noncoherent chaos. Our results demonstrate that especially geometric measures from recurrence network analysis are well suited for tracing transitions between spiral- and screw-type chaos, a common route from phase-coherent to noncoherent chaos also found in other nonlinear oscillators. A detailed explanation of the observed behavior in terms of attractor geometry is given.

Fractal and chaotic laws on seismic dissipated energy in an energy system of engineering structures

NASA Astrophysics Data System (ADS)

Cui, Yu-Hong; Nie, Yong-An; Yan, Zong-Da; Wu, Guo-You

1998-09-01

Fractal and chaotic laws of engineering structures are discussed in this paper, it means that the intrinsic essences and laws on dynamic systems which are made from seismic dissipated energy intensity E d and intensity of seismic dissipated energy moment I e are analyzed. Based on the intrinsic characters of chaotic and fractal dynamic system of E d and I e, three kinds of approximate dynamic models are rebuilt one by one: index autoregressive model, threshold autoregressive model and local-approximate autoregressive model. The innate laws, essences and systematic error of evolutional behavior I e are explained over all, the short-term behavior predictability and long-term behavior probability of which are analyzed in the end. That may be valuable for earthquake-resistant theory and analysis method in practical engineering structures.

Suppression of chaos via control of energy flow

NASA Astrophysics Data System (ADS)

Guo, Shengli; Ma, Jun; Alsaedi, Ahmed

2018-03-01

Continuous energy supply is critical and important to support oscillating behaviour; otherwise, the oscillator will die. For nonlinear and chaotic circuits, enough energy supply is also important to keep electric devices working. In this paper, Hamilton energy is calculated for dimensionless dynamical system (e.g., the chaotic Lorenz system) using Helmholtz's theorem. The Hamilton energy is considered as a new variable and then the dynamical system is controlled by using the scheme of energy feedback. It is found that chaos can be suppressed even when intermittent feedback scheme is applied. This scheme is effective to control chaos and to stabilise other dynamical systems.

On the adaptivity and complexity embedded into differential evolution

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

Senkerik, Roman; Pluhacek, Michal; Jasek, Roman

2016-06-08

This research deals with the comparison of the two modern approaches for evolutionary algorithms, which are the adaptivity and complex chaotic dynamics. This paper aims on the investigations on the chaos-driven Differential Evolution (DE) concept. This paper is aimed at the embedding of discrete dissipative chaotic systems in the form of chaotic pseudo random number generators for the DE and comparing the influence to the performance with the state of the art adaptive representative jDE. This research is focused mainly on the possible disadvantages and advantages of both compared approaches. Repeated simulations for Lozi map driving chaotic systems were performedmore » on the simple benchmark functions set, which are more close to the real optimization problems. Obtained results are compared with the canonical not-chaotic and not adaptive DE. Results show that with used simple test functions, the performance of ChaosDE is better in the most cases than jDE and Canonical DE, furthermore due to the unique sequencing in CPRNG given by the hidden chaotic dynamics, thus better and faster selection of unique individuals from population, ChaosDE is faster.« less

The complexity of proving chaoticity and the Church-Turing thesis

NASA Astrophysics Data System (ADS)

Calude, Cristian S.; Calude, Elena; Svozil, Karl

2010-09-01

Proving the chaoticity of some dynamical systems is equivalent to solving the hardest problems in mathematics. Conversely, classical physical systems may "compute the hard or even the incomputable" by measuring observables which correspond to computationally hard or even incomputable problems.

Implementation of an integrated op-amp based chaotic neuron model and observation of its chaotic dynamics

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

Jung, Jinwoo; Lee, Jewon; Song, Hanjung

2011-03-15

This paper presents a fully integrated circuit implementation of an operational amplifier (op-amp) based chaotic neuron model with a bipolar output function, experimental measurements, and analyses of its chaotic behavior. The proposed chaotic neuron model integrated circuit consists of several op-amps, sample and hold circuits, a nonlinear function block for chaotic signal generation, a clock generator, a nonlinear output function, etc. Based on the HSPICE (circuit program) simulation results, approximated empirical equations for analyses were formulated. Then, the chaotic dynamical responses such as bifurcation diagrams, time series, and Lyapunov exponent were calculated using these empirical equations. In addition, we performedmore » simulations about two chaotic neuron systems with four synapses to confirm neural network connections and got normal behavior of the chaotic neuron such as internal state bifurcation diagram according to the synaptic weight variation. The proposed circuit was fabricated using a 0.8-{mu}m single poly complementary metal-oxide semiconductor technology. Measurements of the fabricated single chaotic neuron with {+-}2.5 V power supplies and a 10 kHz sampling clock frequency were carried out and compared with the simulated results.« less

Experimental Demonstration of Coherent Control in Quantum Chaotic Systems

NASA Astrophysics Data System (ADS)

Bitter, M.; Milner, V.

2017-01-01

We experimentally demonstrate coherent control of a quantum system, whose dynamics is chaotic in the classical limit. Interaction of diatomic molecules with a periodic sequence of ultrashort laser pulses leads to the dynamical localization of the molecular angular momentum, a characteristic feature of the chaotic quantum kicked rotor. By changing the phases of the rotational states in the initially prepared coherent wave packet, we control the rotational distribution of the final localized state and its total energy. We demonstrate the anticipated sensitivity of control to the exact parameters of the kicking field, as well as its disappearance in the classical regime of excitation.

Suppression of chaos at slow variables by rapidly mixing fast dynamics

NASA Astrophysics Data System (ADS)

Abramov, R.

2012-04-01

One of the key questions about chaotic multiscale systems is how the fast dynamics affects chaos at the slow variables, and, therefore, impacts uncertainty and predictability of the slow dynamics. Here we demonstrate that the linear slow-fast coupling with the total energy conservation property promotes the suppression of chaos at the slow variables through the rapid mixing at the fast variables, both theoretically and through numerical simulations. A suitable mathematical framework is developed, connecting the slow dynamics on the tangent subspaces to the infinite-time linear response of the mean state to a constant external forcing at the fast variables. Additionally, it is shown that the uncoupled dynamics for the slow variables may remain chaotic while the complete multiscale system loses chaos and becomes completely predictable at the slow variables through increasing chaos and turbulence at the fast variables. This result contradicts the common sense intuition, where, naturally, one would think that coupling a slow weakly chaotic system with another much faster and much stronger mixing system would result in general increase of chaos at the slow variables.

Hysteresis-induced bifurcation and chaos in a magneto-rheological suspension system under external excitation

NASA Astrophysics Data System (ADS)

Hailong, Zhang; Enrong, Wang; Fuhong, Min; Ning, Zhang

2016-03-01

The magneto-rheological damper (MRD) is a promising device used in vehicle semi-active suspension systems, for its continuous adjustable damping output. However, the innate nonlinear hysteresis characteristic of MRD may cause the nonlinear behaviors. In this work, a two-degree-of-freedom (2-DOF) MR suspension system was established first, by employing the modified Bouc-Wen force-velocity (F-v) hysteretic model. The nonlinear dynamic response of the system was investigated under the external excitation of single-frequency harmonic and bandwidth-limited stochastic road surface. The largest Lyapunov exponent (LLE) was used to detect the chaotic area of the frequency and amplitude of harmonic excitation, and the bifurcation diagrams, time histories, phase portraits, and power spectrum density (PSD) diagrams were used to reveal the dynamic evolution process in detail. Moreover, the LLE and Kolmogorov entropy (K entropy) were used to identify whether the system response was random or chaotic under stochastic road surface. The results demonstrated that the complex dynamical behaviors occur under different external excitation conditions. The oscillating mechanism of alternating periodic oscillations, quasi-periodic oscillations, and chaotic oscillations was observed in detail. The chaotic regions revealed that chaotic motions may appear in conditions of mid-low frequency and large amplitude, as well as small amplitude and all frequency. The obtained parameter regions where the chaotic motions may appear are useful for design of structural parameters of the vibration isolation, and the optimization of control strategy for MR suspension system. Projects supported by the National Natural Science Foundation of China (Grant Nos. 51475246, 51277098, and 51075215), the Research Innovation Program for College Graduates of Jiangsu Province China (Grant No. KYLX15 0725), and the Natural Science Foundation of Jiangsu Province of China (Grant No. BK20131402).

Theory of chaotic orbital variations confirmed by Cretaceous geological evidence

NASA Astrophysics Data System (ADS)

Ma, Chao; Meyers, Stephen R.; Sageman, Bradley B.

2017-02-01

Variations in the Earth’s orbit and spin vector are a primary control on insolation and climate; their recognition in the geological record has revolutionized our understanding of palaeoclimate dynamics, and has catalysed improvements in the accuracy and precision of the geological timescale. Yet the secular evolution of the planetary orbits beyond 50 million years ago remains highly uncertain, and the chaotic dynamical nature of the Solar System predicted by theoretical models has yet to be rigorously confirmed by well constrained (radioisotopically calibrated and anchored) geological data. Here we present geological evidence for a chaotic resonance transition associated with interactions between the orbits of Mars and the Earth, using an integrated radioisotopic and astronomical timescale from the Cretaceous Western Interior Basin of what is now North America. This analysis confirms the predicted chaotic dynamical behaviour of the Solar System, and provides a constraint for refining numerical solutions for insolation, which will enable a more precise and accurate geological timescale to be produced.

Theory of chaotic orbital variations confirmed by Cretaceous geological evidence.

PubMed

Ma, Chao; Meyers, Stephen R; Sageman, Bradley B

2017-02-22

Variations in the Earth's orbit and spin vector are a primary control on insolation and climate; their recognition in the geological record has revolutionized our understanding of palaeoclimate dynamics, and has catalysed improvements in the accuracy and precision of the geological timescale. Yet the secular evolution of the planetary orbits beyond 50 million years ago remains highly uncertain, and the chaotic dynamical nature of the Solar System predicted by theoretical models has yet to be rigorously confirmed by well constrained (radioisotopically calibrated and anchored) geological data. Here we present geological evidence for a chaotic resonance transition associated with interactions between the orbits of Mars and the Earth, using an integrated radioisotopic and astronomical timescale from the Cretaceous Western Interior Basin of what is now North America. This analysis confirms the predicted chaotic dynamical behaviour of the Solar System, and provides a constraint for refining numerical solutions for insolation, which will enable a more precise and accurate geological timescale to be produced.

Experimental Chaos - Proceedings of the 3rd Conference

NASA Astrophysics Data System (ADS)

Harrison, Robert G.; Lu, Weiping; Ditto, William; Pecora, Lou; Spano, Mark; Vohra, Sandeep

1996-10-01

The Table of Contents for the full book PDF is as follows: * Preface * Spatiotemporal Chaos and Patterns * Scale Segregation via Formation of Domains in a Nonlinear Optical System * Laser Dynamics as Hydrodynamics * Spatiotemporal Dynamics of Human Epileptic Seizures * Experimental Transition to Chaos in a Quasi 1D Chain of Oscillators * Measuring Coupling in Spatiotemporal Dynamical Systems * Chaos in Vortex Breakdown * Dynamical Analysis * Radial Basis Function Modelling and Prediction of Time Series * Nonlinear Phenomena in Polyrhythmic Hand Movements * Using Models to Diagnose, Test and Control Chaotic Systems * New Real-Time Analysis of Time Series Data with Physical Wavelets * Control and Synchronization * Measuring and Controlling Chaotic Dynamics in a Slugging Fluidized Bed * Control of Chaos in a Laser with Feedback * Synchronization and Chaotic Diode Resonators * Control of Chaos by Continuous-time Feedback with Delay * A Framework for Communication using Chaos Sychronization * Control of Chaos in Switching Circuits * Astrophysics, Meteorology and Oceanography * Solar-Wind-Magnetospheric Dynamics via Satellite Data * Nonlinear Dynamics of the Solar Atmosphere * Fractal Dimension of Scalar and Vector Variables from Turbulence Measurements in the Atmospheric Surface Layer * Mechanics * Escape and Overturning: Subtle Transient Behavior in Nonlinear Mechanical Models * Organising Centres in the Dynamics of Parametrically Excited Double Pendulums * Intermittent Behaviour in a Heating System Driven by Phase Transitions * Hydrodynamics * Size Segregation in Couette Flow of Granular Material * Routes to Chaos in Rotational Taylor-Couette Flow * Experimental Study of the Laminar-Turbulent Transition in an Open Flow System * Chemistry * Order and Chaos in Excitable Media under External Forcing * A Chemical Wave Propagation with Accelerating Speed Accompanied by Hydrodynamic Flow * Optics * Instabilities in Semiconductor Lasers with Optical Injection * Spatio-Temporal Dynamics of a Bimode CO2 Laser with Saturable Absorber * Chaotic Homoclinic Phenomena in Opto-Thermal Devices * Observation and Characterisation of Low-Frequency Chaos in Semiconductor Lasers with External Feedback * Condensed Matter * The Application of Nonlinear Dynamics in the Study of Ferroelectric Materials * Cellular Convection in a Small Aspect Ratio Liquid Crystal Device * Driven Spin-Wave Dynamics in YIG Films * Quantum Chaology in Quartz * Small Signal Amplification Caused by Nonlinear Properties of Ferroelectrics * Composite Materials Evolved from Chaos * Electronics and Circuits * Controlling a Chaotic Array of Pulse-Coupled Fitzhugh-Nagumo Circuits * Experimental Observation of On-Off Intermittency * Phase Lock-In of Chaotic Relaxation Oscillators * Biology and Medicine * Singular Value Decomposition and Circuit Structure in Invertebrate Ganglia * Nonlinear Forecasting of Spike Trains from Neurons of a Mollusc * Ultradian Rhythm in the Sensitive Plants: Chaos or Coloured Noise? * Chaos and the Crayfish Sixth Ganglion * Hardware Coupled Nonlinear Oscillators as a Model of Retina

Lyapunov exponents from CHUA's circuit time series using artificial neural networks

NASA Technical Reports Server (NTRS)

Gonzalez, J. Jesus; Espinosa, Ismael E.; Fuentes, Alberto M.

1995-01-01

In this paper we present the general problem of identifying if a nonlinear dynamic system has a chaotic behavior. If the answer is positive the system will be sensitive to small perturbations in the initial conditions which will imply that there is a chaotic attractor in its state space. A particular problem would be that of identifying a chaotic oscillator. We present an example of three well known different chaotic oscillators where we have knowledge of the equations that govern the dynamical systems and from there we can obtain the corresponding time series. In a similar example we assume that we only know the time series and, finally, in another example we have to take measurements in the Chua's circuit to obtain sample points of the time series. With the knowledge about the time series the phase plane portraits are plotted and from them, by visual inspection, it is concluded whether or not the system is chaotic. This method has the problem of uncertainty and subjectivity and for that reason a different approach is needed. A quantitative approach is the computation of the Lyapunov exponents. We describe several methods for obtaining them and apply a little known method of artificial neural networks to the different examples mentioned above. We end the paper discussing the importance of the Lyapunov exponents in the interpretation of the dynamic behavior of biological neurons and biological neural networks.

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.

Robust and irreversible development in cell society as a general consequence of intra-inter dynamics

NASA Astrophysics Data System (ADS)

Kaneko, Kunihiko; Furusawa, Chikara

2000-05-01

A dynamical systems scenario for developmental cell biology is proposed, based on numerical studies of a system with interacting units with internal dynamics and reproduction. Diversification, formation of discrete and recursive types, and rules for differentiation are found as a natural consequence of such a system. “Stem cells” that either proliferate or differentiate to different types stochastically are found to appear when intra-cellular dynamics are chaotic. Robustness of the developmental process against microscopic and macroscopic perturbations is shown to be a natural consequence of such intra-inter dynamics, while irreversibility in developmental process is discussed in terms of the gain of stability, loss of diversity and chaotic instability.

Attractors of relaxation discrete-time systems with chaotic dynamics on a fast time scale.

PubMed

Maslennikov, Oleg V; Nekorkin, Vladimir I

2016-07-01

In this work, a new type of relaxation systems is considered. Their prominent feature is that they comprise two distinct epochs, one is slow regular motion and another is fast chaotic motion. Unlike traditionally studied slow-fast systems that have smooth manifolds of slow motions in the phase space and fast trajectories between them, in this new type one observes, apart the same geometric objects, areas of transient chaos. Alternating periods of slow regular motions and fast chaotic ones as well as transitions between them result in a specific chaotic attractor with chaos on a fast time scale. We formulate basic properties of such attractors in the framework of discrete-time systems and consider several examples. Finally, we provide an important application of such systems, the neuronal electrical activity in the form of chaotic spike-burst oscillations.

Coexistence and chaos in complex ecologies [rapid communication

NASA Astrophysics Data System (ADS)

Sprott, J. C.; Vano, J. A.; Wildenberg, J. C.; Anderson, M. B.; Noel, J. K.

2005-02-01

Many complex dynamical systems in ecology, economics, neurology, and elsewhere, in which agents compete for limited resources, exhibit apparently chaotic fluctuations. This Letter proposes a purely deterministic mechanism for evolving robustly but weakly chaotic systems that exhibit adaptation, self-organization, sporadic volatility, and punctuated equilibria.

Chaotic Dynamics of a Josephson Junction with a Ratchet Potential and Current-Modulating Damping

NASA Astrophysics Data System (ADS)

Li, Fei; Li, Wenwu; Xu, Lan

2018-06-01

The chaotic dynamics of a Josephson junction with a ratchet potential and current-modulating damping are studied. Under the first-order approximation, we construct the general solution of the first-order equation whose boundedness condition contains the famous Melnikov chaotic criterion. Based on the general solution, the incomputability and unpredictability of the system's chaotic behavior are discussed. For the case beyond perturbation conditions, the evolution of stroboscopic Poincaré sections shows that the system undergoes a quasi-periodic transition to chaos with an increasing intensity of the rf-current. Through a suitable feedback controlling strategy, the chaos can be effectively suppressed and the intensity of the controller can vary in a large range. It is also found that the current between the two separated superconductors increases monotonously in some specific parameter spaces.

Chaotic Dynamics of a Josephson Junction with a Ratchet Potential and Current-Modulating Damping

NASA Astrophysics Data System (ADS)

Li, Fei; Li, Wenwu; Xu, Lan

2018-04-01

The chaotic dynamics of a Josephson junction with a ratchet potential and current-modulating damping are studied. Under the first-order approximation, we construct the general solution of the first-order equation whose boundedness condition contains the famous Melnikov chaotic criterion. Based on the general solution, the incomputability and unpredictability of the system's chaotic behavior are discussed. For the case beyond perturbation conditions, the evolution of stroboscopic Poincaré sections shows that the system undergoes a quasi-periodic transition to chaos with an increasing intensity of the rf-current. Through a suitable feedback controlling strategy, the chaos can be effectively suppressed and the intensity of the controller can vary in a large range. It is also found that the current between the two separated superconductors increases monotonously in some specific parameter spaces.

On the modeling and nonlinear dynamics of autonomous Silva-Young type chaotic oscillators with flat power spectrum

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

Kengne, Jacques; Kenmogne, Fabien

2014-12-15

The nonlinear dynamics of fourth-order Silva-Young type chaotic oscillators with flat power spectrum recently introduced by Tamaseviciute and collaborators is considered. In this type of oscillators, a pair of semiconductor diodes in an anti-parallel connection acts as the nonlinear component necessary for generating chaotic oscillations. Based on the Shockley diode equation and an appropriate selection of the state variables, a smooth mathematical model (involving hyperbolic sine and cosine functions) is derived for a better description of both the regular and chaotic dynamics of the system. The complex behavior of the oscillator is characterized in terms of its parameters by usingmore » time series, bifurcation diagrams, Lyapunov exponents' plots, Poincaré sections, and frequency spectra. It is shown that the onset of chaos is achieved via the classical period-doubling and symmetry restoring crisis scenarios. Some PSPICE simulations of the nonlinear dynamics of the oscillator are presented in order to confirm the ability of the proposed mathematical model to accurately describe/predict both the regular and chaotic behaviors of the oscillator.« less

Generation of 2N + 1-scroll existence in new three-dimensional chaos systems

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

Liu, Yue; Guan, Jian; Ma, Chunyang

2016-08-15

We propose a systematic methodology for creating 2N + 1-scroll chaotic attractors from a simple three-dimensional system, which is named as the translation chaotic system. It satisfies the condition a{sub 12}a{sub 21} = 0, while the Chua system satisfies a{sub 12}a{sub 21} > 0. In this paper, we also propose a successful (an effective) design and an analytical approach for constructing 2N + 1-scrolls, the translation transformation principle. Also, the dynamics properties of the system are studied in detail. MATLAB simulation results show very sophisticated dynamical behaviors and unique chaotic behaviors of the system. It provides a new approach for 2N + 1-scroll attractors. Finally, to explore the potential usemore » in technological applications, a novel block circuit diagram is also designed for the hardware implementation of 1-, 3-, 5-, and 7-scroll attractors via switching the switches. Translation chaotic system has the merit of convenience and high sensitivity to initial values, emerging potentials in future engineering chaos design.« less

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.

Chaos and Hyperchaos in Coupled Antiphase Driven Toda Oscillators

NASA Astrophysics Data System (ADS)

Stankevich, Nataliya V.; Dvorak, Anton; Astakhov, Vladimir; Jaros, Patrycja; Kapitaniak, Marcin; Perlikowski, Przemysław; Kapitaniak, Tomasz

2018-01-01

The dynamics of two coupled antiphase driven Toda oscillators is studied. We demonstrate three different routes of transition to chaotic dynamics associated with different bifurcations of periodic and quasi-periodic regimes. As a result of these, two types of chaotic dynamics with one and two positive Lyapunov exponents are observed. We argue that the results obtained are robust as they can exist in a wide range of the system parameters.

Generating Random Numbers by Means of Nonlinear Dynamic Systems

ERIC Educational Resources Information Center

Zang, Jiaqi; Hu, Haojie; Zhong, Juhua; Luo, Duanbin; Fang, Yi

2018-01-01

To introduce the randomness of a physical process to students, a chaotic pendulum experiment was opened in East China University of Science and Technology (ECUST) on the undergraduate level in the physics department. It was shown chaotic motion could be initiated through adjusting the operation of a chaotic pendulum. By using the data of the…

Dynamics of coherent states in regular and chaotic regimes of the non-integrable Dicke model

NASA Astrophysics Data System (ADS)

Lerma-Hernández, S.; Chávez-Carlos, J.; Bastarrachea-Magnani, M. A.; López-del-Carpio, B.; Hirsch, J. G.

2018-04-01

The quantum dynamics of initial coherent states is studied in the Dicke model and correlated with the dynamics, regular or chaotic, of their classical limit. Analytical expressions for the survival probability, i.e. the probability of finding the system in its initial state at time t, are provided in the regular regions of the model. The results for regular regimes are compared with those of the chaotic ones. It is found that initial coherent states in regular regions have a much longer equilibration time than those located in chaotic regions. The properties of the distributions for the initial coherent states in the Hamiltonian eigenbasis are also studied. It is found that for regular states the components with no negligible contribution are organized in sequences of energy levels distributed according to Gaussian functions. In the case of chaotic coherent states, the energy components do not have a simple structure and the number of participating energy levels is larger than in the regular cases.

The chaotic saddle of a three degrees of freedom scattering system reconstructed from cross-section data

NASA Astrophysics Data System (ADS)

Drótos, G.; Jung, C.

2016-06-01

The topic of this paper is hyperbolic chaotic scattering in a three degrees of freedom system. We generalize how shadows in the domain of the doubly differential cross-section are found: they are traced out by the appropriately filtered unstable manifolds of the periodic trajectories in the chaotic saddle. These shadows are related to the rainbow singularities in the doubly differential cross-section. As a result of this relation, we discover a method of how to recognize in the cross section a smoothly deformed image of the chaotic saddle, allowing the reconstruction of the symbolic dynamics of the chaotic saddle, its topology and its scaling factors.

Nonlinear dynamics as an engine of computation.

PubMed

Kia, Behnam; Lindner, John F; Ditto, William L

2017-03-06

Control of chaos teaches that control theory can tame the complex, random-like behaviour of chaotic systems. This alliance between control methods and physics-cybernetical physics-opens the door to many applications, including dynamics-based computing. In this article, we introduce nonlinear dynamics and its rich, sometimes chaotic behaviour as an engine of computation. We review our work that has demonstrated how to compute using nonlinear dynamics. Furthermore, we investigate the interrelationship between invariant measures of a dynamical system and its computing power to strengthen the bridge between physics and computation.This article is part of the themed issue 'Horizons of cybernetical physics'. © 2017 The Author(s).

Nonlinear dynamics as an engine of computation

PubMed Central

Lindner, John F.; Ditto, William L.

2017-01-01

Control of chaos teaches that control theory can tame the complex, random-like behaviour of chaotic systems. This alliance between control methods and physics—cybernetical physics—opens the door to many applications, including dynamics-based computing. In this article, we introduce nonlinear dynamics and its rich, sometimes chaotic behaviour as an engine of computation. We review our work that has demonstrated how to compute using nonlinear dynamics. Furthermore, we investigate the interrelationship between invariant measures of a dynamical system and its computing power to strengthen the bridge between physics and computation. This article is part of the themed issue ‘Horizons of cybernetical physics’. PMID:28115619

Experiments of reconstructing discrete atmospheric dynamic models from data (I)

NASA Astrophysics Data System (ADS)

Lin, Zhenshan; Zhu, Yanyu; Deng, Ziwang

1995-03-01

In this paper, we give some experimental results of our study in reconstructing discrete atmospheric dynamic models from data. After a great deal of numerical experiments, we found that the logistic map, x n + 1 = 1- μx {2/n}, could be used in monthly mean temperature prediction when it was approaching the chaotic region, and its predictive results were in reverse states to the practical data. This means that the nonlinear developing behavior of the monthly mean temperature system is bifurcating back into the critical chaotic states from the chaotic ones.

Chaotic itinerancy in the oscillator neural network without Lyapunov functions.

PubMed

Uchiyama, Satoki; Fujisaka, Hirokazu

2004-09-01

Chaotic itinerancy (CI), which is defined as an incessant spontaneous switching phenomenon among attractor ruins in deterministic dynamical systems without Lyapunov functions, is numerically studied in the case of an oscillator neural network model. The model is the pseudoinverse-matrix version of the previous model [S. Uchiyama and H. Fujisaka, Phys. Rev. E 65, 061912 (2002)] that was studied theoretically with the aid of statistical neurodynamics. It is found that CI in neural nets can be understood as the intermittent dynamics of weakly destabilized chaotic retrieval solutions. Copyright 2004 American Institute of Physics

Chaotic trajectories in the standard map. The concept of anti-integrability

NASA Astrophysics Data System (ADS)

Aubry, Serge; Abramovici, Gilles

1990-07-01

A rigorous proof is given in the standard map (associated with a Frenkel-Kontorowa model) for the existence of chaotic trajectories with unbounded momenta for large enough coupling constant k > k0. These chaotic trajectories (with finite entropy per site) are coded by integer sequences { mi} such that the sequence bi = |m i+1 + m i-1-2m i| be bounded by some integer b. The bound k0 in k depends on b and can be lowered for coding sequences { mi} fulfilling more restrictive conditions. The obtained chaotic trajectories correspond to stationary configurations of the Frenkel-Kontorowa model with a finite (non-zero) photon gap (called gap parameter in dimensionless units). This property implies that the trajectory (or the configuration { ui}) can be uniquely continued as a uniformly continuous function of the model parameter k in some neighborhood of the initial configuration. A non-zero gap parameter implies that the Lyapunov coefficient is strictly positive (when it is defined). In addition, the existence of dilating and contracting manifolds is proven for these chaotic trajectories. “Exotic” trajectories such as ballistic trajectories are also proven to exist as a consequence of these theorems. The concept of anti-integrability emerges from these theorems. In the anti-integrable limit which can be only defined for a discrete time dynamical system, the coordinates of the trajectory at time i do not depend on the coordinates at time i - 1. Thus, at this singular limit, the existence of chaotic trajectories is trivial and the dynamical system reduces to a Bernoulli shift. It is well known that the KAM tori of symplectic dynamical originates by continuity from the invariant tori which exists in the integrible limit (under certain conditions). In a similar way, it appears that the chaotic trajectories of dynamical systems originate by continuity from those which exists at the anti-integrable limits (also under certain conditions).

Fast and secure encryption-decryption method based on chaotic dynamics

DOEpatents

Protopopescu, Vladimir A.; Santoro, Robert T.; Tolliver, Johnny S.

1995-01-01

A method and system for the secure encryption of information. The method comprises the steps of dividing a message of length L into its character components; generating m chaotic iterates from m independent chaotic maps; producing an "initial" value based upon the m chaotic iterates; transforming the "initial" value to create a pseudo-random integer; repeating the steps of generating, producing and transforming until a pseudo-random integer sequence of length L is created; and encrypting the message as ciphertext based upon the pseudo random integer sequence. A system for accomplishing the invention is also provided.

Chaotic Mixing in Magmatic Systems: a new experiment

NASA Astrophysics Data System (ADS)

de Campos, C. P.; Perugini, D.; Dingwell, D. B.; Poli, G.; Ertel-Ingrisch, W.; Hess, K.

2007-12-01

Previous studies on magma mixing systems have evidenced that mixing processes could be controlled by chaotic dynamics. These processes are thought to be the source of fractal structures propagating within natural magmatic systems, from meter to the micrometer length scale (Perugini et al., 2006. EPSL, 234: 669-680 and references therein). We have developed a device for experimental studies of chaotic mixing dynamics in silicate melts at high temperatures (up to 1700°C). This device has been inspired by the journal bearing or eccentric cylinder geometry for viscous fluids for the study of chaotic mixing in slow flows (Swanson and Ottino, 1990. J. Fluid Mech., 213:227-249). This geometry is thought to be an ideal system for chaotic studies because a) it is experimentally accessible/feasible for silicate rheologies and b) it is subject to an analytical solution for the stream function. In the journal bearing system the flow region, is confined in the torus between the centers of the two cylinders. Their central axes are parallel but not coincident, i. e. the cylinders are eccentric. In order to generate chaos in a flow, the streamlines must be time dependent, resulting in alternating movements between the two cylinders. This means that at least one of the cylinders has alternating rotation directions. The dimension of this new experimental device follows the required main dimensionless numbers for a chaotic flow. Our first experimental goal is to characterize the mixing process in a prototypical system (haplogranite-haplobasalt)under variable mixing protocols. muenchen.de/

Dynamical transitions in large systems of mean field-coupled Landau-Stuart oscillators: Extensive chaos and cluster states.

PubMed

Ku, Wai Lim; Girvan, Michelle; Ott, Edward

2015-12-01

In this paper, we study dynamical systems in which a large number N of identical Landau-Stuart oscillators are globally coupled via a mean-field. Previously, it has been observed that this type of system can exhibit a variety of different dynamical behaviors. These behaviors include time periodic cluster states in which each oscillator is in one of a small number of groups for which all oscillators in each group have the same state which is different from group to group, as well as a behavior in which all oscillators have different states and the macroscopic dynamics of the mean field is chaotic. We argue that this second type of behavior is "extensive" in the sense that the chaotic attractor in the full phase space of the system has a fractal dimension that scales linearly with N and that the number of positive Lyapunov exponents of the attractor also scales linearly with N. An important focus of this paper is the transition between cluster states and extensive chaos as the system is subjected to slow adiabatic parameter change. We observe discontinuous transitions between the cluster states (which correspond to low dimensional dynamics) and the extensively chaotic states. Furthermore, examining the cluster state, as the system approaches the discontinuous transition to extensive chaos, we find that the oscillator population distribution between the clusters continually evolves so that the cluster state is always marginally stable. This behavior is used to reveal the mechanism of the discontinuous transition. We also apply the Kaplan-Yorke formula to study the fractal structure of the extensively chaotic attractors.

Dynamical transitions in large systems of mean field-coupled Landau-Stuart oscillators: Extensive chaos and cluster states

NASA Astrophysics Data System (ADS)

Ku, Wai Lim; Girvan, Michelle; Ott, Edward

2015-12-01

In this paper, we study dynamical systems in which a large number N of identical Landau-Stuart oscillators are globally coupled via a mean-field. Previously, it has been observed that this type of system can exhibit a variety of different dynamical behaviors. These behaviors include time periodic cluster states in which each oscillator is in one of a small number of groups for which all oscillators in each group have the same state which is different from group to group, as well as a behavior in which all oscillators have different states and the macroscopic dynamics of the mean field is chaotic. We argue that this second type of behavior is "extensive" in the sense that the chaotic attractor in the full phase space of the system has a fractal dimension that scales linearly with N and that the number of positive Lyapunov exponents of the attractor also scales linearly with N. An important focus of this paper is the transition between cluster states and extensive chaos as the system is subjected to slow adiabatic parameter change. We observe discontinuous transitions between the cluster states (which correspond to low dimensional dynamics) and the extensively chaotic states. Furthermore, examining the cluster state, as the system approaches the discontinuous transition to extensive chaos, we find that the oscillator population distribution between the clusters continually evolves so that the cluster state is always marginally stable. This behavior is used to reveal the mechanism of the discontinuous transition. We also apply the Kaplan-Yorke formula to study the fractal structure of the extensively chaotic attractors.

Attractors of relaxation discrete-time systems with chaotic dynamics on a fast time scale

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

Maslennikov, Oleg V.; Nekorkin, Vladimir I.

In this work, a new type of relaxation systems is considered. Their prominent feature is that they comprise two distinct epochs, one is slow regular motion and another is fast chaotic motion. Unlike traditionally studied slow-fast systems that have smooth manifolds of slow motions in the phase space and fast trajectories between them, in this new type one observes, apart the same geometric objects, areas of transient chaos. Alternating periods of slow regular motions and fast chaotic ones as well as transitions between them result in a specific chaotic attractor with chaos on a fast time scale. We formulate basicmore » properties of such attractors in the framework of discrete-time systems and consider several examples. Finally, we provide an important application of such systems, the neuronal electrical activity in the form of chaotic spike-burst oscillations.« less

Dynamic analysis, circuit implementation and passive control of a novel four-dimensional chaotic system with multiscroll attractor and multiple coexisting attractors

NASA Astrophysics Data System (ADS)

Lai, Bang-Cheng; He, Jian-Jun

2018-03-01

In this paper, we construct a novel 4D autonomous chaotic system with four cross-product nonlinear terms and five equilibria. The multiple coexisting attractors and the multiscroll attractor of the system are numerically investigated. Research results show that the system has various types of multiple attractors, including three strange attractors with a limit cycle, three limit cycles, two strange attractors with a pair of limit cycles, two coexisting strange attractors. By using the passive control theory, a controller is designed for controlling the chaos of the system. Both analytical and numerical studies verify that the designed controller can suppress chaotic motion and stabilise the system at the origin. Moreover, an electronic circuit is presented for implementing the chaotic system.

Chaotic coordinates for the Large Helical Device

NASA Astrophysics Data System (ADS)

Hudson, Stuart; Suzuki, Yasuhiro

2014-10-01

The study of dynamical systems is facilitated by a coordinate framework with coordinate surfaces that coincide with invariant structures of the dynamical flow. For axisymmetric systems, a continuous family of invariant surfaces is guaranteed and straight-fieldline coordinates may be constructed. For non-integrable systems, e.g. stellarators, perturbed tokamaks, this continuous family is broken. Nevertheless, coordinates can still be constructed that simplify the description of the dynamics. The Poincare-Birkhoff theorem, the Aubry-Mather theorem, and the KAM theorem show that there are important structures that are invariant under the perturbed dynamics; namely the periodic orbits, the cantori, and the irrational flux surfaces. Coordinates adapted to these invariant sets, which we call chaotic coordinates, provide substantial advantages. The regular motion becomes straight, and the irregular motion is bounded by, and dissected by, coordinate surfaces that coincide with surfaces of locally-minimal magnetic-fieldline flux. The chaotic edge of the magnetic field, as calculated by HINT2 code, in the Large Helical Device (LHD) is examined, and a coordinate system is constructed so that the flux surfaces are ``straight'' and the islands become ``square.''

A Tribute to J. C. Sprott

NASA Astrophysics Data System (ADS)

Nazarimehr, Fahimeh; Jafari, Sajad; Chen, Guanrong; Kapitaniak, Tomasz; Kuznetsov, Nikolay V.; Leonov, Gennady A.; Li, Chunbiao; Wei, Zhouchao

2017-12-01

In honor of his 75th birthday, we review the prominent works of Professor Julien Clinton Sprott in chaos and nonlinear dynamics. We categorize his works into three important groups. The first and most important group is identifying new dynamical systems with special properties. He has proposed different chaotic maps, flows, complex variable systems, nonautonomous systems, partial differential equations, fractional-order systems, delay differential systems, spatiotemporal systems, artificial neural networks, and chaotic electrical circuits. He has also studied dynamical properties of complex systems such as bifurcations and basins of attraction. He has done work on generating fractal art. He has examined models of real-world systems that exhibit chaos. The second group of his works comprise control and synchronization of chaos. Finally, the third group is extracting dynamical properties of systems using time-series analysis. This paper highlights the impact of Sprott’s work on the promotion of nonlinear dynamics.

Improving Ecological Forecasting: Data Assimilation Enhances the Ecological Forecast Horizon of a Complex Food Web

NASA Astrophysics Data System (ADS)

Massoud, E. C.; Huisman, J.; Benincà, E.; Bouten, W.; Vrugt, J. A.

2017-12-01

Species abundances in ecological communities can display chaotic non-equilibrium dynamics. A characteristic feature of chaotic systems is that long-term prediction of the system's trajectory is fundamentally impossible. How then should we make predictions for complex multi-species communities? We explore data assimilation (DA) with the Ensemble Kalman Filter (EnKF) to fuse a two-predator-two-prey model with abundance data from a long term experiment of a plankton community which displays chaotic dynamics. The results show that DA improves substantially the predictability and ecological forecast horizon of complex community dynamics. In addition, we show that DA helps provide guidance on measurement design, for instance on defining the frequency of observations. The study presented here is highly innovative, because DA methods at the current stage are almost unknown in ecology.

Bit-level quantum color image encryption scheme with quantum cross-exchange operation and hyper-chaotic system

NASA Astrophysics Data System (ADS)

Zhou, Nanrun; Chen, Weiwei; Yan, Xinyu; Wang, Yunqian

2018-06-01

In order to obtain higher encryption efficiency, a bit-level quantum color image encryption scheme by exploiting quantum cross-exchange operation and a 5D hyper-chaotic system is designed. Additionally, to enhance the scrambling effect, the quantum channel swapping operation is employed to swap the gray values of corresponding pixels. The proposed color image encryption algorithm has larger key space and higher security since the 5D hyper-chaotic system has more complex dynamic behavior, better randomness and unpredictability than those based on low-dimensional hyper-chaotic systems. Simulations and theoretical analyses demonstrate that the presented bit-level quantum color image encryption scheme outperforms its classical counterparts in efficiency and security.

The onset of dynamical instability and chaos in navigation satellite orbits

NASA Astrophysics Data System (ADS)

Rosengren, Aaron Jay; Daquin, Jérôme; Alessi, Elisa Maria; Valsecchi, Giovanni B.; Rossi, Alessandro; Deleflie, Florent

2015-05-01

Orbital resonances are ubiquitous in the Solar System and are harbingers for the onset of dynamical instability and chaos. It has long been suspected that the Global Navigation Satellite Systems exist in a background of complex resonances and chaotic motion; yet, the precise dynamical character of these phenomena remains elusive. Here we will show that the same underlying physical mechanism, the overlapping of secular resonances, responsible for the eventual destabilization of Mercury and recently proposed to explain the orbital architecture of extrasolar planetary systems (Lithwick Y., Wu Y., 2014, PNAS; Batygin K., Morbidelli A., Holman M.J., 2015, ApJ) is at the heart of the orbital instabilities of seemingly more mundane celestial bodies---the Earth's navigation satellites. We will demonstrate that the occurrence and nature of the secular resonances driving these dynamics depend chiefly on one aspect of the Moon's perturbed motion, the regression of the line of nodes. This talk will present analytical models that accurately reflect the true nature of the resonant interactions, and will show how chaotic diffusion is mediated by the web-like structure of secular resonances. We will also present an atlas of FLI stability maps, showing the extent of the chaotic regions of the phase space, computed through a hierarchy of more realistic, and more complicated, models, and compare the chaotic zones in these charts with the analytical estimation of the width of the chaotic layers from the heuristic Chirikov resonance-overlap criterion. The obtained results have remarkable practical applications for space debris mitigation and for satellite technology, and are both of essential dynamical and theoretical importance, with broad implications for planetary science.

Improved numerical solutions for chaotic-cancer-model

NASA Astrophysics Data System (ADS)

Yasir, Muhammad; Ahmad, Salman; Ahmed, Faizan; Aqeel, Muhammad; Akbar, Muhammad Zubair

2017-01-01

In biological sciences, dynamical system of cancer model is well known due to its sensitivity and chaoticity. Present work provides detailed computational study of cancer model by counterbalancing its sensitive dependency on initial conditions and parameter values. Cancer chaotic model is discretized into a system of nonlinear equations that are solved using the well-known Successive-Over-Relaxation (SOR) method with a proven convergence. This technique enables to solve large systems and provides more accurate approximation which is illustrated through tables, time history maps and phase portraits with detailed analysis.

Influence of the black hole spin on the chaotic particle dynamics within a dipolar halo

NASA Astrophysics Data System (ADS)

Nag, Sankhasubhra; Sinha, Siddhartha; Ananda, Deepika B.; Das, Tapas K.

2017-04-01

We investigate the role of the spin angular momentum of astrophysical black holes in controlling the special relativistic chaotic dynamics of test particles moving under the influence of a post-Newtonian pseudo-Kerr black hole potential, along with a perturbative potential created by an asymmetrically placed (dipolar) halo. Proposing a Lyapunov-like exponent to be the effective measure of the degree of chaos observed in the system under consideration, it has been found that black hole spin anti-correlates with the degree of chaos for the aforementioned dynamics. Our findings have been explained applying the general principles of dynamical systems analysis.

Exact coherent structures and chaotic dynamics in a model of cardiac tissue

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

Byrne, Greg; Marcotte, Christopher D.; Grigoriev, Roman O., E-mail: roman.grigoriev@physics.gatech.edu

Unstable nonchaotic solutions embedded in the chaotic attractor can provide significant new insight into chaotic dynamics of both low- and high-dimensional systems. In particular, in turbulent fluid flows, such unstable solutions are referred to as exact coherent structures (ECS) and play an important role in both initiating and sustaining turbulence. The nature of ECS and their role in organizing spatiotemporally chaotic dynamics, however, is reasonably well understood only for systems on relatively small spatial domains lacking continuous Euclidean symmetries. Construction of ECS on large domains and in the presence of continuous translational and/or rotational symmetries remains a challenge. This ismore » especially true for models of excitable media which display spiral turbulence and for which the standard approach to computing ECS completely breaks down. This paper uses the Karma model of cardiac tissue to illustrate a potential approach that could allow computing a new class of ECS on large domains of arbitrary shape by decomposing them into a patchwork of solutions on smaller domains, or tiles, which retain Euclidean symmetries locally.« less

A new reduced-order observer for the synchronization of nonlinear chaotic systems: An application to secure communications

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

Castro-Ramírez, Joel, E-mail: ingcastro.7@gmail.com; Martínez-Guerra, Rafael, E-mail: rguerra@ctrl.cinvestav.mx; Cruz-Victoria, Juan Crescenciano, E-mail: juancrescenciano.cruz@uptlax.edu.mx

2015-10-15

This paper deals with the master-slave synchronization scheme for partially known nonlinear chaotic systems, where the unknown dynamics is considered as the master system and we propose the slave system structure which estimates the unknown states. It introduced a new reduced order observer, using the concept of Algebraic Observability; we applied the results to a Sundarapandian chaotic system, and by means of some numerical simulations we show the effectiveness of the suggested approach. Finally, the proposed observer is utilized for encryption, where encryption key is the master system and decryption key is the slave system.

Spatiotemporal chaos in mixed linear-nonlinear two-dimensional coupled logistic map lattice

NASA Astrophysics Data System (ADS)

Zhang, Ying-Qian; He, Yi; Wang, Xing-Yuan

2018-01-01

We investigate a new spatiotemporal dynamics with mixing degrees of nonlinear chaotic maps for spatial coupling connections based on 2DCML. Here, the coupling methods are including with linear neighborhood coupling and the nonlinear chaotic map coupling of lattices, and the former 2DCML system is only a special case in the proposed system. In this paper the criteria such Kolmogorov-Sinai entropy density and universality, bifurcation diagrams, space-amplitude and snapshot pattern diagrams are provided in order to investigate the chaotic behaviors of the proposed system. Furthermore, we also investigate the parameter ranges of the proposed system which holds those features in comparisons with those of the 2DCML system and the MLNCML system. Theoretical analysis and computer simulation indicate that the proposed system contains features such as the higher percentage of lattices in chaotic behaviors for most of parameters, less periodic windows in bifurcation diagrams and the larger range of parameters for chaotic behaviors, which is more suitable for cryptography.

Piecewise affine models of chaotic attractors: the Rossler and Lorenz systems.

PubMed

Amaral, Gleison F V; Letellier, Christophe; Aguirre, Luis Antonio

2006-03-01

This paper proposes a procedure by which it is possible to synthesize Rossler [Phys. Lett. A 57, 397-398 (1976)] and Lorenz [J. Atmos. Sci. 20, 130-141 (1963)] dynamics by means of only two affine linear systems and an abrupt switching law. Comparison of different (valid) switching laws suggests that parameters of such a law behave as codimension one bifurcation parameters that can be changed to produce various dynamical regimes equivalent to those observed with the original systems. Topological analysis is used to characterize the resulting attractors and to compare them with the original attractors. The paper provides guidelines that are helpful to synthesize other chaotic dynamics by means of switching affine linear systems.

Security Analysis of a Block Encryption Algorithm Based on Dynamic Sequences of Multiple Chaotic Systems

NASA Astrophysics Data System (ADS)

Du, Mao-Kang; He, Bo; Wang, Yong

2011-01-01

Recently, the cryptosystem based on chaos has attracted much attention. Wang and Yu (Commun. Nonlin. Sci. Numer. Simulat. 14 (2009) 574) proposed a block encryption algorithm based on dynamic sequences of multiple chaotic systems. We analyze the potential flaws in the algorithm. Then, a chosen-plaintext attack is presented. Some remedial measures are suggested to avoid the flaws effectively. Furthermore, an improved encryption algorithm is proposed to resist the attacks and to keep all the merits of the original cryptosystem.

A novel grid multiwing chaotic system with only non-hyperbolic equilibria

NASA Astrophysics Data System (ADS)

Zhang, Sen; Zeng, Yicheng; Li, Zhijun; Wang, Mengjiao; Xiong, Le

2018-05-01

The structure of the chaotic attractor of a system is mainly determined by the nonlinear functions in system equations. By using a new saw-tooth wave function and a new stair function, a novel complex grid multiwing chaotic system which belongs to non-Shil'nikov chaotic system with non-hyperbolic equilibrium points is proposed in this paper. It is particularly interesting that the complex grid multiwing attractors are generated by increasing the number of non-hyperbolic equilibrium points, which are different from the traditional methods of realising multiwing attractors by adding the index-2 saddle-focus equilibrium points in double-wing chaotic systems. The basic dynamical properties of the new system, such as dissipativity, phase portraits, the stability of the equilibria, the time-domain waveform, power spectrum, bifurcation diagram, Lyapunov exponents, and so on, are investigated by theoretical analysis and numerical simulations. Furthermore, the corresponding electronic circuit is designed and simulated on the Multisim platform. The Multisim simulation results and the hardware experimental results are in good agreement with the numerical simulations of the same system on Matlab platform, which verify the feasibility of this new grid multiwing chaotic system.

Least Squares Shadowing sensitivity analysis of chaotic limit cycle oscillations

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

Wang, Qiqi, E-mail: qiqi@mit.edu; Hu, Rui, E-mail: hurui@mit.edu; Blonigan, Patrick, E-mail: blonigan@mit.edu

2014-06-15

The adjoint method, among other sensitivity analysis methods, can fail in chaotic dynamical systems. The result from these methods can be too large, often by orders of magnitude, when the result is the derivative of a long time averaged quantity. This failure is known to be caused by ill-conditioned initial value problems. This paper overcomes this failure by replacing the initial value problem with the well-conditioned “least squares shadowing (LSS) problem”. The LSS problem is then linearized in our sensitivity analysis algorithm, which computes a derivative that converges to the derivative of the infinitely long time average. We demonstrate ourmore » algorithm in several dynamical systems exhibiting both periodic and chaotic oscillations.« less

Is the normal heart rate ``chaotic'' due to respiration?

NASA Astrophysics Data System (ADS)

Wessel, Niels; Riedl, Maik; Kurths, Jürgen

2009-06-01

The incidence of cardiovascular diseases increases with the growth of the human population and an aging society, leading to very high expenses in the public health system. Therefore, it is challenging to develop sophisticated methods in order to improve medical diagnostics. The question whether the normal heart rate is chaotic or not is an attempt to elucidate the underlying mechanisms of cardiovascular dynamics and therefore a highly controversial topical challenge. In this contribution we demonstrate that linear and nonlinear parameters allow us to separate completely the data sets of the three groups provided for this controversial topic in nonlinear dynamics. The question whether these time series are chaotic or not cannot be answered satisfactorily without investigating the underlying mechanisms leading to them. We give an example of the dominant influence of respiration on heart beat dynamics, which shows that observed fluctuations can be mostly explained by respiratory modulations of heart rate and blood pressure (coefficient of determination: 96%). Therefore, we recommend reformulating the following initial question: "Is the normal heart rate chaotic?" We rather ask the following: "Is the normal heart rate `chaotic' due to respiration?"

Chaotic interactions of self-replicating RNA.

PubMed

Forst, C V

1996-03-01

A general system of high-order differential equations describing complex dynamics of replicating biomolecules is given. Symmetry relations and coordinate transformations of general replication systems leading to topologically equivalent systems are derived. Three chaotic attractors observed in Lotka-Volterra equations of dimension n = 3 are shown to represent three cross-sections of one and the same chaotic regime. Also a fractal torus in a generalized three-dimensional Lotka-Volterra Model has been linked to one of the chaotic attractors. The strange attractors are studied in the equivalent four-dimensional catalytic replicator network. The fractal torus has been examined in adapted Lotka-Volterra equations. Analytic expressions are derived for the Lyapunov exponents of the flow in the replicator system. Lyapunov spectra for different pathways into chaos has been calculated. In the generalized Lotka-Volterra system a second inner rest point--coexisting with (quasi)-periodic orbits--can be observed; with an abundance of different bifurcations. Pathways from chaotic tori, via quasi-periodic tori, via limit cycles, via multi-periodic orbits--emerging out of periodic doubling bifurcations--to "simple" chaotic attractors can be found.

Synchronization of chaotic systems involving fractional operators of Liouville-Caputo type with variable-order

NASA Astrophysics Data System (ADS)

Coronel-Escamilla, A.; Gómez-Aguilar, J. F.; Torres, L.; Escobar-Jiménez, R. F.; Valtierra-Rodríguez, M.

2017-12-01

In this paper, we propose a state-observer-based approach to synchronize variable-order fractional (VOF) chaotic systems. In particular, this work is focused on complete synchronization with a so-called unidirectional master-slave topology. The master is described by a dynamical system in state-space representation whereas the slave is described by a state observer. The slave is composed of a master copy and a correction term which in turn is constituted of an estimation error and an appropriate gain that assures the synchronization. The differential equations of the VOF chaotic system are described by the Liouville-Caputo and Atangana-Baleanu-Caputo derivatives. Numerical simulations involving the synchronization of Rössler oscillators, Chua's systems and multi-scrolls are studied. The simulations show that different chaotic behaviors can be obtained if different smooths functions defined in the interval (0 , 1 ] are used as the variable order of the fractional derivatives. Furthermore, simulations show that the VOF chaotic systems can be synchronized.

Dynamical transitions in large systems of mean field-coupled Landau-Stuart oscillators: Extensive chaos and cluster states

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

Ku, Wai Lim; Girvan, Michelle; Ott, Edward

In this paper, we study dynamical systems in which a large number N of identical Landau-Stuart oscillators are globally coupled via a mean-field. Previously, it has been observed that this type of system can exhibit a variety of different dynamical behaviors. These behaviors include time periodic cluster states in which each oscillator is in one of a small number of groups for which all oscillators in each group have the same state which is different from group to group, as well as a behavior in which all oscillators have different states and the macroscopic dynamics of the mean field ismore » chaotic. We argue that this second type of behavior is “extensive” in the sense that the chaotic attractor in the full phase space of the system has a fractal dimension that scales linearly with N and that the number of positive Lyapunov exponents of the attractor also scales linearly with N. An important focus of this paper is the transition between cluster states and extensive chaos as the system is subjected to slow adiabatic parameter change. We observe discontinuous transitions between the cluster states (which correspond to low dimensional dynamics) and the extensively chaotic states. Furthermore, examining the cluster state, as the system approaches the discontinuous transition to extensive chaos, we find that the oscillator population distribution between the clusters continually evolves so that the cluster state is always marginally stable. This behavior is used to reveal the mechanism of the discontinuous transition. We also apply the Kaplan-Yorke formula to study the fractal structure of the extensively chaotic attractors.« less

Characterization of chaotic dynamics in the human menstrual cycle

NASA Astrophysics Data System (ADS)

Derry, Gregory; Derry, Paula

2010-03-01

The human menstrual cycle exhibits much unexplained variability, which is typically dismissed as random variation. Given the many delayed nonlinear feedbacks in the reproductive endocrine system, however, the menstrual cycle might well be a nonlinear dynamical system in a chaotic trajectory, and that this instead accounts for the observed variability. Here, we test this hypothesis by performing a time series analysis on data for 7438 menstrual cycles from 38 women in the 20-40 year age range, using the database maintained by the Tremin Research Program on Women's Health. Using phase space reconstruction techniques with a maximum embedding dimension of 6, we find appropriate scaling behavior in the correlation sums for this data, indicating low dimensional deterministic dynamics. A correlation dimension of 2.6 is measured in this scaling regime, and this result is confirmed by recalculation using the Takens estimator. These results may be interpreted as offering an approximation to the fractal dimension of a strange attractor governing the chaotic dynamics of the menstrual cycle.

Chaotic behaviour of Zeeman machines at introductory course of mechanics

NASA Astrophysics Data System (ADS)

Nagy, Péter; Tasnádi, Péter

2016-05-01

Investigation of chaotic motions and cooperative systems offers a magnificent opportunity to involve modern physics into the basic course of mechanics taught to engineering students. In the present paper it will be demonstrated that Zeeman Machine can be a versatile and motivating tool for students to get introductory knowledge about chaotic motion via interactive simulations. It works in a relatively simple way and its properties can be understood very easily. Since the machine can be built easily and the simulation of its movement is also simple the experimental investigation and the theoretical description can be connected intuitively. Although Zeeman Machine is known mainly for its quasi-static and catastrophic behaviour, its dynamic properties are also of interest with its typical chaotic features. By means of a periodically driven Zeeman Machine a wide range of chaotic properties of the simple systems can be demonstrated such as bifurcation diagrams, chaotic attractors, transient chaos and so on. The main goal of this paper is the presentation of an interactive learning material for teaching the basic features of the chaotic systems through the investigation of the Zeeman Machine.

Self-organization of complex networks as a dynamical system

NASA Astrophysics Data System (ADS)

Aoki, Takaaki; Yawata, Koichiro; Aoyagi, Toshio

2015-01-01

To understand the dynamics of real-world networks, we investigate a mathematical model of the interplay between the dynamics of random walkers on a weighted network and the link weights driven by a resource carried by the walkers. Our numerical studies reveal that, under suitable conditions, the co-evolving dynamics lead to the emergence of stationary power-law distributions of the resource and link weights, while the resource quantity at each node ceaselessly changes with time. We analyze the network organization as a deterministic dynamical system and find that the system exhibits multistability, with numerous fixed points, limit cycles, and chaotic states. The chaotic behavior of the system leads to the continual changes in the microscopic network dynamics in the absence of any external random noises. We conclude that the intrinsic interplay between the states of the nodes and network reformation constitutes a major factor in the vicissitudes of real-world networks.

Self-organization of complex networks as a dynamical system.

PubMed

Aoki, Takaaki; Yawata, Koichiro; Aoyagi, Toshio

2015-01-01

To understand the dynamics of real-world networks, we investigate a mathematical model of the interplay between the dynamics of random walkers on a weighted network and the link weights driven by a resource carried by the walkers. Our numerical studies reveal that, under suitable conditions, the co-evolving dynamics lead to the emergence of stationary power-law distributions of the resource and link weights, while the resource quantity at each node ceaselessly changes with time. We analyze the network organization as a deterministic dynamical system and find that the system exhibits multistability, with numerous fixed points, limit cycles, and chaotic states. The chaotic behavior of the system leads to the continual changes in the microscopic network dynamics in the absence of any external random noises. We conclude that the intrinsic interplay between the states of the nodes and network reformation constitutes a major factor in the vicissitudes of real-world networks.

Chaotic oscillator containing memcapacitor and meminductor and its dimensionality reduction analysis.

PubMed

Yuan, Fang; Wang, Guangyi; Wang, Xiaowei

2017-03-01

In this paper, smooth curve models of meminductor and memcapacitor are designed, which are generalized from a memristor. Based on these models, a new five-dimensional chaotic oscillator that contains a meminductor and memcapacitor is proposed. By dimensionality reducing, this five-dimensional system can be transformed into a three-dimensional system. The main work of this paper is to give the comparisons between the five-dimensional system and its dimensionality reduction model. To investigate dynamics behaviors of the two systems, equilibrium points and stabilities are analyzed. And the bifurcation diagrams and Lyapunov exponent spectrums are used to explore their properties. In addition, digital signal processing technologies are used to realize this chaotic oscillator, and chaotic sequences are generated by the experimental device, which can be used in encryption applications.

Desktop chaotic systems: Intuition and visualization

NASA Technical Reports Server (NTRS)

Bright, Michelle M.; Melcher, Kevin J.; Qammar, Helen K.; Hartley, Tom T.

1993-01-01

This paper presents a dynamic study of the Wildwood Pendulum, a commercially available desktop system which exhibits a strange attractor. The purpose of studying this chaotic pendulum is twofold: to gain insight in the paradigmatic approach of modeling, simulating, and determining chaos in nonlinear systems; and to provide a desktop model of chaos as a visual tool. For this study, the nonlinear behavior of this chaotic pendulum is modeled, a computer simulation is performed, and an experimental performance is measured. An assessment of the pendulum in the phase plane shows the strange attractor. Through the use of a box-assisted correlation dimension methodology, the attractor dimension is determined for both the model and the experimental pendulum systems. Correlation dimension results indicate that the pendulum and the model are chaotic and their fractal dimensions are similar.

Mixing Silicate Melts with High Viscosity Contrast by Chaotic Dynamics: Results from a New Experimental Device

NASA Astrophysics Data System (ADS)

de Campos, Cristina; Perugini, Diego; Ertel-Ingrisch, Werner; Dingwell, Donald B.; Poli, Giampiero

2010-05-01

A new experimental device has been developed to perform chaotic mixing between high viscosity melts under controlled fluid-dynamic conditions. The apparatus is based on the Journal Bearing System (JBS). It consists of an outer cylinder hosting the melts of interest and an inner cylinder, which is eccentrically located. Both cylinders can be independently moved to generate chaotic streamlines in the mixing system. Two experiments were performed using as end-members different proportions of a peralkaline haplogranite and a mafic melt, corresponding to the 1 atm eutectic composition in the An-Di binary system. The two melts were stirred together in the JBS for ca. two hours, at 1,400° C and under laminar fluid dynamic condition (Re of the order of 10-7). The viscosity ratio between the two melts, at the beginning of the experiment, was of the order of 103. Optical analyses of experimental samples revealed, at short length scale (of the order of μm), a complex pattern of mixed structures. These consisted of an intimate distribution of filaments; a complex inter-fingering of the two melts. Such features are typically observed in rocks thought to be produced by magma mixing processes. Stretching and folding dynamics between the melts induced chaotic flow fields and generated wide compositional interfaces. In this way, chemical diffusion processes become more efficient, producing melts with highly heterogeneous compositions. A remarkable modulation of compositional fields has been obtained by performing short time-scale experiments and using melts with a high viscosity ratio. This indicates that chaotic mixing of magmas can be a very efficient process in modulating compositional variability in igneous systems, especially under high viscosity ratios and laminar fluid-dynamic regimes. Our experimental device may replicate magma mixing features, observed in natural rocks, and therefore open new frontiers in the study of this important petrologic and volcanological process.

Long-term influence of asteroids on planet longitudes and chaotic dynamics of the solar system

NASA Astrophysics Data System (ADS)

Woillez, E.; Bouchet, F.

2017-11-01

Over timescales much longer than an orbital period, the solar system exhibits large-scale chaotic behavior and can thus be viewed as a stochastic dynamical system. The aim of the present paper is to compare different sources of stochasticity in the solar system. More precisely we studied the importance of the long term influence of asteroids on the chaotic dynamics of the solar system. We show that the effects of asteroids on planets is similar to a white noise process, when those effects are considered on a timescale much larger than the correlation time τϕ ≃ 104 yr of asteroid trajectories. We computed the timescale τe after which the effects of the stochastic evolution of the asteroids lead to a loss of information for the initial conditions of the perturbed Laplace-Lagrange secular dynamics. The order of magnitude of this timescale is precisely determined by theoretical argument, and we find that τe ≃ 104 Myr. Although comparable to the full main-sequence lifetime of the sun, this timescale is considerably longer than the Lyapunov time τI ≃ 10 Myr of the solar system without asteroids. This shows that the external sources of chaos arise as a small perturbation in the stochastic secular behavior of the solar system, rather due to intrinsic chaos.

Chaotic diffusion in the Gliese-876 planetary system

NASA Astrophysics Data System (ADS)

Martí, J. G.; Cincotta, P. M.; Beaugé, C.

2016-07-01

Chaotic diffusion is supposed to be responsible for orbital instabilities in planetary systems after the dissipation of the protoplanetary disc, and a natural consequence of irregular motion. In this paper, we show that resonant multiplanetary systems, despite being highly chaotic, not necessarily exhibit significant diffusion in phase space, and may still survive virtually unchanged over time-scales comparable to their age. Using the GJ-876 system as an example, we analyse the chaotic diffusion of the outermost (and less massive) planet. We construct a set of stability maps in the surrounding regions of the Laplace resonance. We numerically integrate ensembles of close initial conditions, compute Poincaré maps and estimate the chaotic diffusion present in this system. Our results show that, the Laplace resonance contains two different regions: an inner domain characterized by low chaoticity and slow diffusion, and an outer one displaying larger values of dynamical indicators. In the outer resonant domain, the stochastic borders of the Laplace resonance seem to prevent the complete destruction of the system. We characterize the diffusion for small ensembles along the parameters of the outermost planet. Finally, we perform a stability analysis of the inherent chaotic, albeit stable Laplace resonance, by linking the behaviour of the resonant variables of the configurations to the different sub-structures inside the three-body resonance.

Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability.

PubMed

Zhang, Sen; Zeng, Yicheng; Li, Zhijun; Wang, Mengjiao; Xiong, Le

2018-01-01

By using a simple state feedback controller in a three-dimensional chaotic system, a novel 4D chaotic system is derived in this paper. The system state equations are composed of nine terms including only one constant term. Depending on the different values of the constant term, this new proposed system has a line of equilibrium points or no equilibrium points. Compared with other similar chaotic systems, the newly presented system owns more abundant and complicated dynamic properties. What interests us is the observation that if the value of the constant term of the system is nonzero, it has no equilibria, and therefore, the Shil'nikov theorem is not suitable to verify the existence of chaos for the lack of heteroclinic or homoclinic trajectory. However, one-wing, two-wing, three-wing, and four-wing hidden attractors can be obtained from this new system. In addition, various coexisting hidden attractors are obtained and the complex transient transition behaviors are also observed. More interestingly, the unusual and striking dynamic behavior of the coexistence of infinitely many hidden attractors is revealed by selecting the different initial values of the system, which means that extreme multistability arises. The rich and complex hidden dynamic characteristics of this system are investigated by phase portraits, bifurcation diagrams, Lyapunov exponents, and so on. Finally, the new system is implemented by an electronic circuit. A very good agreement is observed between the experimental results and the numerical simulations of the same system on the Matlab platform.

Generating one to four-wing hidden attractors in a novel 4D no-equilibrium chaotic system with extreme multistability

NASA Astrophysics Data System (ADS)

Zhang, Sen; Zeng, Yicheng; Li, Zhijun; Wang, Mengjiao; Xiong, Le

2018-01-01

By using a simple state feedback controller in a three-dimensional chaotic system, a novel 4D chaotic system is derived in this paper. The system state equations are composed of nine terms including only one constant term. Depending on the different values of the constant term, this new proposed system has a line of equilibrium points or no equilibrium points. Compared with other similar chaotic systems, the newly presented system owns more abundant and complicated dynamic properties. What interests us is the observation that if the value of the constant term of the system is nonzero, it has no equilibria, and therefore, the Shil'nikov theorem is not suitable to verify the existence of chaos for the lack of heteroclinic or homoclinic trajectory. However, one-wing, two-wing, three-wing, and four-wing hidden attractors can be obtained from this new system. In addition, various coexisting hidden attractors are obtained and the complex transient transition behaviors are also observed. More interestingly, the unusual and striking dynamic behavior of the coexistence of infinitely many hidden attractors is revealed by selecting the different initial values of the system, which means that extreme multistability arises. The rich and complex hidden dynamic characteristics of this system are investigated by phase portraits, bifurcation diagrams, Lyapunov exponents, and so on. Finally, the new system is implemented by an electronic circuit. A very good agreement is observed between the experimental results and the numerical simulations of the same system on the Matlab platform.

Terminal Transient Phase of Chaotic Transients

NASA Astrophysics Data System (ADS)

Lilienkamp, Thomas; Parlitz, Ulrich

2018-03-01

Transient chaos in spatially extended systems can be characterized by the length of the transient phase, which typically grows quickly with the system size (supertransients). For a large class of these systems, the chaotic phase terminates abruptly, without any obvious precursors in commonly used observables. Here we investigate transient spatiotemporal chaos in two different models of this class. By probing the state space using perturbed trajectories we show the existence of a "terminal transient phase," which occurs prior to the abrupt collapse of chaotic dynamics. During this phase the impact of perturbations is significantly different from the earlier transient and particular patterns of (non)susceptible regions in state space occur close to the chaotic trajectories. We therefore hypothesize that even without perturbations proper precursors for the collapse of chaotic transients exist, which might be highly relevant for coping with spatiotemporal chaos in cardiac arrhythmias or brain functionality, for example.

Chaos and multi-scroll attractors in RCL-shunted junction coupled Jerk circuit connected by memristor

PubMed Central

Zhou, Ping; Ahmad, Bashir; Ren, Guodong; Wang, Chunni

2018-01-01

In this paper, a new four-variable dynamical system is proposed to set chaotic circuit composed of memristor and Josephson junction, and the dependence of chaotic behaviors on nonlinearity is investigated. A magnetic flux-controlled memristor is used to couple with the RCL-shunted junction circuit, and the dynamical behaviors can be modulated by changing the coupling intensity between the memristor and the RCL-shunted junction. Bifurcation diagram and Lyapunov exponent are calculated to confirm the emergence of chaos in the improved dynamical system. The outputs and dynamical behaviors can be controlled by the initial setting and external stimulus as well. As a result, chaos can be suppressed and spiking occurs in the sampled outputs under negative feedback, while applying positive feedback type via memristor can be effective to trigger chaos. Furthermore, it is found that the number of multi-attractors in the Jerk circuit can be modulated when memristor coupling is applied on the circuit. These results indicate that memristor coupling can be effective to control chaotic circuits and it is also useful to reproduce dynamical behaviors for neuronal activities. PMID:29342178

Chaos and multi-scroll attractors in RCL-shunted junction coupled Jerk circuit connected by memristor.

PubMed

Ma, Jun; Zhou, Ping; Ahmad, Bashir; Ren, Guodong; Wang, Chunni

2018-01-01

In this paper, a new four-variable dynamical system is proposed to set chaotic circuit composed of memristor and Josephson junction, and the dependence of chaotic behaviors on nonlinearity is investigated. A magnetic flux-controlled memristor is used to couple with the RCL-shunted junction circuit, and the dynamical behaviors can be modulated by changing the coupling intensity between the memristor and the RCL-shunted junction. Bifurcation diagram and Lyapunov exponent are calculated to confirm the emergence of chaos in the improved dynamical system. The outputs and dynamical behaviors can be controlled by the initial setting and external stimulus as well. As a result, chaos can be suppressed and spiking occurs in the sampled outputs under negative feedback, while applying positive feedback type via memristor can be effective to trigger chaos. Furthermore, it is found that the number of multi-attractors in the Jerk circuit can be modulated when memristor coupling is applied on the circuit. These results indicate that memristor coupling can be effective to control chaotic circuits and it is also useful to reproduce dynamical behaviors for neuronal activities.

Transition to Chaos in Random Neuronal Networks

NASA Astrophysics Data System (ADS)

Kadmon, Jonathan; Sompolinsky, Haim

2015-10-01

Firing patterns in the central nervous system often exhibit strong temporal irregularity and considerable heterogeneity in time-averaged response properties. Previous studies suggested that these properties are the outcome of the intrinsic chaotic dynamics of the neural circuits. Indeed, simplified rate-based neuronal networks with synaptic connections drawn from Gaussian distribution and sigmoidal nonlinearity are known to exhibit chaotic dynamics when the synaptic gain (i.e., connection variance) is sufficiently large. In the limit of an infinitely large network, there is a sharp transition from a fixed point to chaos, as the synaptic gain reaches a critical value. Near the onset, chaotic fluctuations are slow, analogous to the ubiquitous, slow irregular fluctuations observed in the firing rates of many cortical circuits. However, the existence of a transition from a fixed point to chaos in neuronal circuit models with more realistic architectures and firing dynamics has not been established. In this work, we investigate rate-based dynamics of neuronal circuits composed of several subpopulations with randomly diluted connections. Nonzero connections are either positive for excitatory neurons or negative for inhibitory ones, while single neuron output is strictly positive with output rates rising as a power law above threshold, in line with known constraints in many biological systems. Using dynamic mean field theory, we find the phase diagram depicting the regimes of stable fixed-point, unstable-dynamic, and chaotic-rate fluctuations. We focus on the latter and characterize the properties of systems near this transition. We show that dilute excitatory-inhibitory architectures exhibit the same onset to chaos as the single population with Gaussian connectivity. In these architectures, the large mean excitatory and inhibitory inputs dynamically balance each other, amplifying the effect of the residual fluctuations. Importantly, the existence of a transition to chaos and its critical properties depend on the shape of the single-neuron nonlinear input-output transfer function, near firing threshold. In particular, for nonlinear transfer functions with a sharp rise near threshold, the transition to chaos disappears in the limit of a large network; instead, the system exhibits chaotic fluctuations even for small synaptic gain. Finally, we investigate transition to chaos in network models with spiking dynamics. We show that when synaptic time constants are slow relative to the mean inverse firing rates, the network undergoes a transition from fast spiking fluctuations with constant rates to a state where the firing rates exhibit chaotic fluctuations, similar to the transition predicted by rate-based dynamics. Systems with finite synaptic time constants and firing rates exhibit a smooth transition from a regime dominated by stationary firing rates to a regime of slow rate fluctuations. This smooth crossover obeys scaling properties, similar to crossover phenomena in statistical mechanics. The theoretical results are supported by computer simulations of several neuronal architectures and dynamics. Consequences for cortical circuit dynamics are discussed. These results advance our understanding of the properties of intrinsic dynamics in realistic neuronal networks and their functional consequences.

Coexisting multiple attractors and riddled basins of a memristive system.

PubMed

Wang, Guangyi; Yuan, Fang; Chen, Guanrong; Zhang, Yu

2018-01-01

In this paper, a new memristor-based chaotic system is designed, analyzed, and implemented. Multistability, multiple attractors, and complex riddled basins are observed from the system, which are investigated along with other dynamical behaviors such as equilibrium points and their stabilities, symmetrical bifurcation diagrams, and sustained chaotic states. With different sets of system parameters, the system can also generate various multi-scroll attractors. Finally, the system is realized by experimental circuits.

Chaotic dynamics in nanoscale NbO2 Mott memristors for analogue computing

NASA Astrophysics Data System (ADS)

Kumar, Suhas; Strachan, John Paul; Williams, R. Stanley

2017-08-01

At present, machine learning systems use simplified neuron models that lack the rich nonlinear phenomena observed in biological systems, which display spatio-temporal cooperative dynamics. There is evidence that neurons operate in a regime called the edge of chaos that may be central to complexity, learning efficiency, adaptability and analogue (non-Boolean) computation in brains. Neural networks have exhibited enhanced computational complexity when operated at the edge of chaos, and networks of chaotic elements have been proposed for solving combinatorial or global optimization problems. Thus, a source of controllable chaotic behaviour that can be incorporated into a neural-inspired circuit may be an essential component of future computational systems. Such chaotic elements have been simulated using elaborate transistor circuits that simulate known equations of chaos, but an experimental realization of chaotic dynamics from a single scalable electronic device has been lacking. Here we describe niobium dioxide (NbO2) Mott memristors each less than 100 nanometres across that exhibit both a nonlinear-transport-driven current-controlled negative differential resistance and a Mott-transition-driven temperature-controlled negative differential resistance. Mott materials have a temperature-dependent metal-insulator transition that acts as an electronic switch, which introduces a history-dependent resistance into the device. We incorporate these memristors into a relaxation oscillator and observe a tunable range of periodic and chaotic self-oscillations. We show that the nonlinear current transport coupled with thermal fluctuations at the nanoscale generates chaotic oscillations. Such memristors could be useful in certain types of neural-inspired computation by introducing a pseudo-random signal that prevents global synchronization and could also assist in finding a global minimum during a constrained search. We specifically demonstrate that incorporating such memristors into the hardware of a Hopfield computing network can greatly improve the efficiency and accuracy of converging to a solution for computationally difficult problems.

Multistability and hidden attractors in a relay system with hysteresis

NASA Astrophysics Data System (ADS)

Zhusubaliyev, Zhanybai T.; Mosekilde, Erik; Rubanov, Vasily G.; Nabokov, Roman A.

2015-06-01

For nonlinear dynamic systems with switching control, the concept of a "hidden attractor" naturally applies to a stable dynamic state that either (1) coexists with the stable switching cycle or (2), if the switching cycle is unstable, has a basin of attraction that does not intersect with the neighborhood of that cycle. We show how the equilibrium point of a relay system disappears in a boundary-equilibrium bifurcation as the system enters the region of autonomous switching dynamics and demonstrate experimentally how a relay system can exhibit large amplitude chaotic oscillations at high values of the supply voltage. By investigating a four-dimensional model of the experimental relay system we finally show how a variety of hidden periodic, quasiperiodic and chaotic attractors arise, transform and disappear through different bifurcations.

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.

Role of Orbital Dynamics in Spin Relaxation and Weak Antilocalization in Quantum Dots

NASA Astrophysics Data System (ADS)

Zaitsev, Oleg; Frustaglia, Diego; Richter, Klaus

2005-01-01

We develop a semiclassical theory for spin-dependent quantum transport to describe weak (anti)localization in quantum dots with spin-orbit coupling. This allows us to distinguish different types of spin relaxation in systems with chaotic, regular, and diffusive orbital classical dynamics. We find, in particular, that for typical Rashba spin-orbit coupling strengths, integrable ballistic systems can exhibit weak localization, while corresponding chaotic systems show weak antilocalization. We further calculate the magnetoconductance and analyze how the weak antilocalization is suppressed with decreasing quantum dot size and increasing additional in-plane magnetic field.

Enhancing synchrony in chaotic oscillators by dynamic relaying

NASA Astrophysics Data System (ADS)

Banerjee, Ranjib; Ghosh, Dibakar; Padmanaban, E.; Ramaswamy, R.; Pecora, L. M.; Dana, Syamal K.

2012-02-01

In a chain of mutually coupled oscillators, the coupling threshold for synchronization between the outermost identical oscillators decreases when a type of impurity (in terms of parameter mismatch) is introduced in the inner oscillator(s). The outer oscillators interact indirectly via dynamic relaying, mediated by the inner oscillator(s). We confirm this enhancing of critical coupling in the chaotic regimes of the Lorenz system, in the Rössler system in the absence of coupling delay, and in the Mackey-Glass system with delay coupling. The enhancing effect is experimentally verified in the electronic circuit of Rössler oscillators.

A solution for two-dimensional mazes with use of chaotic dynamics in a recurrent neural network model.

PubMed

Suemitsu, Yoshikazu; Nara, Shigetoshi

2004-09-01

Chaotic dynamics introduced into a neural network model is applied to solving two-dimensional mazes, which are ill-posed problems. A moving object moves from the position at t to t + 1 by simply defined motion function calculated from firing patterns of the neural network model at each time step t. We have embedded several prototype attractors that correspond to the simple motion of the object orienting toward several directions in two-dimensional space in our neural network model. Introducing chaotic dynamics into the network gives outputs sampled from intermediate state points between embedded attractors in a state space, and these dynamics enable the object to move in various directions. System parameter switching between a chaotic and an attractor regime in the state space of the neural network enables the object to move to a set target in a two-dimensional maze. Results of computer simulations show that the success rate for this method over 300 trials is higher than that of random walk. To investigate why the proposed method gives better performance, we calculate and discuss statistical data with respect to dynamical structure.

A Non-Intrusive Algorithm for Sensitivity Analysis of Chaotic Flow Simulations

NASA Technical Reports Server (NTRS)

Blonigan, Patrick J.; Wang, Qiqi; Nielsen, Eric J.; Diskin, Boris

2017-01-01

We demonstrate a novel algorithm for computing the sensitivity of statistics in chaotic flow simulations to parameter perturbations. The algorithm is non-intrusive but requires exposing an interface. Based on the principle of shadowing in dynamical systems, this algorithm is designed to reduce the effect of the sampling error in computing sensitivity of statistics in chaotic simulations. We compare the effectiveness of this method to that of the conventional finite difference method.

Regular-to-Chaotic Tunneling Rates: From the Quantum to the Semiclassical Regime

NASA Astrophysics Data System (ADS)

Löck, Steffen; Bäcker, Arnd; Ketzmerick, Roland; Schlagheck, Peter

2010-03-01

We derive a prediction of dynamical tunneling rates from regular to chaotic phase-space regions combining the direct regular-to-chaotic tunneling mechanism in the quantum regime with an improved resonance-assisted tunneling theory in the semiclassical regime. We give a qualitative recipe for identifying the relevance of nonlinear resonances in a given ℏ regime. For systems with one or multiple dominant resonances we find excellent agreement to numerics.

Competitive Modes for the Detection of Chaotic Parameter Regimes in the General Chaotic Bilinear System of Lorenz Type

NASA Astrophysics Data System (ADS)

Mallory, Kristina; van Gorder, Robert A.

We study chaotic behavior of solutions to the bilinear system of Lorenz type developed by Celikovsky and Vanecek [1994] through an application of competitive modes. This bilinear system of Lorenz type is one possible canonical form holding the Lorenz equation as a special case. Using a competitive modes analysis, which is a completely analytical method allowing one to identify parameter regimes for which chaos may occur, we are able to demonstrate a number of parameter regimes which admit a variety of distinct chaotic behaviors. Indeed, we are able to draw some interesting conclusions which relate the behavior of the mode frequencies arising from writing the state variables for the Celikovsky-Vanecek model as coupled oscillators, and the types of emergent chaotic behaviors observed. The competitive modes analysis is particularly useful if all but one of the model parameters are fixed, and the remaining free parameter is used to modify the chaos observed, in a manner analogous to a bifurcation parameter. Through a thorough application of the method, we are able to identify several parameter regimes which give new dynamics (such as specific forms of chaos) which were not observed or studied previously in the Celikovsky-Vanecek model. Therefore, the results demonstrate the advantage of the competitive modes approach for detecting new parameter regimes leading to chaos in third-order dynamical systems.

THE ASTEROID BELT AS A RELIC FROM A CHAOTIC EARLY SOLAR SYSTEM

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

Izidoro, André; Raymond, Sean N.; Pierens, Arnaud

The orbital structure of the asteroid belt holds a record of the solar system’s dynamical history. The current belt only contains ∼10{sup −3} Earth masses yet the asteroids’ orbits are dynamically excited, with a large spread in eccentricity and inclination. In the context of models of terrestrial planet formation, the belt may have been excited by Jupiter’s orbital migration. The terrestrial planets can also be reproduced without invoking a migrating Jupiter; however, as it requires a severe mass deficit beyond Earth’s orbit, this model systematically under-excites the asteroid belt. Here we show that the orbits of the asteroids may havemore » been excited to their current state if Jupiter’s and Saturn’s early orbits were chaotic. Stochastic variations in the gas giants’ orbits cause resonances to continually jump across the main belt and excite the asteroids’ orbits on a timescale of tens of millions of years. While hydrodynamical simulations show that the gas giants were likely in mean motion resonance at the end of the gaseous disk phase, small perturbations could have driven them into a chaotic but stable state. The gas giants’ current orbits were achieved later, during an instability in the outer solar system. Although it is well known that the present-day solar system exhibits chaotic behavior, our results suggest that the early solar system may also have been chaotic.« less

Chaotic itinerancy and power-law residence time distribution in stochastic dynamical systems.

PubMed

Namikawa, Jun

2005-08-01

Chaotic itinerant motion among varieties of ordered states is described by a stochastic model based on the mechanism of chaotic itinerancy. The model consists of a random walk on a half-line and a Markov chain with a transition probability matrix. The stability of attractor ruin in the model is investigated by analyzing the residence time distribution of orbits at attractor ruins. It is shown that the residence time distribution averaged over all attractor ruins can be described by the superposition of (truncated) power-law distributions if the basin of attraction for each attractor ruin has a zero measure. This result is confirmed by simulation of models exhibiting chaotic itinerancy. Chaotic itinerancy is also shown to be absent in coupled Milnor attractor systems if the transition probability among attractor ruins can be represented as a Markov chain.

The role of model dynamics in ensemble Kalman filter performance for chaotic systems

USGS Publications Warehouse

Ng, G.-H.C.; McLaughlin, D.; Entekhabi, D.; Ahanin, A.

2011-01-01

The ensemble Kalman filter (EnKF) is susceptible to losing track of observations, or 'diverging', when applied to large chaotic systems such as atmospheric and ocean models. Past studies have demonstrated the adverse impact of sampling error during the filter's update step. We examine how system dynamics affect EnKF performance, and whether the absence of certain dynamic features in the ensemble may lead to divergence. The EnKF is applied to a simple chaotic model, and ensembles are checked against singular vectors of the tangent linear model, corresponding to short-term growth and Lyapunov vectors, corresponding to long-term growth. Results show that the ensemble strongly aligns itself with the subspace spanned by unstable Lyapunov vectors. Furthermore, the filter avoids divergence only if the full linearized long-term unstable subspace is spanned. However, short-term dynamics also become important as non-linearity in the system increases. Non-linear movement prevents errors in the long-term stable subspace from decaying indefinitely. If these errors then undergo linear intermittent growth, a small ensemble may fail to properly represent all important modes, causing filter divergence. A combination of long and short-term growth dynamics are thus critical to EnKF performance. These findings can help in developing practical robust filters based on model dynamics. ?? 2011 The Authors Tellus A ?? 2011 John Wiley & Sons A/S.

Chaos and Forecasting - Proceedings of the Royal Society Discussion Meeting

NASA Astrophysics Data System (ADS)

Tong, Howell

1995-04-01

The Table of Contents for the full book PDF is as follows: * Preface * Orthogonal Projection, Embedding Dimension and Sample Size in Chaotic Time Series from a Statistical Perspective * A Theory of Correlation Dimension for Stationary Time Series * On Prediction and Chaos in Stochastic Systems * Locally Optimized Prediction of Nonlinear Systems: Stochastic and Deterministic * A Poisson Distribution for the BDS Test Statistic for Independence in a Time Series * Chaos and Nonlinear Forecastability in Economics and Finance * Paradigm Change in Prediction * Predicting Nonuniform Chaotic Attractors in an Enzyme Reaction * Chaos in Geophysical Fluids * Chaotic Modulation of the Solar Cycle * Fractal Nature in Earthquake Phenomena and its Simple Models * Singular Vectors and the Predictability of Weather and Climate * Prediction as a Criterion for Classifying Natural Time Series * Measuring and Characterising Spatial Patterns, Dynamics and Chaos in Spatially-Extended Dynamical Systems and Ecologies * Non-Linear Forecasting and Chaos in Ecology and Epidemiology: Measles as a Case Study

Time-delayed chameleon: Analysis, synchronization and FPGA implementation

NASA Astrophysics Data System (ADS)

Rajagopal, Karthikeyan; Jafari, Sajad; Laarem, Guessas

2017-12-01

In this paper we report a time-delayed chameleon-like chaotic system which can belong to different families of chaotic attractors depending on the choices of parameters. Such a characteristic of self-excited and hidden chaotic flows in a simple 3D system with time delay has not been reported earlier. Dynamic analysis of the proposed time-delayed systems are analysed in time-delay space and parameter space. A novel adaptive modified functional projective lag synchronization algorithm is derived for synchronizing identical time-delayed chameleon systems with uncertain parameters. The proposed time-delayed systems and the synchronization algorithm with controllers and parameter estimates are then implemented in FPGA using hardware-software co-simulation and the results are presented.

Chaos in high-dimensional dissipative dynamical systems

PubMed Central

Ispolatov, Iaroslav; Madhok, Vaibhav; Allende, Sebastian; Doebeli, Michael

2015-01-01

For dissipative dynamical systems described by a system of ordinary differential equations, we address the question of how the probability of chaotic dynamics increases with the dimensionality of the phase space. We find that for a system of d globally coupled ODE’s with quadratic and cubic non-linearities with randomly chosen coefficients and initial conditions, the probability of a trajectory to be chaotic increases universally from ~10−5 − 10−4 for d = 3 to essentially one for d ~ 50. In the limit of large d, the invariant measure of the dynamical systems exhibits universal scaling that depends on the degree of non-linearity, but not on the choice of coefficients, and the largest Lyapunov exponent converges to a universal scaling limit. Using statistical arguments, we provide analytical explanations for the observed scaling, universality, and for the probability of chaos. PMID:26224119

A secure communication using cascade chaotic computing systems on clinical decision support.

PubMed

Koksal, Ahmet Sertol; Er, Orhan; Evirgen, Hayrettin; Yumusak, Nejat

2016-06-01

Clinical decision support systems (C-DSS) provide supportive tools to the expert for the determination of the disease. Today, many of the support systems, which have been developed for a better and more accurate diagnosis, have reached a dynamic structure due to artificial intelligence techniques. However, in cases when important diagnosis studies should be performed in secret, a secure communication system is required. In this study, secure communication of a DSS is examined through a developed double layer chaotic communication system. The developed communication system consists of four main parts: random number generator, cascade chaotic calculation layer, PCM, and logical mixer layers. Thanks to this system, important patient data created by DSS will be conveyed to the center through a secure communication line.

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.

Chaotic dynamics in the (47171) Lempo triple system

NASA Astrophysics Data System (ADS)

Correia, Alexandre C. M.

2018-05-01

We investigate the dynamics of the (47171) Lempo triple system, also known by 1999 TC36. We derive a full 3D N-body model that takes into account the orbital and spin evolution of all bodies, which are assumed triaxial ellipsoids. We show that, for reasonable values of the shapes and rotational periods, the present best fitted orbital solution for the Lempo system is chaotic and unstable in short time-scales. The formation mechanism of this system is unknown, but the orbits can be stabilised when tidal dissipation is taken into account. The dynamics of the Lempo system is very rich, but depends on many parameters that are presently unknown. A better understanding of this systems thus requires more observations, which also need to be fitted with a complete model like the one presented here.

Precisely cyclic sand: self-organization of periodically sheared frictional grains.

PubMed

Royer, John R; Chaikin, Paul M

2015-01-06

The disordered static structure and chaotic dynamics of frictional granular matter has occupied scientists for centuries, yet there are few organizational principles or guiding rules for this highly hysteretic, dissipative material. We show that cyclic shear of a granular material leads to dynamic self-organization into several phases with different spatial and temporal order. Using numerical simulations, we present a phase diagram in strain-friction space that shows chaotic dispersion, crystal formation, vortex patterns, and most unusually a disordered phase in which each particle precisely retraces its unique path. However, the system is not reversible. Rather, the trajectory of each particle, and the entire frictional, many-degrees-of-freedom system, organizes itself into a limit cycle absorbing state. Of particular note is that fact that the cyclic states are spatially disordered, whereas the ordered states are chaotic.

Precisely cyclic sand: Self-organization of periodically sheared frictional grains

PubMed Central

Royer, John R.; Chaikin, Paul M.

2015-01-01

The disordered static structure and chaotic dynamics of frictional granular matter has occupied scientists for centuries, yet there are few organizational principles or guiding rules for this highly hysteretic, dissipative material. We show that cyclic shear of a granular material leads to dynamic self-organization into several phases with different spatial and temporal order. Using numerical simulations, we present a phase diagram in strain–friction space that shows chaotic dispersion, crystal formation, vortex patterns, and most unusually a disordered phase in which each particle precisely retraces its unique path. However, the system is not reversible. Rather, the trajectory of each particle, and the entire frictional, many–degrees-of-freedom system, organizes itself into a limit cycle absorbing state. Of particular note is that fact that the cyclic states are spatially disordered, whereas the ordered states are chaotic. PMID:25538298

Chaotic patterns of autonomic activity during hypnotic recall.

PubMed

Bob, Petr; Siroka, Ivana; Susta, Marek

2009-01-01

Chaotic neural dynamics likely emerge in cognitive processes and may present time periods that are extremely sensitive to influences affecting the neural system. Recent findings suggest that this sensitivity may increase during retrieval of stressful emotional experiences reflecting underlying mechanism related to consolidation of traumatic memories. In this context, hypnotic recall of anxiety memories in 10 patients, simultaneously with ECG measurement was performed. The same measurement was performed during control cognitive task in 8 anxiety patients and 22 healthy controls. Nonlinear data analysis of ECG records indicates significant increase in the degree of chaos during retrieval of stressful memory in all the patients. The results suggest a role of chaotic neural dynamics during processing of anxiety-related stressful memories.

Stochastic dynamics and combinatorial optimization

NASA Astrophysics Data System (ADS)

Ovchinnikov, Igor V.; Wang, Kang L.

2017-11-01

Natural dynamics is often dominated by sudden nonlinear processes such as neuroavalanches, gamma-ray bursts, solar flares, etc., that exhibit scale-free statistics much in the spirit of the logarithmic Ritcher scale for earthquake magnitudes. On phase diagrams, stochastic dynamical systems (DSs) exhibiting this type of dynamics belong to the finite-width phase (N-phase for brevity) that precedes ordinary chaotic behavior and that is known under such names as noise-induced chaos, self-organized criticality, dynamical complexity, etc. Within the recently proposed supersymmetric theory of stochastic dynamics, the N-phase can be roughly interpreted as the noise-induced “overlap” between integrable and chaotic deterministic dynamics. As a result, the N-phase dynamics inherits the properties of the both. Here, we analyze this unique set of properties and conclude that the N-phase DSs must naturally be the most efficient optimizers: on one hand, N-phase DSs have integrable flows with well-defined attractors that can be associated with candidate solutions and, on the other hand, the noise-induced attractor-to-attractor dynamics in the N-phase is effectively chaotic or aperiodic so that a DS must avoid revisiting solutions/attractors thus accelerating the search for the best solution. Based on this understanding, we propose a method for stochastic dynamical optimization using the N-phase DSs. This method can be viewed as a hybrid of the simulated and chaotic annealing methods. Our proposition can result in a new generation of hardware devices for efficient solution of various search and/or combinatorial optimization problems.

Transient chaos in the Lorenz-type map with periodic forcing.

PubMed

Maslennikov, Oleg V; Nekorkin, Vladimir I; Kurths, Jürgen

2018-03-01

We consider a case study of perturbing a system with a boundary crisis of a chaotic attractor by periodic forcing. In the static case, the system exhibits persistent chaos below the critical value of the control parameter but transient chaos above the critical value. We discuss what happens to the system and particularly to the transient chaotic dynamics if the control parameter periodically oscillates. We find a non-exponential decaying behavior of the survival probability function, study the impact of the forcing frequency and amplitude on the escape rate, analyze the phase-space image of the observed dynamics, and investigate the influence of initial conditions.

Transient chaos in the Lorenz-type map with periodic forcing

NASA Astrophysics Data System (ADS)

Maslennikov, Oleg V.; Nekorkin, Vladimir I.; Kurths, Jürgen

2018-03-01

We consider a case study of perturbing a system with a boundary crisis of a chaotic attractor by periodic forcing. In the static case, the system exhibits persistent chaos below the critical value of the control parameter but transient chaos above the critical value. We discuss what happens to the system and particularly to the transient chaotic dynamics if the control parameter periodically oscillates. We find a non-exponential decaying behavior of the survival probability function, study the impact of the forcing frequency and amplitude on the escape rate, analyze the phase-space image of the observed dynamics, and investigate the influence of initial conditions.

Nonlinear modeling of chaotic time series: Theory and applications

NASA Astrophysics Data System (ADS)

Casdagli, M.; Eubank, S.; Farmer, J. D.; Gibson, J.; Desjardins, D.; Hunter, N.; Theiler, J.

We review recent developments in the modeling and prediction of nonlinear time series. In some cases, apparent randomness in time series may be due to chaotic behavior of a nonlinear but deterministic system. In such cases, it is possible to exploit the determinism to make short term forecasts that are much more accurate than one could make from a linear stochastic model. This is done by first reconstructing a state space, and then using nonlinear function approximation methods to create a dynamical model. Nonlinear models are valuable not only as short term forecasters, but also as diagnostic tools for identifying and quantifying low-dimensional chaotic behavior. During the past few years, methods for nonlinear modeling have developed rapidly, and have already led to several applications where nonlinear models motivated by chaotic dynamics provide superior predictions to linear models. These applications include prediction of fluid flows, sunspots, mechanical vibrations, ice ages, measles epidemics, and human speech.

Semiconductor lasers driven by self-sustained chaotic electronic oscillators and applications to optical chaos cryptography.

PubMed

Kingni, Sifeu Takougang; Mbé, Jimmi Hervé Talla; Woafo, Paul

2012-09-01

In this work, we numerically study the dynamics of vertical cavity surface emitting laser (VCSEL) firstly when it is driven by Chua's oscillator, secondly in case where it is driven by a broad frequency spectral bandwidth chaotic oscillator developed by Nana et al. [Commun. Nonlinear Sci. Numer. Simul. 14, 2266 (2009)]. We demonstrated that the VCSEL generated robust chaotic dynamics compared to the ones found in VCSEL subject to a sinusoidally modulated current and therefore it is more suitable for chaos encryption techniques. The synchronization characteristics and the communication performances of unidirectional coupled VCSEL driven by the broad frequency spectral bandwidth chaotic oscillators are investigated numerically. The results show that high-quality synchronization and transmission of messages can be realized for suitable system parameters. Chaos shift keying method is successfully applied to encrypt a message at a high bitrate.

Control of complex dynamics and chaos in distributed parameter systems

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

Chakravarti, S.; Marek, M.; Ray, W.H.

This paper discusses a methodology for controlling complex dynamics and chaos in distributed parameter systems. The reaction-diffusion system with Brusselator kinetics, where the torus-doubling or quasi-periodic (two characteristic incommensurate frequencies) route to chaos exists in a defined range of parameter values, is used as an example. Poincare maps are used for characterization of quasi-periodic and chaotic attractors. The dominant modes or topos, which are inherent properties of the system, are identified by means of the Singular Value Decomposition. Tested modal feedback control schemas based on identified dominant spatial modes confirm the possibility of stabilization of simple quasi-periodic trajectories in themore » complex quasi-periodic or chaotic spatiotemporal patterns.« less

A qualitative numerical study of high dimensional dynamical systems

NASA Astrophysics Data System (ADS)

Albers, David James

Since Poincare, the father of modern mathematical dynamical systems, much effort has been exerted to achieve a qualitative understanding of the physical world via a qualitative understanding of the functions we use to model the physical world. In this thesis, we construct a numerical framework suitable for a qualitative, statistical study of dynamical systems using the space of artificial neural networks. We analyze the dynamics along intervals in parameter space, separating the set of neural networks into roughly four regions: the fixed point to the first bifurcation; the route to chaos; the chaotic region; and a transition region between chaos and finite-state neural networks. The study is primarily with respect to high-dimensional dynamical systems. We make the following general conclusions as the dimension of the dynamical system is increased: the probability of the first bifurcation being of type Neimark-Sacker is greater than ninety-percent; the most probable route to chaos is via a cascade of bifurcations of high-period periodic orbits, quasi-periodic orbits, and 2-tori; there exists an interval of parameter space such that hyperbolicity is violated on a countable, Lebesgue measure 0, "increasingly dense" subset; chaos is much more likely to persist with respect to parameter perturbation in the chaotic region of parameter space as the dimension is increased; moreover, as the number of positive Lyapunov exponents is increased, the likelihood that any significant portion of these positive exponents can be perturbed away decreases with increasing dimension. The maximum Kaplan-Yorke dimension and the maximum number of positive Lyapunov exponents increases linearly with dimension. The probability of a dynamical system being chaotic increases exponentially with dimension. The results with respect to the first bifurcation and the route to chaos comment on previous results of Newhouse, Ruelle, Takens, Broer, Chenciner, and Iooss. Moreover, results regarding the high-dimensional chaotic region of parameter space is interpreted and related to the closing lemma of Pugh, the windows conjecture of Barreto, the stable ergodicity theorem of Pugh and Shub, and structural stability theorem of Robbin, Robinson, and Mane.

Random Matrix Theory Approach to Chaotic Coherent Perfect Absorbers

NASA Astrophysics Data System (ADS)

Li, Huanan; Suwunnarat, Suwun; Fleischmann, Ragnar; Schanz, Holger; Kottos, Tsampikos

2017-01-01

We employ random matrix theory in order to investigate coherent perfect absorption (CPA) in lossy systems with complex internal dynamics. The loss strength γCPA and energy ECPA, for which a CPA occurs, are expressed in terms of the eigenmodes of the isolated cavity—thus carrying over the information about the chaotic nature of the target—and their coupling to a finite number of scattering channels. Our results are tested against numerical calculations using complex networks of resonators and chaotic graphs as CPA cavities.

Dynamical phases of the Hindmarsh-Rose neuronal model: studies of the transition from bursting to spiking chaos.

PubMed

Innocenti, Giacomo; Morelli, Alice; Genesio, Roberto; Torcini, Alessandro

2007-12-01

The dynamical phases of the Hindmarsh-Rose neuronal model are analyzed in detail by varying the external current I. For increasing current values, the model exhibits a peculiar cascade of nonchaotic and chaotic period-adding bifurcations leading the system from the silent regime to a chaotic state dominated by bursting events. At higher I-values, this phase is substituted by a regime of continuous chaotic spiking and finally via an inverse period doubling cascade the system returns to silence. The analysis is focused on the transition between the two chaotic phases displayed by the model: one dominated by spiking dynamics and the other by bursts. At the transition an abrupt shrinking of the attractor size associated with a sharp peak in the maximal Lyapunov exponent is observable. However, the transition appears to be continuous and smoothed out over a finite current interval, where bursts and spikes coexist. The beginning of the transition (from the bursting side) is signaled from a structural modification in the interspike interval return map. This change in the map shape is associated with the disappearance of the family of solutions responsible for the onset of the bursting chaos. The successive passage from bursting to spiking chaos is associated with a progressive pruning of unstable long-lasting bursts.

Network-induced chaos in integrate-and-fire neuronal ensembles.

PubMed

Zhou, Douglas; Rangan, Aaditya V; Sun, Yi; Cai, David

2009-09-01

It has been shown that a single standard linear integrate-and-fire (IF) neuron under a general time-dependent stimulus cannot possess chaotic dynamics despite the firing-reset discontinuity. Here we address the issue of whether conductance-based, pulsed-coupled network interactions can induce chaos in an IF neuronal ensemble. Using numerical methods, we demonstrate that all-to-all, homogeneously pulse-coupled IF neuronal networks can indeed give rise to chaotic dynamics under an external periodic current drive. We also provide a precise characterization of the largest Lyapunov exponent for these high dimensional nonsmooth dynamical systems. In addition, we present a stable and accurate numerical algorithm for evaluating the largest Lyapunov exponent, which can overcome difficulties encountered by traditional methods for these nonsmooth dynamical systems with degeneracy induced by, e.g., refractoriness of neurons.

Reducing the Dynamical Degradation by Bi-Coupling Digital Chaotic Maps

NASA Astrophysics Data System (ADS)

Liu, Lingfeng; Liu, Bocheng; Hu, Hanping; Miao, Suoxia

A chaotic map which is realized on a computer will suffer dynamical degradation. Here, a coupled chaotic model is proposed to reduce the dynamical degradation. In this model, the state variable of one digital chaotic map is used to control the parameter of the other digital map. This coupled model is universal and can be used for all chaotic maps. In this paper, two coupled models (one is coupled by two logistic maps, the other is coupled by Chebyshev map and Baker map) are performed, and the numerical experiments show that the performances of these two coupled chaotic maps are greatly improved. Furthermore, a simple pseudorandom bit generator (PRBG) based on coupled digital logistic maps is proposed as an application for our method.

Intermittent and sustained periodic windows in networked chaotic Rössler oscillators

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

He, Zhiwei; Sun, Yong; University of the Chinese Academy of Sciences, Beijing 100049

Route to chaos (or periodicity) in dynamical systems is one of fundamental problems. Here, dynamical behaviors of coupled chaotic Rössler oscillators on complex networks are investigated and two different types of periodic windows with the variation of coupling strength are found. Under a moderate coupling, the periodic window is intermittent, and the attractors within the window extremely sensitively depend on the initial conditions, coupling parameter, and topology of the network. Therefore, after adding or removing one edge of network, the periodic attractor can be destroyed and substituted by a chaotic one, or vice versa. In contrast, under an extremely weakmore » coupling, another type of periodic window appears, which insensitively depends on the initial conditions, coupling parameter, and network. It is sustained and unchanged for different types of network structure. It is also found that the phase differences of the oscillators are almost discrete and randomly distributed except that directly linked oscillators more likely have different phases. These dynamical behaviors have also been generally observed in other networked chaotic oscillators.« less

A method of recovering the initial vectors of globally coupled map lattices based on symbolic dynamics

NASA Astrophysics Data System (ADS)

Sun, Li-Sha; Kang, Xiao-Yun; Zhang, Qiong; Lin, Lan-Xin

2011-12-01

Based on symbolic dynamics, a novel computationally efficient algorithm is proposed to estimate the unknown initial vectors of globally coupled map lattices (CMLs). It is proved that not all inverse chaotic mapping functions are satisfied for contraction mapping. It is found that the values in phase space do not always converge on their initial values with respect to sufficient backward iteration of the symbolic vectors in terms of global convergence or divergence (CD). Both CD property and the coupling strength are directly related to the mapping function of the existing CML. Furthermore, the CD properties of Logistic, Bernoulli, and Tent chaotic mapping functions are investigated and compared. Various simulation results and the performances of the initial vector estimation with different signal-to-noise ratios (SNRs) are also provided to confirm the proposed algorithm. Finally, based on the spatiotemporal chaotic characteristics of the CML, the conditions of estimating the initial vectors using symbolic dynamics are discussed. The presented method provides both theoretical and experimental results for better understanding and characterizing the behaviours of spatiotemporal chaotic systems.

Quantum-classical correspondence in the vicinity of periodic orbits

NASA Astrophysics Data System (ADS)

Kumari, Meenu; Ghose, Shohini

2018-05-01

Quantum-classical correspondence in chaotic systems is a long-standing problem. We describe a method to quantify Bohr's correspondence principle and calculate the size of quantum numbers for which we can expect to observe quantum-classical correspondence near periodic orbits of Floquet systems. Our method shows how the stability of classical periodic orbits affects quantum dynamics. We demonstrate our method by analyzing quantum-classical correspondence in the quantum kicked top (QKT), which exhibits both regular and chaotic behavior. We use our correspondence conditions to identify signatures of classical bifurcations even in a deep quantum regime. Our method can be used to explain the breakdown of quantum-classical correspondence in chaotic systems.

Recurrence Quantification of Fractal Structures

PubMed Central

Webber, Charles L.

2012-01-01

By definition, fractal structures possess recurrent patterns. At different levels repeating patterns can be visualized at higher magnifications. The purpose of this chapter is threefold. First, general characteristics of dynamical systems are addressed from a theoretical mathematical perspective. Second, qualitative and quantitative recurrence analyses are reviewed in brief, but the reader is directed to other sources for explicit details. Third, example mathematical systems that generate strange attractors are explicitly defined, giving the reader the ability to reproduce the rich dynamics of continuous chaotic flows or discrete chaotic iterations. The challenge is then posited for the reader to study for themselves the recurrent structuring of these different dynamics. With a firm appreciation of the power of recurrence analysis, the reader will be prepared to turn their sights on real-world systems (physiological, psychological, mechanical, etc.). PMID:23060808

Quantification of scaling exponents and dynamical complexity of microwave refractivity in a tropical climate

NASA Astrophysics Data System (ADS)

Fuwape, Ibiyinka A.; Ogunjo, Samuel T.

2016-12-01

Radio refractivity index is used to quantify the effect of atmospheric parameters in communication systems. Scaling and dynamical complexities of radio refractivity across different climatic zones of Nigeria have been studied. Scaling property of the radio refractivity across Nigeria was estimated from the Hurst Exponent obtained using two different scaling methods namely: The Rescaled Range (R/S) and the detrended fluctuation analysis(DFA). The delay vector variance (DVV), Largest Lyapunov Exponent (λ1) and Correlation Dimension (D2) methods were used to investigate nonlinearity and the results confirm the presence of deterministic nonlinear profile in the radio refractivity time series. The recurrence quantification analysis (RQA) was used to quantify the degree of chaoticity in the radio refractivity across the different climatic zones. RQA was found to be a good measure for identifying unique fingerprint and signature of chaotic time series data. Microwave radio refractivity was found to be persistent and chaotic in all the study locations. The dynamics of radio refractivity increases in complexity and chaoticity from the Coastal region towards the Sahelian climate. The design, development and deployment of robust and reliable microwave communication link in the region will be greatly affected by the chaotic nature of radio refractivity in the region.

Study on a new chaotic bitwise dynamical system and its FPGA implementation

NASA Astrophysics Data System (ADS)

Wang, Qian-Xue; Yu, Si-Min; Guyeux, C.; Bahi, J.; Fang, Xiao-Le

2015-06-01

In this paper, the structure of a new chaotic bitwise dynamical system (CBDS) is described. Compared to our previous research work, it uses various random bitwise operations instead of only one. The chaotic behavior of CBDS is mathematically proven according to the Devaney's definition, and its statistical properties are verified both for uniformity and by a comprehensive, reputed and stringent battery of tests called TestU01. Furthermore, a systematic methodology developing the parallel computations is proposed for FPGA platform-based realization of this CBDS. Experiments finally validate the proposed systematic methodology. Project supported by China Postdoctoral Science Foundation (Grant No. 2014M552175), the Scientific Research Foundation for the Returned Overseas Chinese Scholars, Chinese Education Ministry, the National Natural Science Foundation of China (Grant No. 61172023), and the Specialized Research Foundation of Doctoral Subjects of Chinese Education Ministry (Grant No. 20114420110003).

Impact of Noise on a Dynamical System: Prediction and Uncertainties from a Swarm-Optimized Neural Network

PubMed Central

López-Caraballo, C. H.; Lazzús, J. A.; Salfate, I.; Rojas, P.; Rivera, M.; Palma-Chilla, L.

2015-01-01

An artificial neural network (ANN) based on particle swarm optimization (PSO) was developed for the time series prediction. The hybrid ANN+PSO algorithm was applied on Mackey-Glass chaotic time series in the short-term x(t + 6). The performance prediction was evaluated and compared with other studies available in the literature. Also, we presented properties of the dynamical system via the study of chaotic behaviour obtained from the predicted time series. Next, the hybrid ANN+PSO algorithm was complemented with a Gaussian stochastic procedure (called stochastic hybrid ANN+PSO) in order to obtain a new estimator of the predictions, which also allowed us to compute the uncertainties of predictions for noisy Mackey-Glass chaotic time series. Thus, we studied the impact of noise for several cases with a white noise level (σ N) from 0.01 to 0.1. PMID:26351449

Impact of Noise on a Dynamical System: Prediction and Uncertainties from a Swarm-Optimized Neural Network.

PubMed

López-Caraballo, C H; Lazzús, J A; Salfate, I; Rojas, P; Rivera, M; Palma-Chilla, L

2015-01-01

An artificial neural network (ANN) based on particle swarm optimization (PSO) was developed for the time series prediction. The hybrid ANN+PSO algorithm was applied on Mackey-Glass chaotic time series in the short-term x(t + 6). The performance prediction was evaluated and compared with other studies available in the literature. Also, we presented properties of the dynamical system via the study of chaotic behaviour obtained from the predicted time series. Next, the hybrid ANN+PSO algorithm was complemented with a Gaussian stochastic procedure (called stochastic hybrid ANN+PSO) in order to obtain a new estimator of the predictions, which also allowed us to compute the uncertainties of predictions for noisy Mackey-Glass chaotic time series. Thus, we studied the impact of noise for several cases with a white noise level (σ(N)) from 0.01 to 0.1.

Dynamics of some fictitious satellites of Venus and Mars

NASA Astrophysics Data System (ADS)

Yokoyama, Tadashi

1999-05-01

The dynamics of some fictitious satellites of Venus and Mars are studied considering only solar perturbation and the oblateness of the planet, as disturbing forces. Several numerical integrations of the averaged system, taking different values of the obliquity of ecliptic (ε), show the existence of strong chaotic motion, provided that the semi major axis is near a critical value. As a consequence, large increase of eccentricities occur and the satellites may collide with the planet or cross possible internal orbits. Even starting from almost circular and equatorial orbits, most satellites can easily reach prohibitive values. The extension of the chaotic zone depends clearly on the value of ε, so that, previous regular regions may become chaotic, provided ε increases sufficiently.

Emergence of chimeras through induced multistability

NASA Astrophysics Data System (ADS)

Ujjwal, Sangeeta Rani; Punetha, Nirmal; Prasad, Awadhesh; Ramaswamy, Ramakrishna

2017-03-01

Chimeras, namely coexisting desynchronous and synchronized dynamics, are formed in an ensemble of identically coupled identical chaotic oscillators when the coupling induces multiple stable attractors, and further when the basins of the different attractors are intertwined in a complex manner. When there is coupling-induced multistability, an ensemble of identical chaotic oscillators—with global coupling, or also under the influence of common noise or an external drive (chaotic, periodic, or quasiperiodic)—inevitably exhibits chimeric behavior. Induced multistability in the system leads to the formation of distinct subpopulations, one or more of which support synchronized dynamics, while in others the motion is asynchronous or incoherent. We study the mechanism for the emergence of such chimeric states, and we discuss the generality of our results.

Controlling effect of geometrically defined local structural changes on chaotic Hamiltonian systems.

PubMed

Ben Zion, Yossi; Horwitz, Lawrence

2010-04-01

An effective characterization of chaotic conservative Hamiltonian systems in terms of the curvature associated with a Riemannian metric tensor derived from the structure of the Hamiltonian has been extended to a wide class of potential models of standard form through definition of a conformal metric. The geodesic equations reproduce the Hamilton equations of the original potential model through an inverse map in the tangent space. The second covariant derivative of the geodesic deviation in this space generates a dynamical curvature, resulting in (energy-dependent) criteria for unstable behavior different from the usual Lyapunov criteria. We show here that this criterion can be constructively used to modify locally the potential of a chaotic Hamiltonian model in such a way that stable motion is achieved. Since our criterion for instability is local in coordinate space, these results provide a minimal method for achieving control of a chaotic system.

Numerical Study of Sound Emission by 2D Regular and Chaotic Vortex Configurations

NASA Astrophysics Data System (ADS)

Knio, Omar M.; Collorec, Luc; Juvé, Daniel

1995-02-01

The far-field noise generated by a system of three Gaussian vortices lying over a flat boundary is numerically investigated using a two-dimensional vortex element method. The method is based on the discretization of the vorticity field into a finite number of smoothed vortex elements of spherical overlapping cores. The elements are convected in a Lagrangian reference along particle trajectories using the local velocity vector, given in terms of a desingularized Biot-Savart law. The initial structure of the vortex system is triangular; a one-dimensional family of initial configurations is constructed by keeping one side of the triangle fixed and vertical, and varying the abscissa of the centroid of the remaining vortex. The inviscid dynamics of this vortex configuration are first investigated using non-deformable vortices. Depending on the aspect ratio of the initial system, regular or chaotic motion occurs. Due to wall-related symmetries, the far-field sound always exhibits a time-independent quadrupolar directivity with maxima parallel end perpendicular to the wall. When regular motion prevails, the noise spectrum is dominated by discrete frequencies which correspond to the fundamental system frequency and its superharmonics. For chaotic motion, a broadband spectrum is obtained; computed soundlevels are substantially higher than in non-chaotic systems. A more sophisticated analysis is then performed which accounts for vortex core dynamics. Results show that the vortex cores are susceptible to inviscid instability which leads to violent vorticity reorganization within the core. This phenomenon has little effect on the large-scale features of the motion of the system or on low frequency sound emission. However, it leads to the generation of a high-frequency noise band in the acoustic pressure spectrum. The latter is observed in both regular and chaotic system simulations.

Finding equilibrium in the spatiotemporal chaos of the complex Ginzburg-Landau equation

NASA Astrophysics Data System (ADS)

Ballard, Christopher C.; Esty, C. Clark; Egolf, David A.

2016-11-01

Equilibrium statistical mechanics allows the prediction of collective behaviors of large numbers of interacting objects from just a few system-wide properties; however, a similar theory does not exist for far-from-equilibrium systems exhibiting complex spatial and temporal behavior. We propose a method for predicting behaviors in a broad class of such systems and apply these ideas to an archetypal example, the spatiotemporal chaotic 1D complex Ginzburg-Landau equation in the defect chaos regime. Building on the ideas of Ruelle and of Cross and Hohenberg that a spatiotemporal chaotic system can be considered a collection of weakly interacting dynamical units of a characteristic size, the chaotic length scale, we identify underlying, mesoscale, chaotic units and effective interaction potentials between them. We find that the resulting equilibrium Takahashi model accurately predicts distributions of particle numbers. These results suggest the intriguing possibility that a class of far-from-equilibrium systems may be well described at coarse-grained scales by the well-established theory of equilibrium statistical mechanics.

Finding equilibrium in the spatiotemporal chaos of the complex Ginzburg-Landau equation.

PubMed

Ballard, Christopher C; Esty, C Clark; Egolf, David A

2016-11-01

Equilibrium statistical mechanics allows the prediction of collective behaviors of large numbers of interacting objects from just a few system-wide properties; however, a similar theory does not exist for far-from-equilibrium systems exhibiting complex spatial and temporal behavior. We propose a method for predicting behaviors in a broad class of such systems and apply these ideas to an archetypal example, the spatiotemporal chaotic 1D complex Ginzburg-Landau equation in the defect chaos regime. Building on the ideas of Ruelle and of Cross and Hohenberg that a spatiotemporal chaotic system can be considered a collection of weakly interacting dynamical units of a characteristic size, the chaotic length scale, we identify underlying, mesoscale, chaotic units and effective interaction potentials between them. We find that the resulting equilibrium Takahashi model accurately predicts distributions of particle numbers. These results suggest the intriguing possibility that a class of far-from-equilibrium systems may be well described at coarse-grained scales by the well-established theory of equilibrium statistical mechanics.

Efficient topological chaos embedded in the blinking vortex system.

PubMed

Kin, Eiko; Sakajo, Takashi

2005-06-01

We consider the particle mixing in the plane by two vortex points appearing one after the other, called the blinking vortex system. Mathematical and numerical studies of the system reveal that the chaotic particle mixing, i.e., the chaotic advection, is observed due to the homoclinic chaos, but the mixing region is restricted locally in the neighborhood of the vortex points. The present article shows that it is possible to realize a global and efficient chaotic advection in the blinking vortex system with the help of the Thurston-Nielsen theory, which classifies periodic orbits for homeomorphisms in the plane into three types: periodic, reducible, and pseudo-Anosov (pA). It is mathematically shown that periodic orbits of pA type generate a complicated dynamics, which is called topological chaos. We show that the combination of the local chaotic mixing due to the topological chaos and the dipole-like return orbits realize an efficient and global particle mixing in the blinking vortex system.

Robust optimization with transiently chaotic dynamical systems

NASA Astrophysics Data System (ADS)

Sumi, R.; Molnár, B.; Ercsey-Ravasz, M.

2014-05-01

Efficiently solving hard optimization problems has been a strong motivation for progress in analog computing. In a recent study we presented a continuous-time dynamical system for solving the NP-complete Boolean satisfiability (SAT) problem, with a one-to-one correspondence between its stable attractors and the SAT solutions. While physical implementations could offer great efficiency, the transiently chaotic dynamics raises the question of operability in the presence of noise, unavoidable on analog devices. Here we show that the probability of finding solutions is robust to noise intensities well above those present on real hardware. We also developed a cellular neural network model realizable with analog circuits, which tolerates even larger noise intensities. These methods represent an opportunity for robust and efficient physical implementations.

On the spin-axis dynamics of a Moonless Earth

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

Li, Gongjie; Batygin, Konstantin, E-mail: gli@cfa.harvard.edu

2014-07-20

The variation of a planet's obliquity is influenced by the existence of satellites with a high mass ratio. For instance, Earth's obliquity is stabilized by the Moon and would undergo chaotic variations in the Moon's absence. In turn, such variations can lead to large-scale changes in the atmospheric circulation, rendering spin-axis dynamics a central issue for understanding climate. The relevant quantity for dynamically forced climate change is the rate of chaotic diffusion. Accordingly, here we re-examine the spin-axis evolution of a Moonless Earth within the context of a simplified perturbative framework. We present analytical estimates of the characteristic Lyapunov coefficientmore » as well as the chaotic diffusion rate and demonstrate that even in absence of the Moon, the stochastic change in Earth's obliquity is sufficiently slow to not preclude long-term habitability. Our calculations are consistent with published numerical experiments and illustrate the putative system's underlying dynamical structure in a simple and intuitive manner.« less

Terminal Model Of Newtonian Dynamics

NASA Technical Reports Server (NTRS)

Zak, Michail

1994-01-01

Paper presents study of theory of Newtonian dynamics of terminal attractors and repellers, focusing on issues of reversibility vs. irreversibility and deterministic evolution vs. probabilistic or chaotic evolution of dynamic systems. Theory developed called "terminal dynamics" emphasizes difference between it and classical Newtonian dynamics. Also holds promise for explaining irreversibility, unpredictability, probabilistic behavior, and chaos in turbulent flows, in thermodynamic phenomena, and in other dynamic phenomena and systems.

Generating random numbers by means of nonlinear dynamic systems

NASA Astrophysics Data System (ADS)

Zang, Jiaqi; Hu, Haojie; Zhong, Juhua; Luo, Duanbin; Fang, Yi

2018-07-01

To introduce the randomness of a physical process to students, a chaotic pendulum experiment was opened in East China University of Science and Technology (ECUST) on the undergraduate level in the physics department. It was shown chaotic motion could be initiated through adjusting the operation of a chaotic pendulum. By using the data of the angular displacements of chaotic motion, random binary numerical arrays can be generated. To check the randomness of generated numerical arrays, the NIST Special Publication 800-20 method was adopted. As a result, it was found that all the random arrays which were generated by the chaotic motion could pass the validity criteria and some of them were even better than the quality of pseudo-random numbers generated by a computer. Through the experiments, it is demonstrated that chaotic pendulum can be used as an efficient mechanical facility in generating random numbers, and can be applied in teaching random motion to the students.

A review of human factors challenges of complex adaptive systems: discovering and understanding chaos in human performance.

PubMed

Karwowski, Waldemar

2012-12-01

In this paper, the author explores a need for a greater understanding of the true nature of human-system interactions from the perspective of the theory of complex adaptive systems, including the essence of complexity, emergent properties of system behavior, nonlinear systems dynamics, and deterministic chaos. Human performance, more often than not, constitutes complex adaptive phenomena with emergent properties that exhibit nonlinear dynamical (chaotic) behaviors. The complexity challenges in the design and management of contemporary work systems, including service systems, are explored. Examples of selected applications of the concepts of nonlinear dynamics to the study of human physical performance are provided. Understanding and applications of the concepts of theory of complex adaptive and dynamical systems should significantly improve the effectiveness of human-centered design efforts of a large system of systems. Performance of many contemporary work systems and environments may be sensitive to the initial conditions and may exhibit dynamic nonlinear properties and chaotic system behaviors. Human-centered design of emergent human-system interactions requires application of the theories of nonlinear dynamics and complex adaptive system. The success of future human-systems integration efforts requires the fusion of paradigms, knowledge, design principles, and methodologies of human factors and ergonomics with those of the science of complex adaptive systems as well as modern systems engineering.

Time series analyses of breathing patterns of lung cancer patients using nonlinear dynamical system theory.

PubMed

Tewatia, D K; Tolakanahalli, R P; Paliwal, B R; Tomé, W A

2011-04-07

The underlying requirements for successful implementation of any efficient tumour motion management strategy are regularity and reproducibility of a patient's breathing pattern. The physiological act of breathing is controlled by multiple nonlinear feedback and feed-forward couplings. It would therefore be appropriate to analyse the breathing pattern of lung cancer patients in the light of nonlinear dynamical system theory. The purpose of this paper is to analyse the one-dimensional respiratory time series of lung cancer patients based on nonlinear dynamics and delay coordinate state space embedding. It is very important to select a suitable pair of embedding dimension 'm' and time delay 'τ' when performing a state space reconstruction. Appropriate time delay and embedding dimension were obtained using well-established methods, namely mutual information and the false nearest neighbour method, respectively. Establishing stationarity and determinism in a given scalar time series is a prerequisite to demonstrating that the nonlinear dynamical system that gave rise to the scalar time series exhibits a sensitive dependence on initial conditions, i.e. is chaotic. Hence, once an appropriate state space embedding of the dynamical system has been reconstructed, we show that the time series of the nonlinear dynamical systems under study are both stationary and deterministic in nature. Once both criteria are established, we proceed to calculate the largest Lyapunov exponent (LLE), which is an invariant quantity under time delay embedding. The LLE for all 16 patients is positive, which along with stationarity and determinism establishes the fact that the time series of a lung cancer patient's breathing pattern is not random or irregular, but rather it is deterministic in nature albeit chaotic. These results indicate that chaotic characteristics exist in the respiratory waveform and techniques based on state space dynamics should be employed for tumour motion management.

Adaptive fuzzy dynamic surface control for the chaotic permanent magnet synchronous motor using Nussbaum gain

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

Luo, Shaohua

This paper is concerned with the problem of adaptive fuzzy dynamic surface control (DSC) for the permanent magnet synchronous motor (PMSM) system with chaotic behavior, disturbance and unknown control gain and parameters. Nussbaum gain is adopted to cope with the situation that the control gain is unknown. And the unknown items can be estimated by fuzzy logic system. The proposed controller guarantees that all the signals in the closed-loop system are bounded and the system output eventually converges to a small neighborhood of the desired reference signal. Finally, the numerical simulations indicate that the proposed scheme can suppress the chaosmore » of PMSM and show the effectiveness and robustness of the proposed method.« less

Adaptive fuzzy dynamic surface control for the chaotic permanent magnet synchronous motor using Nussbaum gain.

PubMed

Luo, Shaohua

2014-09-01

This paper is concerned with the problem of adaptive fuzzy dynamic surface control (DSC) for the permanent magnet synchronous motor (PMSM) system with chaotic behavior, disturbance and unknown control gain and parameters. Nussbaum gain is adopted to cope with the situation that the control gain is unknown. And the unknown items can be estimated by fuzzy logic system. The proposed controller guarantees that all the signals in the closed-loop system are bounded and the system output eventually converges to a small neighborhood of the desired reference signal. Finally, the numerical simulations indicate that the proposed scheme can suppress the chaos of PMSM and show the effectiveness and robustness of the proposed method.

The dynamical structure of the MEO region: long-term stability, chaos, and transport

NASA Astrophysics Data System (ADS)

Daquin, Jérôme; Rosengren, Aaron J.; Alessi, Elisa Maria; Deleflie, Florent; Valsecchi, Giovanni B.; Rossi, Alessandro

2016-04-01

It has long been suspected that the Global Navigation Satellite Systems exist in a background of complex resonances and chaotic motion; yet, the precise dynamical character of these phenomena remains elusive. Recent studies have shown that the occurrence and nature of the resonances driving these dynamics depend chiefly on the frequencies of nodal and apsidal precession and the rate of regression of the Moon's nodes. Woven throughout the inclination and eccentricity phase space is an exceedingly complicated web-like structure of lunisolar secular resonances, which become particularly dense near the inclinations of the navigation satellite orbits. A clear picture of the physical significance of these resonances is of considerable practical interest for the design of disposal strategies for the four constellations. Here we present analytical and semi-analytical models that accurately reflect the true nature of the resonant interactions, and trace the topological organization of the manifolds on which the chaotic motions take place. We present an atlas of FLI stability maps, showing the extent of the chaotic regions of the phase space, computed through a hierarchy of more realistic, and more complicated, models, and compare the chaotic zones in these charts with the analytical estimation of the width of the chaotic layers from the heuristic Chirikov resonance-overlap criterion. As the semi-major axis of the satellite is receding, we observe a transition from stable Nekhoroshev-like structures at three Earth radii, where regular orbits dominate, to a Chirikov regime where resonances overlap at five Earth radii. From a numerical estimation of the Lyapunov times, we find that many of the inclined, nearly circular orbits of the navigation satellites are strongly chaotic and that their dynamics are unpredictable on decadal timescales.

Fractal analysis of GPS time series for early detection of disastrous seismic events

NASA Astrophysics Data System (ADS)

Filatov, Denis M.; Lyubushin, Alexey A.

2017-03-01

A new method of fractal analysis of time series for estimating the chaoticity of behaviour of open stochastic dynamical systems is developed. The method is a modification of the conventional detrended fluctuation analysis (DFA) technique. We start from analysing both methods from the physical point of view and demonstrate the difference between them which results in a higher accuracy of the new method compared to the conventional DFA. Then, applying the developed method to estimate the measure of chaoticity of a real dynamical system - the Earth's crust, we reveal that the latter exhibits two distinct mechanisms of transition to a critical state: while the first mechanism has already been known due to numerous studies of other dynamical systems, the second one is new and has not previously been described. Using GPS time series, we demonstrate efficiency of the developed method in identification of critical states of the Earth's crust. Finally we employ the method to solve a practically important task: we show how the developed measure of chaoticity can be used for early detection of disastrous seismic events and provide a detailed discussion of the numerical results, which are shown to be consistent with outcomes of other researches on the topic.

Generalized correlation integral vectors: A distance concept for chaotic dynamical systems

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

Haario, Heikki, E-mail: heikki.haario@lut.fi; Kalachev, Leonid, E-mail: KalachevL@mso.umt.edu; Hakkarainen, Janne

2015-06-15

Several concepts of fractal dimension have been developed to characterise properties of attractors of chaotic dynamical systems. Numerical approximations of them must be calculated by finite samples of simulated trajectories. In principle, the quantities should not depend on the choice of the trajectory, as long as it provides properly distributed samples of the underlying attractor. In practice, however, the trajectories are sensitive with respect to varying initial values, small changes of the model parameters, to the choice of a solver, numeric tolerances, etc. The purpose of this paper is to present a statistically sound approach to quantify this variability. Wemore » modify the concept of correlation integral to produce a vector that summarises the variability at all selected scales. The distribution of this stochastic vector can be estimated, and it provides a statistical distance concept between trajectories. Here, we demonstrate the use of the distance for the purpose of estimating model parameters of a chaotic dynamic model. The methodology is illustrated using computational examples for the Lorenz 63 and Lorenz 95 systems, together with a framework for Markov chain Monte Carlo sampling to produce posterior distributions of model parameters.« less

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.

The geometry of chaotic dynamics — a complex network perspective

NASA Astrophysics Data System (ADS)

Donner, R. V.; Heitzig, J.; Donges, J. F.; Zou, Y.; Marwan, N.; Kurths, J.

2011-12-01

Recently, several complex network approaches to time series analysis have been developed and applied to study a wide range of model systems as well as real-world data, e.g., geophysical or financial time series. Among these techniques, recurrence-based concepts and prominently ɛ-recurrence networks, most faithfully represent the geometrical fine structure of the attractors underlying chaotic (and less interestingly non-chaotic) time series. In this paper we demonstrate that the well known graph theoretical properties local clustering coefficient and global (network) transitivity can meaningfully be exploited to define two new local and two new global measures of dimension in phase space: local upper and lower clustering dimension as well as global upper and lower transitivity dimension. Rigorous analytical as well as numerical results for self-similar sets and simple chaotic model systems suggest that these measures are well-behaved in most non-pathological situations and that they can be estimated reasonably well using ɛ-recurrence networks constructed from relatively short time series. Moreover, we study the relationship between clustering and transitivity dimensions on the one hand, and traditional measures like pointwise dimension or local Lyapunov dimension on the other hand. We also provide further evidence that the local clustering coefficients, or equivalently the local clustering dimensions, are useful for identifying unstable periodic orbits and other dynamically invariant objects from time series. Our results demonstrate that ɛ-recurrence networks exhibit an important link between dynamical systems and graph theory.

Introduction: Second Language Development as a Dynamic Process

ERIC Educational Resources Information Center

De Bot, Kees

2008-01-01

In this contribution, some of the basic characteristics of complex adaptive systems, collectively labeled Dynamic Systems Theory (DST), are discussed. Such systems are self-organizing, dependent on initial conditions, sometimes chaotic, and they show emergent properties. The focus in DST is on development over time. Language is seen as a dynamic…

Security scheme in IMDD-OFDM-PON system with the chaotic pilot interval and scrambling

NASA Astrophysics Data System (ADS)

Chen, Qianghua; Bi, Meihua; Fu, Xiaosong; Lu, Yang; Zeng, Ran; Yang, Guowei; Yang, Xuelin; Xiao, Shilin

2018-01-01

In this paper, a random chaotic pilot interval and permutations scheme without any requirement of redundant sideband information is firstly proposed for the physical layer security-enhanced intensity modulation direct detection orthogonal frequency division multiplexing passive optical network (IMDD-OFDM-PON) system. With the help of the position feature of inserting the pilot, a simple logistic chaos map is used to generate the random pilot interval and scramble the chaotic subcarrier allocation of each column pilot data for improving the physical layer confidentiality. Due to the dynamic chaotic permutations of pilot data, the enhanced key space of ∼103303 is achieved in OFDM-PON. Moreover, the transmission experiment of 10-Gb/s 16-QAM encrypted OFDM data is successfully demonstrated over 20-km single-mode fiber, which indicates that the proposed scheme not only improves the system security, but also can achieve the same performance as in the common IMDD-OFDM-PON system without encryption scheme.

Monte Carlo Sampling in Fractal Landscapes

NASA Astrophysics Data System (ADS)

Leitão, Jorge C.; Lopes, J. M. Viana Parente; Altmann, Eduardo G.

2013-05-01

We design a random walk to explore fractal landscapes such as those describing chaotic transients in dynamical systems. We show that the random walk moves efficiently only when its step length depends on the height of the landscape via the largest Lyapunov exponent of the chaotic system. We propose a generalization of the Wang-Landau algorithm which constructs not only the density of states (transient time distribution) but also the correct step length. As a result, we obtain a flat-histogram Monte Carlo method which samples fractal landscapes in polynomial time, a dramatic improvement over the exponential scaling of traditional uniform-sampling methods. Our results are not limited by the dimensionality of the landscape and are confirmed numerically in chaotic systems with up to 30 dimensions.

Topics in quantum chaos

NASA Astrophysics Data System (ADS)

Jordan, Andrew Noble

2002-09-01

In this dissertation, we study the quantum mechanics of classically chaotic dynamical systems. We begin by considering the decoherence effects a quantum chaotic system has on a simple quantum few state system. Typical time evolution of a quantum system whose classical limit is chaotic generates structures in phase space whose size is much smaller than Planck's constant. A naive application of Heisenberg's uncertainty principle indicates that these structures are not physically relevant. However, if we take the quantum chaotic system in question to be an environment which interacts with a simple two state quantum system (qubit), we show that these small phase-space structures cause the qubit to generically lose quantum coherence if and only if the environment has many degrees of freedom, such as a dilute gas. This implies that many-body environments may be crucial for the phenomenon of quantum decoherence. Next, we turn to an analysis of statistical properties of time correlation functions and matrix elements of quantum chaotic systems. A semiclassical evaluation of matrix elements of an operator indicates that the dominant contribution will be related to a classical time correlation function over the energy surface. For a highly chaotic class of dynamics, these correlation functions may be decomposed into sums of Ruelle resonances, which control exponential decay to the ergodic distribution. The theory is illustrated both numerically and theoretically on the Baker map. For this system, we are able to isolate individual Ruelle modes. We further consider dynamical systems whose approach to ergodicity is given by a power law rather than an exponential in time. We propose a billiard with diffusive boundary conditions, whose classical solution may be calculated analytically. We go on to compare the exact solution with an approximation scheme, as well calculate asympotic corrections. Quantum spectral statistics are calculated assuming the validity of the Again, Altshuler and Andreev ansatz. We find singular behavior of the two point spectral correlator in the limit of small spacing. Finally, we analyse the effect that slow decay to ergodicity has on the structure of the quantum propagator, as well as wavefunction localization. We introduce a statistical quantum description of systems that are composed of both an orderly region and a random region. By averaging over the random region only, we find that measures of localization in momentum space semiclassically diverge with the dimension of the Hilbert space. We illustrate this numerically with quantum maps and suggest various other systems where this behavior should be important.

Synchronization of mobile chaotic oscillator networks

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

Fujiwara, Naoya, E-mail: fujiwara@csis.u-tokyo.ac.jp; Kurths, Jürgen; Díaz-Guilera, Albert

We study synchronization of systems in which agents holding chaotic oscillators move in a two-dimensional plane and interact with nearby ones forming a time dependent network. Due to the uncertainty in observing other agents' states, we assume that the interaction contains a certain amount of noise that turns out to be relevant for chaotic dynamics. We find that a synchronization transition takes place by changing a control parameter. But this transition depends on the relative dynamic scale of motion and interaction. When the topology change is slow, we observe an intermittent switching between laminar and burst states close to themore » transition due to small noise. This novel type of synchronization transition and intermittency can happen even when complete synchronization is linearly stable in the absence of noise. We show that the linear stability of the synchronized state is not a sufficient condition for its stability due to strong fluctuations of the transverse Lyapunov exponent associated with a slow network topology change. Since this effect can be observed within the linearized dynamics, we can expect such an effect in the temporal networks with noisy chaotic oscillators, irrespective of the details of the oscillator dynamics. When the topology change is fast, a linearized approximation describes well the dynamics towards synchrony. These results imply that the fluctuations of the finite-time transverse Lyapunov exponent should also be taken into account to estimate synchronization of the mobile contact networks.« less

Dynamic Regimes of El Niño Southern Oscillation and Influenza Pandemic Timing

PubMed Central

Oluwole, Olusegun Steven Ayodele

2017-01-01

El Niño southern oscillation (ENSO) dynamics has been shown to drive seasonal influenza dynamics. Severe seasonal influenza epidemics and the 2009–2010 pandemic were coincident with chaotic regime of ENSO dynamics. ENSO dynamics from 1876 to 2016 were characterized to determine if influenza pandemics are coupled to chaotic regimes. Time-varying spectra of southern oscillation index (SOI) and sea surface temperature (SST) were compared. SOI and SST were decomposed to components using the algorithm of noise-assisted multivariate empirical mode decomposition. The components were Hilbert transformed to generate instantaneous amplitudes and phases. The trajectories and attractors of components were characterized in polar coordinates and state space. Influenza pandemics were mapped to dynamic regimes of SOI and SST joint recurrence of annual components. State space geometry of El Niños lagged by influenza pandemics were characterized and compared with other El Niños. Timescales of SOI and SST components ranged from sub-annual to multidecadal. The trajectories of SOI and SST components and the joint recurrence of annual components were dissipative toward chaotic attractors. Periodic, quasi-periodic, and chaotic regimes were present in the recurrence of trajectories, but chaos–chaos transitions dominated. Influenza pandemics occurred during chaotic regimes of significantly low transitivity dimension (p < 0.0001). El Niños lagged by influenza pandemics had distinct state space geometry (p < 0.0001). Chaotic dynamics explains the aperiodic timing, and varying duration and strength of El Niños. Coupling of all influenza pandemics of the past 140 years to chaotic regimes of low transitivity indicate that ENSO dynamics drives influenza pandemic dynamics. Forecasts models from ENSO dynamics should compliment surveillance for novel influenza viruses. PMID:29218303

On the Chaotic Vibrations of Electrostatically Actuated Arch Micro/Nano Resonators: A Parametric Study

NASA Astrophysics Data System (ADS)

Tajaddodianfar, Farid; Hairi Yazdi, Mohammad Reza; Pishkenari, Hossein Nejat

Motivated by specific applications, electrostatically actuated bistable arch shaped micro-nano resonators have attracted growing attention in the research community in recent years. Nevertheless, some issues relating to their nonlinear dynamics, including the possibility of chaos, are still not well known. In this paper, we investigate the chaotic vibrations of a bistable resonator comprised of a double clamped initially curved microbeam under combined harmonic AC and static DC distributed electrostatic actuation. A reduced order equation obtained by the application of the Galerkin method to the nonlinear partial differential equation of motion, given in the framework of Euler-Bernoulli beam theory, is used for the investigation in this paper. We numerically integrate the obtained equation to study the chaotic vibrations of the proposed system. Moreover, we investigate the effects of various parameters including the arch curvature, the actuation parameters and the quality factor of the resonator, which are effective in the formation of both static and dynamic behaviors of the system. Using appropriate numerical tools, including Poincaré maps, bifurcation diagrams, Fourier spectrum and Lyapunov exponents we scrutinize the effects of various parameters on the formation of chaotic regions in the parametric space of the resonator. Results of this work provide better insight into the problem of nonlinear dynamics of the investigated family of bistable micro/nano resonators, and facilitate the design of arch resonators for applications such as filters.

Nonlinear modeling of chaotic time series: Theory and applications

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

Casdagli, M.; Eubank, S.; Farmer, J.D.

1990-01-01

We review recent developments in the modeling and prediction of nonlinear time series. In some cases apparent randomness in time series may be due to chaotic behavior of a nonlinear but deterministic system. In such cases it is possible to exploit the determinism to make short term forecasts that are much more accurate than one could make from a linear stochastic model. This is done by first reconstructing a state space, and then using nonlinear function approximation methods to create a dynamical model. Nonlinear models are valuable not only as short term forecasters, but also as diagnostic tools for identifyingmore » and quantifying low-dimensional chaotic behavior. During the past few years methods for nonlinear modeling have developed rapidly, and have already led to several applications where nonlinear models motivated by chaotic dynamics provide superior predictions to linear models. These applications include prediction of fluid flows, sunspots, mechanical vibrations, ice ages, measles epidemics and human speech. 162 refs., 13 figs.« less

Noise induced stabilization of chaotic free-running laser diode

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

Virte, Martin, E-mail: mvirte@b-phot.org

In this paper, we investigate theoretically the stabilization of a free-running vertical-cavity surface-emitting laser exhibiting polarization chaos dynamics. We report the existence of a boundary isolating the chaotic attractor on one side and a steady-state on the other side and identify the unstable periodic orbit playing the role of separatrix. In addition, we highlight a small range of parameters where the chaotic attractor passes through this boundary, and therefore where chaos only appears as a transient behaviour. Then, including the effect of spontaneous emission noise in the laser, we demonstrate that, for realistic levels of noise, the system is systematicallymore » pushed over the separating solution. As a result, we show that the chaotic dynamics cannot be sustained unless the steady-state on the other side of the separatrix becomes unstable. Finally, we link the stability of this steady-state to a small value of the birefringence in the laser cavity and discuss the significance of this result on future experimental work.« less

Granular chaos and mixing: Whirled in a grain of sand.

PubMed

Shinbrot, Troy

2015-09-01

In this paper, we overview examples of chaos in granular flows. We begin by reviewing several remarkable behaviors that have intrigued researchers over the past few decades, and we then focus on three areas in which chaos plays an intrinsic role in granular behavior. First, we discuss pattern formation in vibrated beds, which we show is a direct result of chaotic scattering combined with dynamical dissipation. Next, we consider stick-slip motion, which involves chaotic scattering on the micro-scale, and which results in complex and as yet unexplained peculiarities on the macro-scale. Finally, we examine granular mixing, which we show combines micro-scale chaotic scattering and macro-scale stick-slip motion into behaviors that are well described by dynamical systems tools, such as iterative mappings.

Granular chaos and mixing: Whirled in a grain of sand

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

Shinbrot, Troy, E-mail: shinbrot@rutgers.edu

2015-09-15

In this paper, we overview examples of chaos in granular flows. We begin by reviewing several remarkable behaviors that have intrigued researchers over the past few decades, and we then focus on three areas in which chaos plays an intrinsic role in granular behavior. First, we discuss pattern formation in vibrated beds, which we show is a direct result of chaotic scattering combined with dynamical dissipation. Next, we consider stick-slip motion, which involves chaotic scattering on the micro-scale, and which results in complex and as yet unexplained peculiarities on the macro-scale. Finally, we examine granular mixing, which we show combinesmore » micro-scale chaotic scattering and macro-scale stick-slip motion into behaviors that are well described by dynamical systems tools, such as iterative mappings.« less

Chaos and Correlated Avalanches in Excitatory Neural Networks with Synaptic Plasticity

NASA Astrophysics Data System (ADS)

Pittorino, Fabrizio; Ibáñez-Berganza, Miguel; di Volo, Matteo; Vezzani, Alessandro; Burioni, Raffaella

2017-03-01

A collective chaotic phase with power law scaling of activity events is observed in a disordered mean field network of purely excitatory leaky integrate-and-fire neurons with short-term synaptic plasticity. The dynamical phase diagram exhibits two transitions from quasisynchronous and asynchronous regimes to the nontrivial, collective, bursty regime with avalanches. In the homogeneous case without disorder, the system synchronizes and the bursty behavior is reflected into a period doubling transition to chaos for a two dimensional discrete map. Numerical simulations show that the bursty chaotic phase with avalanches exhibits a spontaneous emergence of persistent time correlations and enhanced Kolmogorov complexity. Our analysis reveals a mechanism for the generation of irregular avalanches that emerges from the combination of disorder and deterministic underlying chaotic dynamics.

Chaos of radiative heat-loss-induced flame front instability.

PubMed

Kinugawa, Hikaru; Ueda, Kazuhiro; Gotoda, Hiroshi

2016-03-01

We are intensively studying the chaos via the period-doubling bifurcation cascade in radiative heat-loss-induced flame front instability by analytical methods based on dynamical systems theory and complex networks. Significant changes in flame front dynamics in the chaotic region, which cannot be seen in the bifurcation diagrams, were successfully extracted from recurrence quantification analysis and nonlinear forecasting and from the network entropy. The temporal dynamics of the fuel concentration in the well-developed chaotic region is much more complicated than that of the flame front temperature. It exhibits self-affinity as a result of the scale-free structure in the constructed visibility graph.

Reconstruction of dynamical systems from resampled point processes produced by neuron models

NASA Astrophysics Data System (ADS)

Pavlova, Olga N.; Pavlov, Alexey N.

2018-04-01

Characterization of dynamical features of chaotic oscillations from point processes is based on embedding theorems for non-uniformly sampled signals such as the sequences of interspike intervals (ISIs). This theoretical background confirms the ability of attractor reconstruction from ISIs generated by chaotically driven neuron models. The quality of such reconstruction depends on the available length of the analyzed dataset. We discuss how data resampling improves the reconstruction for short amount of data and show that this effect is observed for different types of mechanisms for spike generation.

Features of Chaotic Transients in Excitable Media Governed by Spiral and Scroll Waves

NASA Astrophysics Data System (ADS)

Lilienkamp, Thomas; Christoph, Jan; Parlitz, Ulrich

2017-08-01

In excitable media, chaotic dynamics governed by spiral or scroll waves is often not persistent but transient. Using extensive simulations employing different mathematical models we identify a specific type-II supertransient by an exponential increase of transient lifetimes with the system size in 2D and an investigation of the dynamics (number and lifetime of spiral waves, Kaplan-Yorke dimension). In 3D, simulations exhibit an increase of transient lifetimes and filament lengths only above a critical thickness. Finally, potential implications for understanding cardiac arrhythmias are discussed.

Transition from Exponential to Power Law Income Distributions in a Chaotic Market

NASA Astrophysics Data System (ADS)

Pellicer-Lostao, Carmen; Lopez-Ruiz, Ricardo

Economy is demanding new models, able to understand and predict the evolution of markets. To this respect, Econophysics offers models of markets as complex systems, that try to comprehend macro-, system-wide states of the economy from the interaction of many agents at micro-level. One of these models is the gas-like model for trading markets. This tries to predict money distributions in closed economies and quite simply, obtains the ones observed in real economies. However, it reveals technical hitches to explain the power law distribution, observed in individuals with high incomes. In this work, nonlinear dynamics is introduced in the gas-like model in an effort to overcomes these flaws. A particular chaotic dynamics is used to break the pairing symmetry of agents (i, j) ⇔ (j, i). The results demonstrate that a "chaotic gas-like model" can reproduce the Exponential and Power law distributions observed in real economies. Moreover, it controls the transition between them. This may give some insight of the micro-level causes that originate unfair distributions of money in a global society. Ultimately, the chaotic model makes obvious the inherent instability of asymmetric scenarios, where sinks of wealth appear and doom the market to extreme inequality.

Global Optimal Trajectory in Chaos and NP-Hardness

NASA Astrophysics Data System (ADS)

Latorre, Vittorio; Gao, David Yang

This paper presents an unconventional theory and method for solving general nonlinear dynamical systems. Instead of the direct iterative methods, the discretized nonlinear system is first formulated as a global optimization problem via the least squares method. A newly developed canonical duality theory shows that this nonconvex minimization problem can be solved deterministically in polynomial time if a global optimality condition is satisfied. The so-called pseudo-chaos produced by linear iterative methods are mainly due to the intrinsic numerical error accumulations. Otherwise, the global optimization problem could be NP-hard and the nonlinear system can be really chaotic. A conjecture is proposed, which reveals the connection between chaos in nonlinear dynamics and NP-hardness in computer science. The methodology and the conjecture are verified by applications to the well-known logistic equation, a forced memristive circuit and the Lorenz system. Computational results show that the canonical duality theory can be used to identify chaotic systems and to obtain realistic global optimal solutions in nonlinear dynamical systems. The method and results presented in this paper should bring some new insights into nonlinear dynamical systems and NP-hardness in computational complexity theory.

Exploiting chaos for applications.

PubMed

Ditto, William L; Sinha, Sudeshna

2015-09-01

We discuss how understanding the nature of chaotic dynamics allows us to control these systems. A controlled chaotic system can then serve as a versatile pattern generator that can be used for a range of application. Specifically, we will discuss the application of controlled chaos to the design of novel computational paradigms. Thus, we present an illustrative research arc, starting with ideas of control, based on the general understanding of chaos, moving over to applications that influence the course of building better devices.

Adjoint sensitivity analysis of chaotic dynamical systems with non-intrusive least squares shadowing

NASA Astrophysics Data System (ADS)

Blonigan, Patrick J.

2017-11-01

This paper presents a discrete adjoint version of the recently developed non-intrusive least squares shadowing (NILSS) algorithm, which circumvents the instability that conventional adjoint methods encounter for chaotic systems. The NILSS approach involves solving a smaller minimization problem than other shadowing approaches and can be implemented with only minor modifications to preexisting tangent and adjoint solvers. Adjoint NILSS is demonstrated on a small chaotic ODE, a one-dimensional scalar PDE, and a direct numerical simulation (DNS) of the minimal flow unit, a turbulent channel flow on a small spatial domain. This is the first application of an adjoint shadowing-based algorithm to a three-dimensional turbulent flow.

Menstruation, perimenopause, and chaos theory.

PubMed

Derry, Paula S; Derry, Gregory N

2012-01-01

This article argues that menstruation, including the transition to menopause, results from a specific kind of complex system, namely, one that is nonlinear, dynamical, and chaotic. A complexity-based perspective changes how we think about and research menstruation-related health problems and positive health. Chaotic systems are deterministic but not predictable, characterized by sensitivity to initial conditions and strange attractors. Chaos theory provides a coherent framework that qualitatively accounts for puzzling results from perimenopause research. It directs attention to variability within and between women, adaptation, lifespan development, and the need for complex explanations of disease. Whether the menstrual cycle is chaotic can be empirically tested, and a summary of our research on 20- to 40-year-old women is provided.

Chaotic Traversal (CHAT): Very Large Graphs Traversal Using Chaotic Dynamics

NASA Astrophysics Data System (ADS)

Changaival, Boonyarit; Rosalie, Martin; Danoy, Grégoire; Lavangnananda, Kittichai; Bouvry, Pascal

2017-12-01

Graph Traversal algorithms can find their applications in various fields such as routing problems, natural language processing or even database querying. The exploration can be considered as a first stepping stone into knowledge extraction from the graph which is now a popular topic. Classical solutions such as Breadth First Search (BFS) and Depth First Search (DFS) require huge amounts of memory for exploring very large graphs. In this research, we present a novel memoryless graph traversal algorithm, Chaotic Traversal (CHAT) which integrates chaotic dynamics to traverse large unknown graphs via the Lozi map and the Rössler system. To compare various dynamics effects on our algorithm, we present an original way to perform the exploration of a parameter space using a bifurcation diagram with respect to the topological structure of attractors. The resulting algorithm is an efficient and nonresource demanding algorithm, and is therefore very suitable for partial traversal of very large and/or unknown environment graphs. CHAT performance using Lozi map is proven superior than the, commonly known, Random Walk, in terms of number of nodes visited (coverage percentage) and computation time where the environment is unknown and memory usage is restricted.

Chaotic Zones around Rotating Small Bodies

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

Lages, José; Shevchenko, Ivan I.; Shepelyansky, Dima L., E-mail: jose.lages@utinam.cnrs.fr

Small bodies of the solar system, like asteroids, trans-Neptunian objects, cometary nuclei, and planetary satellites, with diameters smaller than 1000 km usually have irregular shapes, often resembling dumb-bells or contact binaries. The spinning of such a gravitating dumb-bell creates around it a zone of chaotic orbits. We determine its extent analytically and numerically. We find that the chaotic zone swells significantly if the rotation rate is decreased; in particular, the zone swells more than twice if the rotation rate is decreased 10 times with respect to the “centrifugal breakup” threshold. We illustrate the properties of the chaotic orbital zones in examples ofmore » the global orbital dynamics about asteroid 243 Ida (which has a moon, Dactyl, orbiting near the edge of the chaotic zone) and asteroid 25143 Itokawa.« less

Synthesis of a fully-integrated digital signal source for communications from chaotic dynamics-based oscillations

NASA Astrophysics Data System (ADS)

Glenn, Chance Michael, Sr.

This work is the conceptualization, derivation, analysis, and fabrication of a fully practical digital signal source designed from a chaotic oscillator. In it we show how a simple electronic circuit based upon the Colpitts oscillator, can be made to produce highly complex signals capable of carrying digital information. We show a direct relationship between the continuous-time chaotic oscillations produced by the circuit and the logistic map, which is discrete-time, one-dimensional map that is a fundamental paradigm for the study of chaotic systems. We demonstrate the direct encoding of binary information into the oscillations of the chaotic circuit. We demonstrate a new concept in power amplification, called syncrodyne amplification , which uses fundamental properties of chaotic oscillators to provide high-efficiency, high gain amplification of standard communication waveforms as well as typical chaotic oscillations. We show modeling results of this system providing nearly 60-dB power gain and 80% PAE for communications waveforms conforming to GMSK modulation. Finally we show results from a fabricated syncrodyne amplifier circuit operating at 2 MHz, providing over 40-dB power gain and 72% PAE, and propose design criteria for an 824--850 MHz circuit utilizing heterojunction bipolar transistors (HBTs), providing the basis for microwave frequency realization.

Nonlinear Dynamics, Chaotic and Complex Systems

NASA Astrophysics Data System (ADS)

Infeld, E.; Zelazny, R.; Galkowski, A.

2011-04-01

Part I. Dynamic Systems Bifurcation Theory and Chaos: 1. Chaos in random dynamical systems V. M. Gunldach; 2. Controlling chaos using embedded unstable periodic orbits: the problem of optimal periodic orbits B. R. Hunt and E. Ott; 3. Chaotic tracer dynamics in open hydrodynamical flows G. Karolyi, A. Pentek, T. Tel and Z. Toroczkai; 4. Homoclinic chaos L. P. Shilnikov; Part II. Spatially Extended Systems: 5. Hydrodynamics of relativistic probability flows I. Bialynicki-Birula; 6. Waves in ionic reaction-diffusion-migration systems P. Hasal, V. Nevoral, I. Schreiber, H. Sevcikova, D. Snita, and M. Marek; 7. Anomalous scaling in turbulence: a field theoretical approach V. Lvov and I. Procaccia; 8. Abelian sandpile cellular automata M. Markosova; 9. Transport in an incompletely chaotic magnetic field F. Spineanu; Part III. Dynamical Chaos Quantum Physics and Foundations Of Statistical Mechanics: 10. Non-equilibrium statistical mechanics and ergodic theory L. A. Bunimovich; 11. Pseudochaos in statistical physics B. Chirikov; 12. Foundations of non-equilibrium statistical mechanics J. P. Dougherty; 13. Thermomechanical particle simulations W. G. Hoover, H. A. Posch, C. H. Dellago, O. Kum, C. G. Hoover, A. J. De Groot and B. L. Holian; 14. Quantum dynamics on a Markov background and irreversibility B. Pavlov; 15. Time chaos and the laws of nature I. Prigogine and D. J. Driebe; 16. Evolutionary Q and cognitive systems: dynamic entropies and predictability of evolutionary processes W. Ebeling; 17. Spatiotemporal chaos information processing in neural networks H. Szu; 18. Phase transitions and learning in neural networks C. Van den Broeck; 19. Synthesis of chaos A. Vanecek and S. Celikovsky; 20. Computational complexity of continuous problems H. Wozniakowski; Part IV. Complex Systems As An Interface Between Natural Sciences and Environmental Social and Economic Sciences: 21. Stochastic differential geometry in finance studies V. G. Makhankov; Part V. Conference Banquet Speech: Where will the future go? M. J. Feigenbaum.

Chaos and crises in a model for cooperative hunting: a symbolic dynamics approach.

PubMed

Duarte, Jorge; Januário, Cristina; Martins, Nuno; Sardanyés, Josep

2009-12-01

In this work we investigate the population dynamics of cooperative hunting extending the McCann and Yodzis model for a three-species food chain system with a predator, a prey, and a resource species. The new model considers that a given fraction sigma of predators cooperates in prey's hunting, while the rest of the population 1-sigma hunts without cooperation. We use the theory of symbolic dynamics to study the topological entropy and the parameter space ordering of the kneading sequences associated with one-dimensional maps that reproduce significant aspects of the dynamics of the species under several degrees of cooperative hunting. Our model also allows us to investigate the so-called deterministic extinction via chaotic crisis and transient chaos in the framework of cooperative hunting. The symbolic sequences allow us to identify a critical boundary in the parameter spaces (K,C(0)) and (K,sigma) which separates two scenarios: (i) all-species coexistence and (ii) predator's extinction via chaotic crisis. We show that the crisis value of the carrying capacity K(c) decreases at increasing sigma, indicating that predator's populations with high degree of cooperative hunting are more sensitive to the chaotic crises. We also show that the control method of Dhamala and Lai [Phys. Rev. E 59, 1646 (1999)] can sustain the chaotic behavior after the crisis for systems with cooperative hunting. We finally analyze and quantify the inner structure of the target regions obtained with this control method for wider parameter values beyond the crisis, showing a power law dependence of the extinction transients on such critical parameters.

Improving the quality of extracting dynamics from interspike intervals via a resampling approach

NASA Astrophysics Data System (ADS)

Pavlova, O. N.; Pavlov, A. N.

2018-04-01

We address the problem of improving the quality of characterizing chaotic dynamics based on point processes produced by different types of neuron models. Despite the presence of embedding theorems for non-uniformly sampled dynamical systems, the case of short data analysis requires additional attention because the selection of algorithmic parameters may have an essential influence on estimated measures. We consider how the preliminary processing of interspike intervals (ISIs) can increase the precision of computing the largest Lyapunov exponent (LE). We report general features of characterizing chaotic dynamics from point processes and show that independently of the selected mechanism for spike generation, the performed preprocessing reduces computation errors when dealing with a limited amount of data.

On the hypothesis that quantum mechanism manifests classical mechanics: Numerical approach to the correspondence in search of quantum chaos

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

Lee, Sang-Bong

1993-09-01

Quantum manifestation of classical chaos has been one of the extensively studied subjects for more than a decade. Yet clear understanding of its nature still remains to be an open question partly due to the lack of a canonical definition of quantum chaos. The classical definition seems to be unsuitable in quantum mechanics partly because of the Heisenberg quantum uncertainty. In this regard, quantum chaos is somewhat misleading and needs to be clarified at the very fundamental level of physics. Since it is well known that quantum mechanics is more fundamental than classical mechanics, the quantum description of classically chaoticmore » nature should be attainable in the limit of large quantum numbers. The focus of my research, therefore, lies on the correspondence principle for classically chaotic systems. The chaotic damped driven pendulum is mainly studied numerically using the split operator method that solves the time-dependent Schroedinger equation. For classically dissipative chaotic systems in which (multi)fractal strange attractors often emerge, several quantum dissipative mechanisms are also considered. For instance, Hoover`s and Kubo-Fox-Keizer`s approaches are studied with some computational analyses. But the notion of complex energy with non-Hermiticity is extensively applied. Moreover, the Wigner and Husimi distribution functions are examined with an equivalent classical distribution in phase-space, and dynamical properties of the wave packet in configuration and momentum spaces are also explored. The results indicate that quantum dynamics embraces classical dynamics although the classicalquantum correspondence fails to be observed in the classically chaotic regime. Even in the semi-classical limits, classically chaotic phenomena would eventually be suppressed by the quantum uncertainty.« less

Mathematical Models to Determine Stable Behavior of Complex Systems

NASA Astrophysics Data System (ADS)

Sumin, V. I.; Dushkin, A. V.; Smolentseva, T. E.

2018-05-01

The paper analyzes a possibility to predict functioning of a complex dynamic system with a significant amount of circulating information and a large number of random factors impacting its functioning. Functioning of the complex dynamic system is described as a chaotic state, self-organized criticality and bifurcation. This problem may be resolved by modeling such systems as dynamic ones, without applying stochastic models and taking into account strange attractors.

Chaotic oscillation and random-number generation based on nanoscale optical-energy transfer.

PubMed

Naruse, Makoto; Kim, Song-Ju; Aono, Masashi; Hori, Hirokazu; Ohtsu, Motoichi

2014-08-12

By using nanoscale energy-transfer dynamics and density matrix formalism, we demonstrate theoretically and numerically that chaotic oscillation and random-number generation occur in a nanoscale system. The physical system consists of a pair of quantum dots (QDs), with one QD smaller than the other, between which energy transfers via optical near-field interactions. When the system is pumped by continuous-wave radiation and incorporates a timing delay between two energy transfers within the system, it emits optical pulses. We refer to such QD pairs as nano-optical pulsers (NOPs). Irradiating an NOP with external periodic optical pulses causes the oscillating frequency of the NOP to synchronize with the external stimulus. We find that chaotic oscillation occurs in the NOP population when they are connected by an external time delay. Moreover, by evaluating the time-domain signals by statistical-test suites, we confirm that the signals are sufficiently random to qualify the system as a random-number generator (RNG). This study reveals that even relatively simple nanodevices that interact locally with each other through optical energy transfer at scales far below the wavelength of irradiating light can exhibit complex oscillatory dynamics. These findings are significant for applications such as ultrasmall RNGs.

Various Attractors, Coexisting Attractors and Antimonotonicity in a Simple Fourth-Order Memristive Twin-T Oscillator

NASA Astrophysics Data System (ADS)

Zhou, Ling; Wang, Chunhua; Zhang, Xin; Yao, Wei

By replacing the resistor in a Twin-T network with a generalized flux-controlled memristor, this paper proposes a simple fourth-order memristive Twin-T oscillator. Rich dynamical behaviors can be observed in the dynamical system. The most striking feature is that this system has various periodic orbits and various chaotic attractors generated by adjusting parameter b. At the same time, coexisting attractors and antimonotonicity are also detected (especially, two full Feigenbaum remerging trees in series are observed in such autonomous chaotic systems). Their dynamical features are analyzed by phase portraits, Lyapunov exponents, bifurcation diagrams and basin of attraction. Moreover, hardware experiments on a breadboard are carried out. Experimental measurements are in accordance with the simulation results. Finally, a multi-channel random bit generator is designed for encryption applications. Numerical results illustrate the usefulness of the random bit generator.

Complex dynamics of a delayed discrete neural network of two nonidentical neurons

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

Chen, Yuanlong; Huang, Tingwen; Huang, Yu, E-mail: stshyu@mail.sysu.edu.cn

2014-03-15

In this paper, we discover that a delayed discrete Hopfield neural network of two nonidentical neurons with self-connections and no self-connections can demonstrate chaotic behaviors. To this end, we first transform the model, by a novel way, into an equivalent system which has some interesting properties. Then, we identify the chaotic invariant set for this system and show that the dynamics of this system within this set is topologically conjugate to the dynamics of the full shift map with two symbols. This confirms chaos in the sense of Devaney. Our main results generalize the relevant results of Huang and Zoumore » [J. Nonlinear Sci. 15, 291–303 (2005)], Kaslik and Balint [J. Nonlinear Sci. 18, 415–432 (2008)] and Chen et al. [Sci. China Math. 56(9), 1869–1878 (2013)]. We also give some numeric simulations to verify our theoretical results.« less

Complex dynamics of a delayed discrete neural network of two nonidentical neurons.

PubMed

Chen, Yuanlong; Huang, Tingwen; Huang, Yu

2014-03-01

In this paper, we discover that a delayed discrete Hopfield neural network of two nonidentical neurons with self-connections and no self-connections can demonstrate chaotic behaviors. To this end, we first transform the model, by a novel way, into an equivalent system which has some interesting properties. Then, we identify the chaotic invariant set for this system and show that the dynamics of this system within this set is topologically conjugate to the dynamics of the full shift map with two symbols. This confirms chaos in the sense of Devaney. Our main results generalize the relevant results of Huang and Zou [J. Nonlinear Sci. 15, 291-303 (2005)], Kaslik and Balint [J. Nonlinear Sci. 18, 415-432 (2008)] and Chen et al. [Sci. China Math. 56(9), 1869-1878 (2013)]. We also give some numeric simulations to verify our theoretical results.

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.

Various Types of Coexisting Attractors in a New 4D Autonomous Chaotic System

NASA Astrophysics Data System (ADS)

Lai, Qiang; Akgul, Akif; Zhao, Xiao-Wen; Pei, Huiqin

An unique 4D autonomous chaotic system with signum function term is proposed in this paper. The system has four unstable equilibria and various types of coexisting attractors appear. Four-wing and four-scroll strange attractors are observed in the system and they will be broken into two coexisting butterfly attractors and two coexisting double-scroll attractors with the variation of the parameters. Numerical simulation shows that the system has various types of multiple coexisting attractors including two butterfly attractors with four limit cycles, two double-scroll attractors with a limit cycle, four single-scroll strange attractors, four limit cycles with regard to different parameters and initial values. The coexistence of the attractors is determined by the bifurcation diagrams. The chaotic and hyperchaotic properties of the attractors are verified by the Lyapunov exponents. Moreover, we present an electronic circuit to experimentally realize the dynamic behavior of the system.

Periodic orbit analysis of a system with continuous symmetry--A tutorial.

PubMed

Budanur, Nazmi Burak; Borrero-Echeverry, Daniel; Cvitanović, Predrag

2015-07-01

Dynamical systems with translational or rotational symmetry arise frequently in studies of spatially extended physical systems, such as Navier-Stokes flows on periodic domains. In these cases, it is natural to express the state of the fluid in terms of a Fourier series truncated to a finite number of modes. Here, we study a 4-dimensional model with chaotic dynamics and SO(2) symmetry similar to those that appear in fluid dynamics problems. A crucial step in the analysis of such a system is symmetry reduction. We use the model to illustrate different symmetry-reduction techniques. The system's relative equilibria are conveniently determined by rewriting the dynamics in terms of a symmetry-invariant polynomial basis. However, for the analysis of its chaotic dynamics, the "method of slices," which is applicable to very high-dimensional problems, is preferable. We show that a Poincaré section taken on the "slice" can be used to further reduce this flow to what is for all practical purposes a unimodal map. This enables us to systematically determine all relative periodic orbits and their symbolic dynamics up to any desired period. We then present cycle averaging formulas adequate for systems with continuous symmetry and use them to compute dynamical averages using relative periodic orbits. The convergence of such computations is discussed.

A Huygens principle for diffusion and anomalous diffusion in spatially extended systems

PubMed Central

Gottwald, Georg A.; Melbourne, Ian

2013-01-01

We present a universal view on diffusive behavior in chaotic spatially extended systems for anisotropic and isotropic media. For anisotropic systems, strong chaos leads to diffusive behavior (Brownian motion with drift) and weak chaos leads to superdiffusive behavior (Lévy processes with drift). For isotropic systems, the drift term vanishes and strong chaos again leads to Brownian motion. We establish the existence of a nonlinear Huygens principle for weakly chaotic systems in isotropic media whereby the dynamics behaves diffusively in even space dimension and exhibits superdiffusive behavior in odd space dimensions. PMID:23653481

Dynamics of rotationally fissioned asteroids: Source of observed small asteroid systems

NASA Astrophysics Data System (ADS)

Jacobson, Seth A.; Scheeres, Daniel J.

2011-07-01

We present a model of near-Earth asteroid (NEA) rotational fission and ensuing dynamics that describes the creation of synchronous binaries and all other observed NEA systems including: doubly synchronous binaries, high- e binaries, ternary systems, and contact binaries. Our model only presupposes the Yarkovsky-O'Keefe-Radzievskii-Paddack (YORP) effect, "rubble pile" asteroid geophysics, and gravitational interactions. The YORP effect torques a "rubble pile" asteroid until the asteroid reaches its fission spin limit and the components enter orbit about each other (Scheeres, D.J. [2007]. Icarus 189, 370-385). Non-spherical gravitational potentials couple the spin states to the orbit state and chaotically drive the system towards the observed asteroid classes along two evolutionary tracks primarily distinguished by mass ratio. Related to this is a new binary process termed secondary fission - the secondary asteroid of the binary system is rotationally accelerated via gravitational torques until it fissions, thus creating a chaotic ternary system. The initially chaotic binary can be stabilized to create a synchronous binary by components of the fissioned secondary asteroid impacting the primary asteroid, solar gravitational perturbations, and mutual body tides. These results emphasize the importance of the initial component size distribution and configuration within the parent asteroid. NEAs may go through multiple binary cycles and many YORP-induced rotational fissions during their approximately 10 Myr lifetime in the inner Solar System. Rotational fission and the ensuing dynamics are responsible for all NEA systems including the most commonly observed synchronous binaries.

Cooling of a magmatic system under thermal chaotic mixing

NASA Astrophysics Data System (ADS)

Petrelli, Maurizio; El Omari, Kamal; Le Guer, Yves; Perugini, Diego

2015-04-01

The cooling of a melt undergoing chaotic advection is studied numerically for a magma with a temperature-dependent viscosity in a 2D cavity with moving boundary. Different statistical mixing and energy indicators are used to characterize the efficiency of cooling by thermal chaotic mixing. We show that different cooling rates can be obtained during the thermal mixing even of a single basaltic magmatic batch undergoing chaotic advection. This process can induce complex temperature patterns inside the magma chamber. The emergence of chaotic dynamics strongly affects the temperature field during time and greatly increases the cooling rates. This mechanism has implications for the lifetime of a magmatic body and may favor the appearance of chemical heterogeneities in igneous systems as a result of different crystallization rates. Results from this study also highlight that even a single magma batch can develop, under chaotic thermal advection, complex thermal and therefore compositional patterns resulting from different cooling rates, which can account for some natural features that, to date, have received unsatisfactory explanations. Among them, the production of magmatic enclaves showing completely different cooling histories compared with the host magma, compositional zoning in mineral phases, and the generation of large-scale compositionally zoning observed in many plutons worldwide.

A challenge to chaotic itinerancy from brain dynamics

NASA Astrophysics Data System (ADS)

Kay, Leslie M.

2003-09-01

Brain hermeneutics and chaotic itinerancy proposed by Tsuda are attractive characterizations of perceptual dynamics in the mammalian olfactory system. This theory proposes that perception occurs at the interface between itinerant neural representation and interaction with the environment. Quantifiable application of these dynamics has been hampered by the lack of definable history and action processes which characterize the changes induced by behavioral state, attention, and learning. Local field potentials measured from several brain areas were used to characterize dynamic activity patterns for their use as representations of history and action processes. The signals were recorded from olfactory areas (olfactory bulb, OB, and pyriform cortex) and hippocampal areas (entorhinal cortex and dentate gyrus, DG) in the brains of rats. During odor-guided behavior the system shows dynamics at three temporal scales. Short time-scale changes are system-wide and can occur in the space of a single sniff. They are predictable, associated with learned shifts in behavioral state and occur periodically on the scale of the intertrial interval. These changes occupy the theta (2-12 Hz), beta (15-30 Hz), and gamma (40-100 Hz) frequency bands within and between all areas. Medium time-scale changes occur relatively unpredictably, manifesting in these data as alterations in connection strength between the OB and DG. These changes are strongly correlated with performance in associated trial blocks (5-10 min) and may be due to fluctuations in attention, mood, or amount of reward received. Long time-scale changes are likely related to learning or decline due to aging or disease. These may be modeled as slow monotonic processes that occur within or across days or even weeks or years. The folding of different time scales is proposed as a mechanism for chaotic itinerancy, represented by dynamic processes instead of static connection strengths. Thus, the individual maintains continuity of experience within the stability of fast periodic and slow monotonic processes, while medium scale events alter experience and performance dramatically but temporarily. These processes together with as yet to be determined action effects from motor system feedback are proposed as an instantiation of brain hermeneutics and chaotic itinerancy.

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

Zhang, Hailong; Vibration Control Lab, School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210042; Zhang, Ning

Magneto-rheological (MR) damper possesses inherent hysteretic characteristics. We investigate the resulting nonlinear behaviors of a two degree-of-freedom (2-DoF) MR vibration isolation system under harmonic external excitation. A MR damper is identified by employing the modified Bouc-wen hysteresis model. By numerical simulation, we characterize the nonlinear dynamic evolution of period-doubling, saddle node bifurcating and inverse period-doubling using bifurcation diagrams of variations in frequency with a fixed amplitude of the harmonic excitation. The strength of chaos is determined by the Lyapunov exponent (LE) spectrum. Semi-physical experiment on the 2-DoF MR vibration isolation system is proposed. We trace the time history and phasemore » trajectory under certain values of frequency of the harmonic excitation to verify the nonlinear dynamical evolution of period-doubling bifurcations to chaos. The largest LEs computed with the experimental data are also presented, confirming the chaotic motion in the experiment. We validate the chaotic motion caused by the hysteresis of the MR damper, and show the transitions between distinct regimes of stable motion and chaotic motion of the 2-DoF MR vibration isolation system for variations in frequency of external excitation.« less

Hash function based on chaotic map lattices.

PubMed

Wang, Shihong; Hu, Gang

2007-06-01

A new hash function system, based on coupled chaotic map dynamics, is suggested. By combining floating point computation of chaos and some simple algebraic operations, the system reaches very high bit confusion and diffusion rates, and this enables the system to have desired statistical properties and strong collision resistance. The chaos-based hash function has its advantages for high security and fast performance, and it serves as one of the most highly competitive candidates for practical applications of hash function for software realization and secure information communications in computer networks.

Transport properties in nontwist area-preserving maps

DOE PAGES

Szezech Jr., J. D.; Caldas, I. L.; Lopes, S. R.; ...

2009-10-23

Nontwist systems, common in the dynamical descriptions of fluids and plasmas, possess a shearless curve with a concomitant transport barrier that eliminates or reduces chaotic transport, even after its breakdown. In order to investigate the transport properties of nontwist systems, we analyze the barrier escape time and barrier transmissivity for the standard nontwist map, a paradigm of such systems. We interpret the sensitive dependence of these quantities upon map parameters by investigating chaotic orbit stickiness and the associated role played by the dominant crossing of stable and unstable manifolds.

Hash function based on chaotic map lattices

NASA Astrophysics Data System (ADS)

Wang, Shihong; Hu, Gang

2007-06-01

A new hash function system, based on coupled chaotic map dynamics, is suggested. By combining floating point computation of chaos and some simple algebraic operations, the system reaches very high bit confusion and diffusion rates, and this enables the system to have desired statistical properties and strong collision resistance. The chaos-based hash function has its advantages for high security and fast performance, and it serves as one of the most highly competitive candidates for practical applications of hash function for software realization and secure information communications in computer networks.

Correlations in electrically coupled chaotic lasers.

PubMed

Rosero, E J; Barbosa, W A S; Martinez Avila, J F; Khoury, A Z; Rios Leite, J R

2016-09-01

We show how two electrically coupled semiconductor lasers having optical feedback can present simultaneous antiphase correlated fast power fluctuations, and strong in-phase synchronized spikes of chaotic power drops. This quite counterintuitive phenomenon is demonstrated experimentally and confirmed by numerical solutions of a deterministic dynamical system of rate equations. The occurrence of negative and positive cross correlation between parts of a complex system according to time scales, as proved in our simple arrangement, is relevant for the understanding and characterization of collective properties in complex networks.

Chaos in the gauge/gravity correspondence

NASA Astrophysics Data System (ADS)

Pando Zayas, Leopoldo A.; Terrero-Escalante, César A.

2010-09-01

We study the motion of a string in the background of the Schwarzschild black hole in AdS 5 by applying the standard arsenal of dynamical systems. Our description of the phase space includes: the power spectrum, the largest Lyapunov exponent, Poincare sections and basins of attractions. We find convincing evidence that the motion is chaotic. We discuss the implications of some of the quantities associated with chaotic systems for aspects of the gauge/gravity correspondence. In particular, we suggest some potential relevance for the information loss paradox.

Analysis of Hepatic Blood Flow Using Chaotic Models

PubMed Central

Cohen, M. E.; Moazamipour, H.; Hudson, D. L.; Anderson, M. F.

1990-01-01

The study of chaos in physical systems is an important new theoretical development in modeling which has emerged in the last fifteen years. It is particularly useful in explaining phenomena which arise in nonlinear dynamic systems, for which previous mathematical models produced results with intractable solutions. Analysis of blood flow is such an application. In the work described here, chaotic models are used to analyze hepatic artery and portal vein blood flow obtained from a pulsed Doppler ultrasonic flowmeter implanted in dogs. ImagesFigure 3

Chaotic Motions in the Real Fuzzy Electronic Circuits (Preprint)

DTIC Science & Technology

2012-12-01

the research field of secure communications, the original source should be blended with other complex signals. Chaotic signals are one of the good... blending of the linear system models. Consider a continuous-time nonlinear dynamic system as follows: Rule i: IF )(1 tx is ...1iM and )(txn is...Chaos Solitons Fractals, vol. 21, no. 4, pp. 957–965, 2004. 29. L. M. Tam and W. M. SiTou, “Parametric study of the fractional order Chen–Lee

Some new surprises in chaos.

PubMed

Bunimovich, Leonid A; Vela-Arevalo, Luz V

2015-09-01

"Chaos is found in greatest abundance wherever order is being sought.It always defeats order, because it is better organized"Terry PratchettA brief review is presented of some recent findings in the theory of chaotic dynamics. We also prove a statement that could be naturally considered as a dual one to the Poincaré theorem on recurrences. Numerical results demonstrate that some parts of the phase space of chaotic systems are more likely to be visited earlier than other parts. A new class of chaotic focusing billiards is discussed that clearly violates the main condition considered to be necessary for chaos in focusing billiards.

Applied Chaos: From Oxymoron to Reality.

NASA Astrophysics Data System (ADS)

Ditto, William

1996-11-01

The rapidly emerging field of chaotic dynamics has presented the applied scientist with intriguing new tools to understand and manipulate systems that behave chaotically. An overview will be presented which will answer the questions: What is Chaos? and What can you do with Chaos? Examples of recent applications of chaos theory to the physical and biological sciences will be presented covering applications that range from encryption in communications to control of chaotically beating human hearts. Part A of program listing

Chaotic Dynamics in the Planar Gravitational Many-Body Problem with Rigid Body Rotations

NASA Astrophysics Data System (ADS)

Kwiecinski, James A.; Kovacs, Attila; Krause, Andrew L.; Planella, Ferran Brosa; van Gorder, Robert A.

The discovery of Pluto’s small moons in the last decade has brought attention to the dynamics of the dwarf planet’s satellites. With such systems in mind, we study a planar N-body system in which all the bodies are point masses, except for a single rigid body. We then present a reduced model consisting of a planar N-body problem with the rigid body treated as a 1D continuum (i.e. the body is treated as a rod with an arbitrary mass distribution). Such a model provides a good approximation to highly asymmetric geometries, such as the recently observed interstellar asteroid ‘Oumuamua, but is also amenable to analysis. We analytically demonstrate the existence of homoclinic chaos in the case where one of the orbits is nearly circular by way of the Melnikov method, and give numerical evidence for chaos when the orbits are more complicated. We show that the extent of chaos in parameter space is strongly tied to the deviations from a purely circular orbit. These results suggest that chaos is ubiquitous in many-body problems when one or more of the rigid bodies exhibits nonspherical and highly asymmetric geometries. The excitation of chaotic rotations does not appear to require tidal dissipation, obliquity variation, or orbital resonance. Such dynamics give a possible explanation for routes to chaotic dynamics observed in N-body systems such as the Pluto system where some of the bodies are highly nonspherical.

Dynamical complexity of short and noisy time series. Compression-Complexity vs. Shannon entropy

NASA Astrophysics Data System (ADS)

Nagaraj, Nithin; Balasubramanian, Karthi

2017-07-01

Shannon entropy has been extensively used for characterizing complexity of time series arising from chaotic dynamical systems and stochastic processes such as Markov chains. However, for short and noisy time series, Shannon entropy performs poorly. Complexity measures which are based on lossless compression algorithms are a good substitute in such scenarios. We evaluate the performance of two such Compression-Complexity Measures namely Lempel-Ziv complexity (LZ) and Effort-To-Compress (ETC) on short time series from chaotic dynamical systems in the presence of noise. Both LZ and ETC outperform Shannon entropy (H) in accurately characterizing the dynamical complexity of such systems. For very short binary sequences (which arise in neuroscience applications), ETC has higher number of distinct complexity values than LZ and H, thus enabling a finer resolution. For two-state ergodic Markov chains, we empirically show that ETC converges to a steady state value faster than LZ. Compression-Complexity measures are promising for applications which involve short and noisy time series.

Memory and betweenness preference in temporal networks induced from time series

NASA Astrophysics Data System (ADS)

Weng, Tongfeng; Zhang, Jie; Small, Michael; Zheng, Rui; Hui, Pan

2017-02-01

We construct temporal networks from time series via unfolding the temporal information into an additional topological dimension of the networks. Thus, we are able to introduce memory entropy analysis to unravel the memory effect within the considered signal. We find distinct patterns in the entropy growth rate of the aggregate network at different memory scales for time series with different dynamics ranging from white noise, 1/f noise, autoregressive process, periodic to chaotic dynamics. Interestingly, for a chaotic time series, an exponential scaling emerges in the memory entropy analysis. We demonstrate that the memory exponent can successfully characterize bifurcation phenomenon, and differentiate the human cardiac system in healthy and pathological states. Moreover, we show that the betweenness preference analysis of these temporal networks can further characterize dynamical systems and separate distinct electrocardiogram recordings. Our work explores the memory effect and betweenness preference in temporal networks constructed from time series data, providing a new perspective to understand the underlying dynamical systems.

Multifractal Approach to the Analysis of Crime Dynamics: Results for Burglary in San Francisco

NASA Astrophysics Data System (ADS)

Melgarejo, Miguel; Obregon, Nelson

This paper provides evidence of fractal, multifractal and chaotic behaviors in urban crime by computing key statistical attributes over a long data register of criminal activity. Fractal and multifractal analyses based on power spectrum, Hurst exponent computation, hierarchical power law detection and multifractal spectrum are considered ways to characterize and quantify the footprint of complexity of criminal activity. Moreover, observed chaos analysis is considered a second step to pinpoint the nature of the underlying crime dynamics. This approach is carried out on a long database of burglary activity reported by 10 police districts of San Francisco city. In general, interarrival time processes of criminal activity in San Francisco exhibit fractal and multifractal patterns. The behavior of some of these processes is close to 1/f noise. Therefore, a characterization as deterministic, high-dimensional, chaotic phenomena is viable. Thus, the nature of crime dynamics can be studied from geometric and chaotic perspectives. Our findings support that crime dynamics may be understood from complex systems theories like self-organized criticality or highly optimized tolerance.

Solar system dynamics

NASA Technical Reports Server (NTRS)

Wisdom, Jack

1987-01-01

The rotational dynamics of irregularly shaped satellites and the origin of Kirkwood Gaps are discussed. The chaotic tumbling of Hyperion and the anomalously low eccentricity of Deimos are examined. The Digital Orrery is used to explore the phase space of the ellipic restricted three body problem near the principal commensurabilities (2/1, 5/2, 3/1, and 3/2). The results for the 3/1 commensurability are in close agreement with those found earlier with the algebraic mapping method. Large chaotic zones are associated with the 3/1, 2/1 and 5/2 resonances, where there are gaps in the distribution of asteroids. The region near the 3/2 resonance, where the Hilda group of asteroids is located, is largely devoid of chaotic behavior. Thus, there is a qualitative agreement between the character of the motion and the distribution of asteroids.

How to control chaotic behaviour and population size with proportional feedback

NASA Astrophysics Data System (ADS)

Liz, Eduardo

2010-01-01

We study the control of chaos in one-dimensional discrete maps as they often occur in modelling population dynamics. For managing the population, we seek to suppress any possible chaotic behavior, leading the system to a stable equilibrium. In this Letter, we make a rigorous analysis of the proportional feedback method under certain conditions fulfilled by a wide family of maps. We show that it is possible to stabilize the chaotic dynamics towards a globally stable positive equilibrium, that can be chosen among a broad range of possible values. In particular, the size of the population can be enhanced by control in form of population reduction. This paradoxical phenomenon is known as the hydra effect, and it has important implications in the design of strategies in such areas as fishing, pest management, and conservation biology.

Investigating chaotic wake dynamics past a flapping airfoil and the role of vortex interactions behind the chaotic transition

NASA Astrophysics Data System (ADS)

Bose, Chandan; Sarkar, Sunetra

2018-04-01

The present study investigates the complex vortex interactions in two-dimensional flow-field behind a symmetric NACA0012 airfoil undergoing a prescribed periodic pitching-plunging motion in low Reynolds number regime. The flow-field transitions from periodic to chaotic through a quasi-periodic route as the plunge amplitude is gradually increased. This study unravels the role of the complex interactions that take place among the main vortex structures in making the unsteady flow-field transition from periodicity to chaos. The leading-edge separation plays a key role in providing the very first trigger for aperiodicity. Subsequent mechanisms like shredding, merging, splitting, and collision of vortices in the near-field that propagate and sustain the disturbance have also been followed and presented. These fundamental mechanisms are seen to give rise to spontaneous and irregular formation of new vortex couples at arbitrary locations, which are the primary agencies for sustaining chaos in the flow-field. The interactions have been studied for each dynamical state to understand the course of transition in the flow-field. The qualitative changes observed in the flow-field are manifestation of changes in the underlying dynamical system. The overall dynamics are established in the present study by means of robust quantitative measures derived from classical and non-classical tools from the dynamical system theory. As the present analysis involves a high fidelity multi-unknown system, non-classical dynamical tools such as recurrence-based time series methods are seen to be very efficient. Moreover, their application is novel in the context of pitch-plunge flapping flight.

Chaotic motion of comets in near-parabolic orbit: Mapping aproaches

NASA Astrophysics Data System (ADS)

Liu, Jie; Sun, Yi-Sui

1994-09-01

There exist many comets with near-parabolic orbits in the solar system. Among various theories proposed to explain their origin, the Oort cloud hypothesis seems to be the most reasonable. The theory assumes that there is a cometary cloud at a distance 103 to 107 from the sun and that perturbing forces from planets or stars make orbits of some of these comets become the near-parabolic type. Concerning the evolution of these orbits under planetary perturbations, we can raise the question: Will they stay in the solar system forever or will they escape from it? This is an attractive dynamical problem. If we go ahead by directly solving the dynamical differential equations, we may encounter the difficulty of long-time computation. For the orbits of these comets are near-parabolic and their periods are too long to study on their long-term evolution. With mapping approaches the difficulty will be overcome. In another aspect, the study of this model has special meaning for chaotic dynamics. We know that in the neighborhood of any separatrix i.e. the trajectory with zero frequency of the uperturbed motion of a Hamiltonian system, some chaotic motions have to be expected. Actually, the simplest example of separatrix is the parabolic trajectory of the two-body problem which separates the bounded and unbounded motion. From this point of view, the dynamical study of near-parabolic motion is very important. Petrosky's elegant but more abstract deduction gives a Kepler mapping which describes the dynamics of the cometary motion. In this paper we derive a similar mapping directly and discuss its dynamical characters.

Neuronal synchrony: Peculiarity and generality

PubMed Central

Nowotny, Thomas; Huerta, Ramon; Rabinovich, Mikhail I.

2008-01-01

Synchronization in neuronal systems is a new and intriguing application of dynamical systems theory. Why are neuronal systems different as a subject for synchronization? (1) Neurons in themselves are multidimensional nonlinear systems that are able to exhibit a wide variety of different activity patterns. Their “dynamical repertoire” includes regular or chaotic spiking, regular or chaotic bursting, multistability, and complex transient regimes. (2) Usually, neuronal oscillations are the result of the cooperative activity of many synaptically connected neurons (a neuronal circuit). Thus, it is necessary to consider synchronization between different neuronal circuits as well. (3) The synapses that implement the coupling between neurons are also dynamical elements and their intrinsic dynamics influences the process of synchronization or entrainment significantly. In this review we will focus on four new problems: (i) the synchronization in minimal neuronal networks with plastic synapses (synchronization with activity dependent coupling), (ii) synchronization of bursts that are generated by a group of nonsymmetrically coupled inhibitory neurons (heteroclinic synchronization), (iii) the coordination of activities of two coupled neuronal networks (partial synchronization of small composite structures), and (iv) coarse grained synchronization in larger systems (synchronization on a mesoscopic scale). PMID:19045493

Chaotic Transport in Circumterrestrial Orbits

NASA Astrophysics Data System (ADS)

Rosengren, Aaron Jay

2018-04-01

The slow deformation of circumterrestrial orbits in the medium region, subject to lunisolar secular resonances, is well approximated by a Hamiltonian system with 2.5 degrees of freedom. This dynamical model is referred to in the astrophysical and celestial dynamics communities as the quadrupolar, secular, hierarchical three-body problem, and, in the non-autonomous case, gives rise to the classical Kozai-Lidov mechanism. In the time-dependent model, brought about in our case by the Moon's perturbed motion, the action variables of the system may experience chaotic variations and large drifts due to the possible overlap of nearby resonances. Using variational chaos indicators, we compute high-resolution portraits of the action space, revealing the existence of tori and structures filling chaotic regions. Our refined and elaborate calculations allow us to isolate precise initial conditions near specific areas of interest and to study their asymptotic behavior in time. We highlight in particular how the drift in phase space is mediated by the complement of the numerically detected KAM tori. Despite their reputed normality, Earth satellite orbits can possess an extraordinarily rich spectrum of dynamical behaviors, and, like the small body remnants of Solar system formation, they have all the complications that make them very interesting candidates for testing the modern tools of chaos theory.

Quantifying chaotic dynamics from integrate-and-fire processes

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

Pavlov, A. N.; Saratov State Technical University, Politehnicheskaya Str. 77, 410054 Saratov; Pavlova, O. N.

2015-01-15

Characterizing chaotic dynamics from integrate-and-fire (IF) interspike intervals (ISIs) is relatively easy performed at high firing rates. When the firing rate is low, a correct estimation of Lyapunov exponents (LEs) describing dynamical features of complex oscillations reflected in the IF ISI sequences becomes more complicated. In this work we discuss peculiarities and limitations of quantifying chaotic dynamics from IF point processes. We consider main factors leading to underestimated LEs and demonstrate a way of improving numerical determining of LEs from IF ISI sequences. We show that estimations of the two largest LEs can be performed using around 400 mean periodsmore » of chaotic oscillations in the regime of phase-coherent chaos. Application to real data is discussed.« less

Chaotic dynamics of flexible Euler-Bernoulli beams

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

Awrejcewicz, J., E-mail: awrejcew@p.lodz.pl; Krysko, A. V., E-mail: anton.krysko@gmail.com; Kutepov, I. E., E-mail: iekutepov@gmail.com

2013-12-15

Mathematical modeling and analysis of spatio-temporal chaotic dynamics of flexible simple and curved Euler-Bernoulli beams are carried out. The Kármán-type geometric non-linearity is considered. Algorithms reducing partial differential equations which govern the dynamics of studied objects and associated boundary value problems are reduced to the Cauchy problem through both Finite Difference Method with the approximation of O(c{sup 2}) and Finite Element Method. The obtained Cauchy problem is solved via the fourth and sixth-order Runge-Kutta methods. Validity and reliability of the results are rigorously discussed. Analysis of the chaotic dynamics of flexible Euler-Bernoulli beams for a series of boundary conditions ismore » carried out with the help of the qualitative theory of differential equations. We analyze time histories, phase and modal portraits, autocorrelation functions, the Poincaré and pseudo-Poincaré maps, signs of the first four Lyapunov exponents, as well as the compression factor of the phase volume of an attractor. A novel scenario of transition from periodicity to chaos is obtained, and a transition from chaos to hyper-chaos is illustrated. In particular, we study and explain the phenomenon of transition from symmetric to asymmetric vibrations. Vibration-type charts are given regarding two control parameters: amplitude q{sub 0} and frequency ω{sub p} of the uniformly distributed periodic excitation. Furthermore, we detected and illustrated how the so called temporal-space chaos is developed following the transition from regular to chaotic system dynamics.« less

Stochastic Erosion of Fractal Structure in Nonlinear Dynamical Systems

NASA Astrophysics Data System (ADS)

Agarwal, S.; Wettlaufer, J. S.

2014-12-01

We analyze the effects of stochastic noise on the Lorenz-63 model in the chaotic regime to demonstrate a set of general issues arising in the interpretation of data from nonlinear dynamical systems typical in geophysics. The model is forced using both additive and multiplicative, white and colored noise and it is shown that, through a suitable choice of the noise intensity, both additive and multiplicative noise can produce similar dynamics. We use a recently developed measure, histogram distance, to show the similarity between the dynamics produced by additive and multiplicative forcing. This phenomenon, in a nonlinear fractal structure with chaotic dynamics can be explained by understanding how noise affects the Unstable Periodic Orbits (UPOs) of the system. For delta-correlated noise, the UPOs erode the fractal structure. In the presence of memory in the noise forcing, the time scale of the noise starts to interact with the period of some UPO and, depending on the noise intensity, stochastic resonance may be observed. This also explains the mixing in dissipative dynamical systems in presence of white noise; as the fractal structure is smoothed, the decay of correlations is enhanced, and hence the rate of mixing increases with noise intensity.

A Route to Chaos after Bifurcation in a Two-section Semiconductor Laser Using Opto-electronic Delayed Feedback at Each In-current

NASA Astrophysics Data System (ADS)

Yan, Sen-lin

2014-12-01

We study dynamics in an opto-electronic delayed feedback two-section semiconductor laser. We predict theoretically that the system can result in bistability and bifurcation. We analyze numerically the route to chaos from stability to bifurcation by varying the delayed time, feedback strength and two in-currents. The system displays the four distinct types or modes of stable, periodic pulsed or self-pulsing, undamped oscillating or beating, and chaos. The frequency and intensity varying with the delayed time in the self-pulsation regions are discussed detailedly to find that the pulsing frequency is reduced with the long delayed time while the pulsing intensity is added. And the chaotic pulsing frequency is increased with the large in-current Ja. The laser relaxation oscillation frequency is decreased with the large in-current Jb. One in-current characterize dynamics in the laser to conduce to stable, periodic pulsed, beating and chaotic states by altering its values. The other in-current characterize dynamics in the chaotic laser to be controlled to a stable state after a road to quasi-period by adding the values.

Input reconstruction of chaos sensors.

PubMed

Yu, Dongchuan; Liu, Fang; Lai, Pik-Yin

2008-06-01

Although the sensitivity of sensors can be significantly enhanced using chaotic dynamics due to its extremely sensitive dependence on initial conditions and parameters, how to reconstruct the measured signal from the distorted sensor response becomes challenging. In this paper we suggest an effective method to reconstruct the measured signal from the distorted (chaotic) response of chaos sensors. This measurement signal reconstruction method applies the neural network techniques for system structure identification and therefore does not require the precise information of the sensor's dynamics. We discuss also how to improve the robustness of reconstruction. Some examples are presented to illustrate the measurement signal reconstruction method suggested.

Chaotic dynamics around cometary nuclei

NASA Astrophysics Data System (ADS)

Lages, José; Shevchenko, Ivan I.; Rollin, Guillaume

2018-06-01

We apply a generalized Kepler map theory to describe the qualitative chaotic dynamics around cometary nuclei, based on accessible observational data for five comets whose nuclei are well-documented to resemble dumb-bells. The sizes of chaotic zones around the nuclei and the Lyapunov times of the motion inside these zones are estimated. In the case of Comet 1P/Halley, the circumnuclear chaotic zone seems to engulf an essential part of the Hill sphere, at least for orbits of moderate to high eccentricity.

Chaotic Dynamical Ferromagnetic Phase Induced by Nonequilibrium Quantum Fluctuations

NASA Astrophysics Data System (ADS)

Lerose, Alessio; Marino, Jamir; Žunkovič, Bojan; Gambassi, Andrea; Silva, Alessandro

2018-03-01

We investigate the robustness of a dynamical phase transition against quantum fluctuations by studying the impact of a ferromagnetic nearest-neighbor spin interaction in one spatial dimension on the nonequilibrium dynamical phase diagram of the fully connected quantum Ising model. In particular, we focus on the transient dynamics after a quantum quench and study the prethermal state via a combination of analytic time-dependent spin wave theory and numerical methods based on matrix product states. We find that, upon increasing the strength of the quantum fluctuations, the dynamical critical point fans out into a chaotic dynamical phase within which the asymptotic ordering is characterized by strong sensitivity to the parameters and initial conditions. We argue that such a phenomenon is general, as it arises from the impact of quantum fluctuations on the mean-field out of equilibrium dynamics of any system which exhibits a broken discrete symmetry.

Chaotic Dynamical Ferromagnetic Phase Induced by Nonequilibrium Quantum Fluctuations.

PubMed

Lerose, Alessio; Marino, Jamir; Žunkovič, Bojan; Gambassi, Andrea; Silva, Alessandro

2018-03-30

We investigate the robustness of a dynamical phase transition against quantum fluctuations by studying the impact of a ferromagnetic nearest-neighbor spin interaction in one spatial dimension on the nonequilibrium dynamical phase diagram of the fully connected quantum Ising model. In particular, we focus on the transient dynamics after a quantum quench and study the prethermal state via a combination of analytic time-dependent spin wave theory and numerical methods based on matrix product states. We find that, upon increasing the strength of the quantum fluctuations, the dynamical critical point fans out into a chaotic dynamical phase within which the asymptotic ordering is characterized by strong sensitivity to the parameters and initial conditions. We argue that such a phenomenon is general, as it arises from the impact of quantum fluctuations on the mean-field out of equilibrium dynamics of any system which exhibits a broken discrete symmetry.

Giant Suppression of the Activation Rate in Dynamical Systems Exhibiting Chaotic Transitions

NASA Astrophysics Data System (ADS)

Gac, J. M.; Xafebrowski, J. J.

2008-05-01

The phenomenon of giant suppression of activation, when two or more correlated noise signals act on the system, was found a few years ago in dynamical bistable or metastable systems. When the correlation between these noise signals is strong enough and the amplitudes of the noise are chosen correctly --- the life time of the metastable state may be longer than in the case of the application of only a single noise even by many orders of magnitude. In this paper, we investigate similar phenomena in systems exhibiting several chaotic transitions: Pomeau--Manneville intermittency, boundary crisis and interior crisis induced intermittency. Our goal is to show that, in these systems the application of two noise components with the proper choice of the parameters in the case of intermittency can also lengthen the mean laminar phase length or, in the case of boundary crisis, lengthen the time the trajectory spends on the pre-crisis attractor. In systems with crisis induced intermittency, we introduce a new phenomenon we called quasi-deterministic giant suppression of activation in which the lengthening of the laminar phase lengths is caused not by the action of two correlated noise signals but by a single noise term which is correlated with one of the chaotic variables of the system.

Why do Reservoir Computing Networks Predict Chaotic Systems so Well?

NASA Astrophysics Data System (ADS)

Lu, Zhixin; Pathak, Jaideep; Girvan, Michelle; Hunt, Brian; Ott, Edward

Recently a new type of artificial neural network, which is called a reservoir computing network (RCN), has been employed to predict the evolution of chaotic dynamical systems from measured data and without a priori knowledge of the governing equations of the system. The quality of these predictions has been found to be spectacularly good. Here, we present a dynamical-system-based theory for how RCN works. Basically a RCN is thought of as consisting of three parts, a randomly chosen input layer, a randomly chosen recurrent network (the reservoir), and an output layer. The advantage of the RCN framework is that training is done only on the linear output layer, making it computationally feasible for the reservoir dimensionality to be large. In this presentation, we address the underlying dynamical mechanisms of RCN function by employing the concepts of generalized synchronization and conditional Lyapunov exponents. Using this framework, we propose conditions on reservoir dynamics necessary for good prediction performance. By looking at the RCN from this dynamical systems point of view, we gain a deeper understanding of its surprising computational power, as well as insights on how to design a RCN. Supported by Army Research Office Grant Number W911NF1210101.

Relativistic quantum Darwinism in Dirac fermion and graphene systems

NASA Astrophysics Data System (ADS)

Ni, Xuan; Huang, Liang; Lai, Ying-Cheng; Pecora, Louis

2012-02-01

We solve the Dirac equation in two spatial dimensions in the setting of resonant tunneling, where the system consists of two symmetric cavities connected by a finite potential barrier. The shape of the cavities can be chosen to yield both regular and chaotic dynamics in the classical limit. We find that certain pointer states about classical periodic orbits can exist, which are signatures of relativistic quantum Darwinism (RQD). These localized states suppress quantum tunneling, and the effect becomes less severe as the underlying classical dynamics in the cavity is chaotic, leading to regularization of quantum tunneling. Qualitatively similar phenomena have been observed in graphene. A physical theory is developed to explain relativistic quantum Darwinism and its effects based on the spectrum of complex eigenenergies of the non-Hermitian Hamiltonian describing the open cavity system.

Simulations of submonolayer Xe on Pt(111): The case for a chaotic low temperature phase

NASA Astrophysics Data System (ADS)

Novaco, Anthony D.; Bavaresco, Jessica

2018-04-01

Molecular dynamics simulations are reported for the structural and thermodynamic properties of submonolayer xenon adsorbed on the (111) surface of platinum for temperatures up to the (apparently incipient) triple point and beyond. While the motion of the atoms in the surface plane is treated with a standard two-dimensional molecular dynamics simulation, the model takes into consideration the thermal excitation of quantum states associated with surface-normal dynamics in an attempt to describe the apparent smoothing of the corrugation with increasing temperature. We examine the importance of this thermal smoothing to the relative stability of several observed and proposed low-temperature structures. Structure factor calculations are compared to experimental results in an attempt to determine the low temperature structure of this system. These calculations provide strong evidence that, at very low temperatures, the domain wall structure of a xenon monolayer adsorbed on a Pt(111) substrate possesses a chaotic-like nature, exhibiting long-lived meta-stable states with pinned domain walls, these walls having narrow widths and irregular shapes. This result is contrary to the standard wisdom regarding this system, namely, that the very low temperature phase of this system is a striped incommensurate phase. We present the case for further experimental investigation of this and similar systems as possible examples of chaotic low temperature phases in two dimensions.

Chimeralike states in a network of oscillators under attractive and repulsive global coupling.

PubMed

Mishra, Arindam; Hens, Chittaranjan; Bose, Mridul; Roy, Prodyot K; Dana, Syamal K

2015-12-01

We report chimeralike states in an ensemble of oscillators using a type of global coupling consisting of two components: attractive and repulsive mean-field feedback. We identify the existence of two types of chimeralike states in a bistable Liénard system; in one type, both the coherent and the incoherent populations are in chaotic states (which we refer to as chaos-chaos chimeralike states) and, in another type, the incoherent population is in periodic state while the coherent population has irregular small oscillation. We find a metastable state in a parameter regime of the Liénard system where the coherent and noncoherent states migrate in time from one to another subpopulation. The relative size of the incoherent subpopulation, in the chimeralike states, remains almost stable with increasing size of the network. The generality of the coupling configuration in the origin of the chimeralike states is tested, using a second example of bistable system, the van der Pol-Duffing oscillator where the chimeralike states emerge as weakly chaotic in the coherent subpopulation and chaotic in the incoherent subpopulation. Furthermore, we apply the coupling, in a simplified form, to form a network of the chaotic Rössler system where both the noncoherent and the coherent subpopulations show chaotic dynamics.

Periodic, Quasi-periodic and Chaotic Dynamics in Simple Gene Elements with Time Delays

PubMed Central

Suzuki, Yoko; Lu, Mingyang; Ben-Jacob, Eshel; Onuchic, José N.

2016-01-01

Regulatory gene circuit motifs play crucial roles in performing and maintaining vital cellular functions. Frequently, theoretical studies of gene circuits focus on steady-state behaviors and do not include time delays. In this study, the inclusion of time delays is shown to entirely change the time-dependent dynamics for even the simplest possible circuits with one and two gene elements with self and cross regulations. These elements can give rise to rich behaviors including periodic, quasi-periodic, weak chaotic, strong chaotic and intermittent dynamics. We introduce a special power-spectrum-based method to characterize and discriminate these dynamical modes quantitatively. Our simulation results suggest that, while a single negative feedback loop of either one- or two-gene element can only have periodic dynamics, the elements with two positive/negative feedback loops are the minimalist elements to have chaotic dynamics. These elements typically have one negative feedback loop that generates oscillations, and another unit that allows frequent switches among multiple steady states or between oscillatory and non-oscillatory dynamics. Possible dynamical features of several simple one- and two-gene elements are presented in details. Discussion is presented for possible roles of the chaotic behavior in the robustness of cellular functions and diseases, for example, in the context of cancer. PMID:26876008

Periodic, Quasi-periodic and Chaotic Dynamics in Simple Gene Elements with Time Delays

NASA Astrophysics Data System (ADS)

Suzuki, Yoko; Lu, Mingyang; Ben-Jacob, Eshel; Onuchic, José N.

2016-02-01

Regulatory gene circuit motifs play crucial roles in performing and maintaining vital cellular functions. Frequently, theoretical studies of gene circuits focus on steady-state behaviors and do not include time delays. In this study, the inclusion of time delays is shown to entirely change the time-dependent dynamics for even the simplest possible circuits with one and two gene elements with self and cross regulations. These elements can give rise to rich behaviors including periodic, quasi-periodic, weak chaotic, strong chaotic and intermittent dynamics. We introduce a special power-spectrum-based method to characterize and discriminate these dynamical modes quantitatively. Our simulation results suggest that, while a single negative feedback loop of either one- or two-gene element can only have periodic dynamics, the elements with two positive/negative feedback loops are the minimalist elements to have chaotic dynamics. These elements typically have one negative feedback loop that generates oscillations, and another unit that allows frequent switches among multiple steady states or between oscillatory and non-oscillatory dynamics. Possible dynamical features of several simple one- and two-gene elements are presented in details. Discussion is presented for possible roles of the chaotic behavior in the robustness of cellular functions and diseases, for example, in the context of cancer.

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.

Simulations of Technology-Induced and Crisis-Led Stochastic and Chaotic Fluctuations in Higher Education Processes: A Model and a Case Study for Performance and Expected Employment

ERIC Educational Resources Information Center

Ahmet, Kara

2015-01-01

This paper presents a simple model of the provision of higher educational services that considers and exemplifies nonlinear, stochastic, and potentially chaotic processes. I use the methods of system dynamics to simulate these processes in the context of a particular sociologically interesting case, namely that of the Turkish higher education…

Insights on chaotic dynamics: mixing experiments between natural silicate melts from Vulcano island (Aeolian Islands, Italy)

NASA Astrophysics Data System (ADS)

Rossi, Stefano; Morgavi, Daniele; Vetere, Francesco; Petrelli, Maurizio; Perugini, Diego

2017-04-01

keywords: Magma mixing, chaotic dynamics, time series experiments Magma mixing is a petrologic phenomenon which is recognized as potential trigger of highly explosive eruptions and its evidence is commonly observable in natural rocks. Here we tried to replicate the dynamic conditions of mixing performing a set of chaotic mixing experiments between shoshonitic and rhyolitic magmas from Vulcano island. Vulcano is the southernmost island of the Aeolian Archipelago (Aeolian Islands, Italy); it is completely built by volcanic rocks with variable degree of evolution ranging from basalt to rhyolite (e.g. Keller 1980; Ellam et al. 1988; De Astis 1995; De Astis et al. 2013) and its magmatic activity dates back to about 120 ky. Last eruption occurred in 1888-1890. The chaotic mixing experiments were performed by using the new ChaOtic Magma Mixing Apparatus (COMMA), held at the Department of Physics and Geology, University of Perugia. This new experimental device allows to track the evolution of the mixing process and the associated modulation of chemical composition between different magmas. Experiments were performed at 1200°C and atmospheric pressure with a viscosity ratio higher than three orders of magnitude. The experimental protocol was chosen to ensure the occurrence of chaotic dynamics in the system and the run duration was progressively increased (e.g. 10.5 h, 21 h, 42 h). The products of each experiment are crystal-free glasses in which the variation of major elements was investigated along different profiles using electron microprobe (EMPA) at Institute für Mineralogie, Leibniz Universität of Hannover (Germany). The efficiency of the mixing process is estimated by calculating the decrease of concentration variance in time and it is shown that the variance of major elements exponentially decays. Our results confirm and quantify how different chemical elements homogenize in the melt at differing rates. It is also observable that the mixing structures generated during the mixing experiments are topologically identical to those observed in natural mixed volcanic rocks.

Information's role in the estimation of chaotic signals

NASA Astrophysics Data System (ADS)

Drake, Daniel Fred

1998-11-01

Researchers have proposed several methods designed to recover chaotic signals from noise-corrupted observations. While the methods vary, their qualitative performance does not: in low levels of noise all methods effectively recover the underlying signal; in high levels of noise no method can recover the underlying signal to any meaningful degree of accuracy. Of the methods proposed to date, all represent sub-optimal estimators. So: Is the inability to recover the signal in high noise levels simply a consequence of estimator sub-optimality? Or is estimator failure actually a manifestation of some intrinsic property of chaos itself? These questions are answered by deriving an optimal estimator for a class of chaotic systems and noting that it, too, fails in high levels of noise. An exact, closed- form expression for the estimator is obtained for a class of chaotic systems whose signals are solutions to a set of linear (but noncausal) difference equations. The existence of this linear description circumvents the difficulties normally encountered when manipulating the nonlinear (but causal) expressions that govern. chaotic behavior. The reason why even the optimal estimator fails to recover underlying chaotic signals in high levels of noise has its roots in information theory. At such noise levels, the mutual information linking the corrupted observations to the underlying signal is essentially nil, reducing the estimator to a simple guessing strategy based solely on a priori statistics. Entropy, long the common bond between information theory and dynamical systems, is actually one aspect of a far more complete characterization of information sources: the rate distortion function. Determining the rate distortion function associated with the class of chaotic systems considered in this work provides bounds on estimator performance in high levels of noise. Finally, a slight modification of the linear description leads to a method of synthesizing on limited precision platforms ``pseudo-chaotic'' sequences that mimic true chaotic behavior to any finite degree of precision and duration. The use of such a technique in spread-spectrum communications is considered.

Recurrence quantity analysis based on matrix eigenvalues

NASA Astrophysics Data System (ADS)

Yang, Pengbo; Shang, Pengjian

2018-06-01

Recurrence plots is a powerful tool for visualization and analysis of dynamical systems. Recurrence quantification analysis (RQA), based on point density and diagonal and vertical line structures in the recurrence plots, is considered to be alternative measures to quantify the complexity of dynamical systems. In this paper, we present a new measure based on recurrence matrix to quantify the dynamical properties of a given system. Matrix eigenvalues can reflect the basic characteristics of the complex systems, so we show the properties of the system by exploring the eigenvalues of the recurrence matrix. Considering that Shannon entropy has been defined as a complexity measure, we propose the definition of entropy of matrix eigenvalues (EOME) as a new RQA measure. We confirm that EOME can be used as a metric to quantify the behavior changes of the system. As a given dynamical system changes from a non-chaotic to a chaotic regime, the EOME will increase as well. The bigger EOME values imply higher complexity and lower predictability. We also study the effect of some factors on EOME,including data length, recurrence threshold, the embedding dimension, and additional noise. Finally, we demonstrate an application in physiology. The advantage of this measure lies in a high sensitivity and simple computation.

Human brain detects short-time nonlinear predictability in the temporal fine structure of deterministic chaotic sounds

NASA Astrophysics Data System (ADS)

Itoh, Kosuke; Nakada, Tsutomu

2013-04-01

Deterministic nonlinear dynamical processes are ubiquitous in nature. Chaotic sounds generated by such processes may appear irregular and random in waveform, but these sounds are mathematically distinguished from random stochastic sounds in that they contain deterministic short-time predictability in their temporal fine structures. We show that the human brain distinguishes deterministic chaotic sounds from spectrally matched stochastic sounds in neural processing and perception. Deterministic chaotic sounds, even without being attended to, elicited greater cerebral cortical responses than the surrogate control sounds after about 150 ms in latency after sound onset. Listeners also clearly discriminated these sounds in perception. The results support the hypothesis that the human auditory system is sensitive to the subtle short-time predictability embedded in the temporal fine structure of sounds.

Phase synchronization in the forced Lorenz system

NASA Astrophysics Data System (ADS)

Park, Eun-Hyoung; Zaks, Michael A.; Kurths, Jürgen

1999-12-01

We demonstrate that the dynamics of phase synchronization in a chaotic system under weak periodic forcing depends crucially on the distribution of intrinsic characteristic times of this system. Under the external periodic action, the frequency of every unstable periodic orbit is locked to the frequency of the force. In systems which in the autonomous case displays nearly isochronous chaotic rotations, the locking ratio is the same for all periodic orbits; since a typical chaotic orbit wanders between the periodic ones, its phase follows the phase of the force. For the Lorenz attractor with its unbounded times of return onto a Poincaré surface, such state of perfect phase synchronization is inaccessible. Analysis with the help of unstable periodic orbits shows that this state is replaced by another one, which we call ``imperfect phase synchronization,'' and in which we observe alternation of temporal segments, corresponding to different rational values of frequency lockings.

Modified Levenberg-Marquardt Method for RÖSSLER Chaotic System Fuzzy Modeling Training

NASA Astrophysics Data System (ADS)

Wang, Yu-Hui; Wu, Qing-Xian; Jiang, Chang-Sheng; Xue, Ya-Li; Fang, Wei

Generally, fuzzy approximation models require some human knowledge and experience. Operator's experience is involved in the mathematics of fuzzy theory as a collection of heuristic rules. The main goal of this paper is to present a new method for identifying unknown nonlinear dynamics such as Rössler system without any human knowledge. Instead of heuristic rules, the presented method uses the input-output data pairs to identify the Rössler chaotic system. The training algorithm is a modified Levenberg-Marquardt (L-M) method, which can adjust the parameters of each linear polynomial and fuzzy membership functions on line, and do not rely on experts' experience excessively. Finally, it is applied to training Rössler chaotic system fuzzy identification. Comparing this method with the standard L-M method, the convergence speed is accelerated. The simulation results demonstrate the effectiveness of the proposed method.

A family of chaotic pure analog coding schemes based on baker's map function

NASA Astrophysics Data System (ADS)

Liu, Yang; Li, Jing; Lu, Xuanxuan; Yuen, Chau; Wu, Jun

2015-12-01

This paper considers a family of pure analog coding schemes constructed from dynamic systems which are governed by chaotic functions—baker's map function and its variants. Various decoding methods, including maximum likelihood (ML), minimum mean square error (MMSE), and mixed ML-MMSE decoding algorithms, have been developed for these novel encoding schemes. The proposed mirrored baker's and single-input baker's analog codes perform a balanced protection against the fold error (large distortion) and weak distortion and outperform the classical chaotic analog coding and analog joint source-channel coding schemes in literature. Compared to the conventional digital communication system, where quantization and digital error correction codes are used, the proposed analog coding system has graceful performance evolution, low decoding latency, and no quantization noise. Numerical results show that under the same bandwidth expansion, the proposed analog system outperforms the digital ones over a wide signal-to-noise (SNR) range.

A reanalysis of "Two types of asynchronous activity in networks of excitatory and inhibitory spiking neurons".

PubMed

Engelken, Rainer; Farkhooi, Farzad; Hansel, David; van Vreeswijk, Carl; Wolf, Fred

2016-01-01

Neuronal activity in the central nervous system varies strongly in time and across neuronal populations. It is a longstanding proposal that such fluctuations generically arise from chaotic network dynamics. Various theoretical studies predict that the rich dynamics of rate models operating in the chaotic regime can subserve circuit computation and learning. Neurons in the brain, however, communicate via spikes and it is a theoretical challenge to obtain similar rate fluctuations in networks of spiking neuron models. A recent study investigated spiking balanced networks of leaky integrate and fire (LIF) neurons and compared their dynamics to a matched rate network with identical topology, where single unit input-output functions were chosen from isolated LIF neurons receiving Gaussian white noise input. A mathematical analogy between the chaotic instability in networks of rate units and the spiking network dynamics was proposed. Here we revisit the behavior of the spiking LIF networks and these matched rate networks. We find expected hallmarks of a chaotic instability in the rate network: For supercritical coupling strength near the transition point, the autocorrelation time diverges. For subcritical coupling strengths, we observe critical slowing down in response to small external perturbations. In the spiking network, we found in contrast that the timescale of the autocorrelations is insensitive to the coupling strength and that rate deviations resulting from small input perturbations rapidly decay. The decay speed even accelerates for increasing coupling strength. In conclusion, our reanalysis demonstrates fundamental differences between the behavior of pulse-coupled spiking LIF networks and rate networks with matched topology and input-output function. In particular there is no indication of a corresponding chaotic instability in the spiking network.

Characterization of normality of chaotic systems including prediction and detection of anomalies

NASA Astrophysics Data System (ADS)

Engler, Joseph John

Accurate prediction and control pervades domains such as engineering, physics, chemistry, and biology. Often, it is discovered that the systems under consideration cannot be well represented by linear, periodic nor random data. It has been shown that these systems exhibit deterministic chaos behavior. Deterministic chaos describes systems which are governed by deterministic rules but whose data appear to be random or quasi-periodic distributions. Deterministically chaotic systems characteristically exhibit sensitive dependence upon initial conditions manifested through rapid divergence of states initially close to one another. Due to this characterization, it has been deemed impossible to accurately predict future states of these systems for longer time scales. Fortunately, the deterministic nature of these systems allows for accurate short term predictions, given the dynamics of the system are well understood. This fact has been exploited in the research community and has resulted in various algorithms for short term predictions. Detection of normality in deterministically chaotic systems is critical in understanding the system sufficiently to able to predict future states. Due to the sensitivity to initial conditions, the detection of normal operational states for a deterministically chaotic system can be challenging. The addition of small perturbations to the system, which may result in bifurcation of the normal states, further complicates the problem. The detection of anomalies and prediction of future states of the chaotic system allows for greater understanding of these systems. The goal of this research is to produce methodologies for determining states of normality for deterministically chaotic systems, detection of anomalous behavior, and the more accurate prediction of future states of the system. Additionally, the ability to detect subtle system state changes is discussed. The dissertation addresses these goals by proposing new representational techniques and novel prediction methodologies. The value and efficiency of these methods are explored in various case studies. Presented is an overview of chaotic systems with examples taken from the real world. A representation schema for rapid understanding of the various states of deterministically chaotic systems is presented. This schema is then used to detect anomalies and system state changes. Additionally, a novel prediction methodology which utilizes Lyapunov exponents to facilitate longer term prediction accuracy is presented and compared with other nonlinear prediction methodologies. These novel methodologies are then demonstrated on applications such as wind energy, cyber security and classification of social networks.

Strong generalized synchronization with a particular relationship R between the coupled systems

NASA Astrophysics Data System (ADS)

Grácio, Clara; Fernandes, Sara; Mário Lopes, Luís

2018-03-01

The question of the chaotic synchronization of two coupled dynamical systems is an issue that interests researchers in many fields, from biology to psychology, through economics, chemistry, physics, and many others. The different forms of couplings and the different types of synchronization, give rise to many problems, most of them little studied. In this paper we deal with general couplings of two dynamical systems and we study strong generalized synchronization with a particular relationship R between them. Our results include the definition of a window in the domain of the coupling strength, where there is an exponentially stable solution, and the explicit determination of this window. In the case of unidirectional or symmetric couplings, this window is presented in terms of the maximum Lyapunov exponent of the systems. Examples of applications to chaotic systems of dimension one and two are presented.

Controlling chaos faster.

PubMed

Bick, Christian; Kolodziejski, Christoph; Timme, Marc

2014-09-01

Predictive feedback control is an easy-to-implement method to stabilize unknown unstable periodic orbits in chaotic dynamical systems. Predictive feedback control is severely limited because asymptotic convergence speed decreases with stronger instabilities which in turn are typical for larger target periods, rendering it harder to effectively stabilize periodic orbits of large period. Here, we study stalled chaos control, where the application of control is stalled to make use of the chaotic, uncontrolled dynamics, and introduce an adaptation paradigm to overcome this limitation and speed up convergence. This modified control scheme is not only capable of stabilizing more periodic orbits than the original predictive feedback control but also speeds up convergence for typical chaotic maps, as illustrated in both theory and application. The proposed adaptation scheme provides a way to tune parameters online, yielding a broadly applicable, fast chaos control that converges reliably, even for periodic orbits of large period.

Double-well chimeras in 2D lattice of chaotic bistable elements

NASA Astrophysics Data System (ADS)

Shepelev, I. A.; Bukh, A. V.; Vadivasova, T. E.; Anishchenko, V. S.; Zakharova, A.

2018-01-01

We investigate spatio-temporal dynamics of a 2D ensemble of nonlocally coupled chaotic cubic maps in a bistability regime. In particular, we perform a detailed study on the transition ;coherence - incoherence; for varying coupling strength for a fixed interaction radius. For the 2D ensemble we show the appearance of amplitude and phase chimera states previously reported for 1D ensembles of nonlocally coupled chaotic systems. Moreover, we uncover a novel type of chimera state, double-well chimera, which occurs due to the interplay of the bistability of the local dynamics and the 2D ensemble structure. Additionally, we find double-well chimera behavior for steady states which we call double-well chimera death. A distinguishing feature of chimera patterns observed in the lattice is that they mainly combine clusters of different chimera types: phase, amplitude and double-well chimeras.

Slow dynamics and regularization phenomena in ensembles of chaotic neurons

NASA Astrophysics Data System (ADS)

Rabinovich, M. I.; Varona, P.; Torres, J. J.; Huerta, R.; Abarbanel, H. D. I.

1999-02-01

We have explored the role of calcium concentration dynamics in the generation of chaos and in the regularization of the bursting oscillations using a minimal neural circuit of two coupled model neurons. In regions of the control parameter space where the slowest component, namely the calcium concentration in the endoplasmic reticulum, weakly depends on the other variables, this model is analogous to three dimensional systems as found in [1] or [2]. These are minimal models that describe the fundamental characteristics of the chaotic spiking-bursting behavior observed in real neurons. We have investigated different regimes of cooperative behavior in large assemblies of such units using lattice of non-identical Hindmarsh-Rose neurons electrically coupled with parameters chosen randomly inside the chaotic region. We study the regularization mechanisms in large assemblies and the development of several spatio-temporal patterns as a function of the interconnectivity among nearest neighbors.

Chaotic system detection of weak seismic signals

NASA Astrophysics Data System (ADS)

Li, Y.; Yang, B. J.; Badal, J.; Zhao, X. P.; Lin, H. B.; Li, R. L.

2009-09-01

When the signal-to-noise (S/N) ratio is less than -3 dB or even 0 dB, seismic events are generally difficult to identify from a common shot record. To overcome this type of problem we present a method to detect weak seismic signals based on the oscillations described by a chaotic dynamic system in phase space. The basic idea is that a non-linear chaotic oscillator is strongly immune to noise. Such a dynamic system is less influenced by noise, but it is more sensitive to periodic signals, changing from a chaotic state to a large-scale periodic phase state when excited by a weak signal. With the purpose of checking the possible contamination of the signal by noise, we have performed a numerical experiment with an oscillator controlled by the Duffing-Holmes equation, taking a distorted Ricker wavelet sequence as input signal. In doing so, we prove that the oscillator system is able to reach a large-scale periodic phase state in a strong noise environment. In the case of a common shot record with low S/N ratio, the onsets reflected from a same interface are similar to one other and can be put on a single trace with a common reference time and the periodicity of the so-generated signal follows as a consequence of moveout at a particular scanning velocity. This operation, which is called `horizontal dynamic correction' and leads to a nearly periodic signal, is implemented on synthetic wavelet sequences taking various sampling arrival times and scanning velocities. Thereafter, two tests, both in a noisy ambient of -3.7 dB, are done using a chaotic oscillator: the first demonstrates the capability of the method to really detect a weak seismic signal; the second takes care of the fundamental weakness of the dynamic correction coming from the use of a particular scanning velocity, which is investigated from the effect caused by near-surface lateral velocity variation on the periodicity of the reconstructed seismic signal. Finally, we have developed an application of the method to real data acquired in seismic prospecting and then converted into pseudo-periodic signals, which has allowed us to discriminate fuzzy waveforms as multiples, thus illustrating in practice the performance of our working scheme.

A chaotic-dynamical conceptual model to describe fluid flow and contaminant transport in a fractured vadose zone. 1997 progress report and presentations at the annual meeting, Ernest Orlando Lawrence Berkeley National Laboratory, December 3--4, 1997

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

Faybishenko, B.; Doughty, C.; Geller, J.

1998-07-01

Understanding subsurface flow and transport processes is critical for effective assessment, decision-making, and remediation activities for contaminated sites. However, for fluid flow and contaminant transport through fractured vadose zones, traditional hydrogeological approaches are often found to be inadequate. In this project, the authors examine flow and transport through a fractured vadose zone as a deterministic chaotic dynamical process, and develop a model of it in these terms. Initially, the authors examine separately the geometric model of fractured rock and the flow dynamics model needed to describe chaotic behavior. Ultimately they will put the geometry and flow dynamics together to developmore » a chaotic-dynamical model of flow and transport in a fractured vadose zone. They investigate water flow and contaminant transport on several scales, ranging from small-scale laboratory experiments in fracture replicas and fractured cores, to field experiments conducted in a single exposed fracture at a basalt outcrop, and finally to a ponded infiltration test using a pond of 7 by 8 m. In the field experiments, they measure the time-variation of water flux, moisture content, and hydraulic head at various locations, as well as the total inflow rate to the subsurface. Such variations reflect the changes in the geometry and physics of water flow that display chaotic behavior, which they try to reconstruct using the data obtained. In the analysis of experimental data, a chaotic model can be used to predict the long-term bounds on fluid flow and transport behavior, known as the attractor of the system, and to examine the limits of short-term predictability within these bounds. This approach is especially well suited to the need for short-term predictions to support remediation decisions and long-term bounding studies. View-graphs from ten presentations made at the annual meeting held December 3--4, 1997 are included in an appendix to this report.« less

An introduction to chaotic and random time series analysis

NASA Technical Reports Server (NTRS)

Scargle, Jeffrey D.

1989-01-01

The origin of chaotic behavior and the relation of chaos to randomness are explained. Two mathematical results are described: (1) a representation theorem guarantees the existence of a specific time-domain model for chaos and addresses the relation between chaotic, random, and strictly deterministic processes; (2) a theorem assures that information on the behavior of a physical system in its complete state space can be extracted from time-series data on a single observable. Focus is placed on an important connection between the dynamical state space and an observable time series. These two results lead to a practical deconvolution technique combining standard random process modeling methods with new embedded techniques.

Chaos as an intermittently forced linear system.

PubMed

Brunton, Steven L; Brunton, Bingni W; Proctor, Joshua L; Kaiser, Eurika; Kutz, J Nathan

2017-05-30

Understanding the interplay of order and disorder in chaos is a central challenge in modern quantitative science. Approximate linear representations of nonlinear dynamics have long been sought, driving considerable interest in Koopman theory. We present a universal, data-driven decomposition of chaos as an intermittently forced linear system. This work combines delay embedding and Koopman theory to decompose chaotic dynamics into a linear model in the leading delay coordinates with forcing by low-energy delay coordinates; this is called the Hankel alternative view of Koopman (HAVOK) analysis. This analysis is applied to the Lorenz system and real-world examples including Earth's magnetic field reversal and measles outbreaks. In each case, forcing statistics are non-Gaussian, with long tails corresponding to rare intermittent forcing that precedes switching and bursting phenomena. The forcing activity demarcates coherent phase space regions where the dynamics are approximately linear from those that are strongly nonlinear.The huge amount of data generated in fields like neuroscience or finance calls for effective strategies that mine data to reveal underlying dynamics. Here Brunton et al.develop a data-driven technique to analyze chaotic systems and predict their dynamics in terms of a forced linear model.

Combinatorial Optimization by Amoeba-Based Neurocomputer with Chaotic Dynamics

NASA Astrophysics Data System (ADS)

Aono, Masashi; Hirata, Yoshito; Hara, Masahiko; Aihara, Kazuyuki

We demonstrate a computing system based on an amoeba of a true slime mold Physarum capable of producing rich spatiotemporal oscillatory behavior. Our system operates as a neurocomputer because an optical feedback control in accordance with a recurrent neural network algorithm leads the amoeba's photosensitive branches to search for a stable configuration concurrently. We show our system's capability of solving the traveling salesman problem. Furthermore, we apply various types of nonlinear time series analysis to the amoeba's oscillatory behavior in the problem-solving process. The results suggest that an individual amoeba might be characterized as a set of coupled chaotic oscillators.

Relevance of deterministic chaos theory to studies in functioning of dynamical systems

NASA Astrophysics Data System (ADS)

Glagolev, S. N.; Bukhonova, S. M.; Chikina, E. D.

2018-03-01

The paper considers chaotic behavior of dynamical systems typical for social and economic processes. Approaches to analysis and evaluation of system development processes are studies from the point of view of controllability and determinateness. Explanations are given for necessity to apply non-standard mathematical tools to explain states of dynamical social and economic systems on the basis of fractal theory. Features of fractal structures, such as non-regularity, self-similarity, dimensionality and fractionality are considered.

Periodic orbit analysis of a system with continuous symmetry—A tutorial

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

Budanur, Nazmi Burak, E-mail: budanur3@gatech.edu; Cvitanović, Predrag; Borrero-Echeverry, Daniel

2015-07-15

Dynamical systems with translational or rotational symmetry arise frequently in studies of spatially extended physical systems, such as Navier-Stokes flows on periodic domains. In these cases, it is natural to express the state of the fluid in terms of a Fourier series truncated to a finite number of modes. Here, we study a 4-dimensional model with chaotic dynamics and SO(2) symmetry similar to those that appear in fluid dynamics problems. A crucial step in the analysis of such a system is symmetry reduction. We use the model to illustrate different symmetry-reduction techniques. The system's relative equilibria are conveniently determined bymore » rewriting the dynamics in terms of a symmetry-invariant polynomial basis. However, for the analysis of its chaotic dynamics, the “method of slices,” which is applicable to very high-dimensional problems, is preferable. We show that a Poincaré section taken on the 'slice' can be used to further reduce this flow to what is for all practical purposes a unimodal map. This enables us to systematically determine all relative periodic orbits and their symbolic dynamics up to any desired period. We then present cycle averaging formulas adequate for systems with continuous symmetry and use them to compute dynamical averages using relative periodic orbits. The convergence of such computations is discussed.« less

Cooling of a Magmatic System Under Thermal Chaotic Mixing

NASA Astrophysics Data System (ADS)

El Omari, Kamal; Le Guer, Yves; Perugini, Diego; Petrelli, Maurizio

2015-07-01

The cooling of a basaltic melt undergoing chaotic advection is studied numerically for a magma with a temperature-dependent viscosity in a two-dimensional (2D) cavity with moving boundary. Different statistical mixing and energy indicators are used to characterize the efficiency of cooling by thermal chaotic mixing. We show that different cooling rates can be obtained during the thermal mixing of a single basaltic magmatic batch undergoing chaotic advection. This process can induce complex temperature patterns inside the magma chamber. The emergence of chaotic dynamics strongly modulates the temperature fields over time and greatly increases the cooling rates. This mechanism has implications for the thermal lifetime of the magmatic body and may favor the appearance of chemical heterogeneities in the igneous system as a result of different crystallization rates. Results from this study also highlight that even a single magma batch can develop, under chaotic thermal advection, complex thermal and therefore compositional patterns resulting from different cooling rates, which can account for some natural features that, to date, have received unsatisfactory explanations, including the production of magmatic enclaves showing completely different cooling histories compared with the host magma, compositional zoning in mineral phases, and the generation of large-scale compositional zoning observed in many plutons worldwide.

Nonlinear Dynamics and Quantum Transport in Small Systems

DTIC Science & Technology

2012-02-22

2.3 Nonlinear wave and chaos in optical metamaterials 2.3.1 Transient chaos in optical metamaterials We investigated the dynamics of light rays in two...equations can be modeled by a set of ordinary differential equations for light rays . We found that transient chaotic dynamics, hyperbolic or nonhyperbolic...are common in optical metamaterial systems. Due to the analogy between light- ray dynamics in metamaterials and the motion of light and matter as

A New Finite-Time Observer for Nonlinear Systems: Applications to Synchronization of Lorenz-Like Systems.

PubMed

Aguilar-López, Ricardo; Mata-Machuca, Juan L

2016-01-01

This paper proposes a synchronization methodology of two chaotic oscillators under the framework of identical synchronization and master-slave configuration. The proposed methodology is based on state observer design under the frame of control theory; the observer structure provides finite-time synchronization convergence by cancelling the upper bounds of the main nonlinearities of the chaotic oscillator. The above is showed via an analysis of the dynamic of the so called synchronization error. Numerical experiments corroborate the satisfactory results of the proposed scheme.

A New Finite-Time Observer for Nonlinear Systems: Applications to Synchronization of Lorenz-Like Systems

PubMed Central

Aguilar-López, Ricardo

2016-01-01

This paper proposes a synchronization methodology of two chaotic oscillators under the framework of identical synchronization and master-slave configuration. The proposed methodology is based on state observer design under the frame of control theory; the observer structure provides finite-time synchronization convergence by cancelling the upper bounds of the main nonlinearities of the chaotic oscillator. The above is showed via an analysis of the dynamic of the so called synchronization error. Numerical experiments corroborate the satisfactory results of the proposed scheme. PMID:27738651

Dynamical singularities of glassy systems in a quantum quench.

PubMed

Obuchi, Tomoyuki; Takahashi, Kazutaka

2012-11-01

We present a prototype of behavior of glassy systems driven by quantum dynamics in a quenching protocol by analyzing the random energy model in a transverse field. We calculate several types of dynamical quantum amplitude and find a freezing transition at some critical time. The behavior is understood by the partition-function zeros in the complex temperature plane. We discuss the properties of the freezing phase as a dynamical chaotic phase, which are contrasted to those of the spin-glass phase in the static system.

Nonlinear analysis of dynamic signature

NASA Astrophysics Data System (ADS)

Rashidi, S.; Fallah, A.; Towhidkhah, F.

2013-12-01

Signature is a long trained motor skill resulting in well combination of segments like strokes and loops. It is a physical manifestation of complex motor processes. The problem, generally stated, is that how relative simplicity in behavior emerges from considerable complexity of perception-action system that produces behavior within an infinitely variable biomechanical and environmental context. To solve this problem, we present evidences which indicate that motor control dynamic in signing process is a chaotic process. This chaotic dynamic may explain a richer array of time series behavior in motor skill of signature. Nonlinear analysis is a powerful approach and suitable tool which seeks for characterizing dynamical systems through concepts such as fractal dimension and Lyapunov exponent. As a result, they can be analyzed in both horizontal and vertical for time series of position and velocity. We observed from the results that noninteger values for the correlation dimension indicates low dimensional deterministic dynamics. This result could be confirmed by using surrogate data tests. We have also used time series to calculate the largest Lyapunov exponent and obtain a positive value. These results constitute significant evidence that signature data are outcome of chaos in a nonlinear dynamical system of motor control.

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.

Initial condition-dependent dynamics and transient period in memristor-based hypogenetic jerk system with four line equilibria

NASA Astrophysics Data System (ADS)

Bao, Han; Wang, Ning; Bao, Bocheng; Chen, Mo; Jin, Peipei; Wang, Guangyi

2018-04-01

Memristor-based nonlinear dynamical system easily presents the initial condition-dependent dynamical phenomenon of extreme multistability, i.e., coexisting infinitely many attractors, which has been received much attention in recent years. By introducing an ideal and active flux-controlled memristor into an existing hypogenetic chaotic jerk system, an interesting memristor-based chaotic system with hypogenetic jerk equation and circuit forms is proposed. The most striking feature is that this system has four line equilibria and exhibits the extreme multistability phenomenon of coexisting infinitely many attractors. Stability of these line equilibria are analyzed, and coexisting infinitely many attractors' behaviors with the variations of the initial conditions are investigated by bifurcation diagrams, Lyapunov exponent spectra, attraction basins, and phased portraits, upon which the forming mechanism of extreme multistablity in the memristor-based hypogenetic jerk system is explored. Specially, unusual transition behavior of long term transient period with steady chaos, completely different from the phenomenon of transient chaos, can be also found for some initial conditions. Moreover, a hardware circuit is design and fabricated and its experimental results effectively verify the truth of extreme multistablity.

RP and RQA Analysis for Floating Potential Fluctuations in a DC Magnetron Sputtering Plasma

NASA Astrophysics Data System (ADS)

Sabavath, Gopikishan; Banerjee, I.; Mahapatra, S. K.

2016-04-01

The nonlinear dynamics of a direct current magnetron sputtering plasma is visualized using recurrence plot (RP) technique. RP comprises the recurrence quantification analysis (RQA) which is an efficient method to observe critical regime transitions in dynamics. Further, RQA provides insight information about the system’s behavior. We observed the floating potential fluctuations of the plasma as a function of discharge voltage by using Langmuir probe. The system exhibits quasi-periodic-chaotic-quasi-periodic-chaotic transitions. These transitions are quantified from determinism, Lmax, and entropy of RQA. Statistical investigations like kurtosis and skewness also studied for these transitions which are in well agreement with RQA results.

Synchronization in counter-rotating oscillators.

PubMed

Bhowmick, Sourav K; Ghosh, Dibakar; Dana, Syamal K

2011-09-01

An oscillatory system can have opposite senses of rotation, clockwise or anticlockwise. We present a general mathematical description of how to obtain counter-rotating oscillators from the definition of a dynamical system. A type of mixed synchronization emerges in counter-rotating oscillators under diffusive scalar coupling when complete synchronization and antisynchronization coexist in different state variables. We present numerical examples of limit cycle van der Pol oscillator and chaotic Rössler and Lorenz systems. Stability conditions of mixed synchronization are analytically obtained for both Rössler and Lorenz systems. Experimental evidences of counter-rotating limit cycle and chaotic oscillators and mixed synchronization are given in electronic circuits.

Solar System Chaos and its climatic and biogeochemical consequences

NASA Astrophysics Data System (ADS)

Ikeda, M.; Tada, R.; Ozaki, K.; Olsen, P. E.

2017-12-01

Insolation changes caused by changes in Earth's orbital parameters are the main driver of climatic variations, whose pace has been used for astronomically-calibrated geologic time scales of high accuracy to understand Earth system dynamics. However, the astrophysical models beyond several tens of million years ago have large uncertainty due to chaotic behavior of the Solar System, and its impact on amplitude modulation of multi-Myr-scale orbital variations and consequent climate changes has become the subject of debate. Here we show the geologic constraints on the past chaotic behavior of orbital cycles from early Mesozoic monsoon-related records; the 30-Myr-long lake level records of the lacustrine sequence in Newark-Hartford basins (North America) and 70-Myr-long biogenic silica (BSi) burial flux record of pelagic deep-sea chert sequence in Inuyama area (Japan). BSi burial flux of chert could be considered as proportional to the dissolved Si (DSi) input from chemical weathering on timescales longer than the residence time of DSi ( 100 kyr), because chert could represent a major sink for oceanic dissolved silica (Ikeda et al., 2017).These geologic records show multi-Myr cycles with similar frequency modulations of eccentricity solution of astronomical model La2010d (Laskar et al., 2011) compared with other astronomical solutions, but not exactly same. Our geologic records provide convincing evidence for the past chaotic dynamical behaviour of the Solar System and new and challenging additional constraints for astrophysical models. In addition, we find that ˜10 Myr cycle detected in monsoon proxies and their amplitude modulation of ˜2 Myr cycle may be related to the amplitude modulation of ˜2 Myr eccentricity cycle through non-linear process(es) of Earth system dynamics, suggesting possible impact of the chaotic behavior of Solar planets on climate change. Further impact of multi-Myr orbital cycles on global biogeochemical cycles will be discussed.

Chaos as a psychological construct: historical roots, principal findings, and current growth directions.

PubMed

Guastello, Stephen J

2009-07-01

The landmarks in the use of chaos and related constructs in psychology were entwined with the growing use of other nonlinear dynamical constructs, especially catastrophes and self-organization. The growth in substantive applications of chaos in psychology is partially related to the development of methodologies that work within the constraints of psychological data. The psychological literature includes rigorous theory with testable propositions, lighter-weight metaphorical uses of the construct, and colloquial uses of "chaos" with no particular theoretical intent. The current state of the chaos construct and supporting empirical research in psychological theory is summarized in neuroscience, psychophysics, psychomotor skill and other learning phenomena, clinical and abnormal psychology, and group dynamics and organizational behavior. Trends indicate that human systems do not remain chaotic indefinitely; they eventually self-organize, and the concept of the complex adaptive system has become prominent. Chaotic turbulence is generally higher in healthy systems compared to unhealthy systems, although opposite appears true in mood disorders. Group dynamics research shows trends consistent with the complex adaptive system, whereas organizational behavior lags behind in empirical studies relative to the quantity of its theory. Future directions for research involving the chaos construct and other nonlinear dynamics are outlined.

Role of Alternative Food in Controlling Chaotic Dynamics in a Predator-Prey Model with Disease in the Predator

NASA Astrophysics Data System (ADS)

Das, Krishna Pada; Bairagi, Nandadulal; Sen, Prabir

It is generally, but not always, accepted that alternative food plays a stabilizing role in predator-prey interaction. Parasites, on the other hand, have the ability to change both the qualitative and quantitative dynamics of its host population. In recent times, researchers are showing growing interest in formulating models that integrate both the ecological and epidemiological aspects. The present paper deals with the effect of alternative food on a predator-prey system with disease in the predator population. We show that the system, in the absence of alternative food, exhibits different dynamics viz. stable coexistence, limit cycle oscillations, period-doubling bifurcation and chaos when infection rate is gradually increased. However, when predator consumes alternative food coupled with its focal prey, the system returns to regular oscillatory state from chaotic state through period-halving bifurcations. Our study shows that alternative food may have larger impact on the community structure and may increase population persistence.

Perception-action map learning in controlled multiscroll systems applied to robot navigation.

PubMed

Arena, Paolo; De Fiore, Sebastiano; Fortuna, Luigi; Patané, Luca

2008-12-01

In this paper a new technique for action-oriented perception in robots is presented. The paper starts from exploiting the successful implementation of the basic idea that perceptual states can be embedded into chaotic attractors whose dynamical evolution can be associated with sensorial stimuli. In this way, it can be possible to encode, into the chaotic dynamics, environment-dependent patterns. These have to be suitably linked to an action, executed by the robot, to fulfill an assigned mission. This task is addressed here: the action-oriented perception loop is closed by introducing a simple unsupervised learning stage, implemented via a bio-inspired structure based on the motor map paradigm. In this way, perceptual meanings, useful for solving a given task, can be autonomously learned, based on the environment-dependent patterns embedded into the controlled chaotic dynamics. The presented framework has been tested on a simulated robot and the performance have been successfully compared with other traditional navigation control paradigms. Moreover an implementation of the proposed architecture on a Field Programmable Gate Array is briefly outlined and preliminary experimental results on a roving robot are also reported.

Parametric Identification of Nonlinear Dynamical Systems

NASA Technical Reports Server (NTRS)

Feeny, Brian

2002-01-01

In this project, we looked at the application of harmonic balancing as a tool for identifying parameters (HBID) in a nonlinear dynamical systems with chaotic responses. The main idea is to balance the harmonics of periodic orbits extracted from measurements of each coordinate during a chaotic response. The periodic orbits are taken to be approximate solutions to the differential equations that model the system, the form of the differential equations being known, but with unknown parameters to be identified. Below we summarize the main points addressed in this work. The details of the work are attached as drafts of papers, and a thesis, in the appendix. Our study involved the following three parts: (1) Application of the harmonic balance to a simulation case in which the differential equation model has known form for its nonlinear terms, in contrast to a differential equation model which has either power series or interpolating functions to represent the nonlinear terms. We chose a pendulum, which has sinusoidal nonlinearities; (2) Application of the harmonic balance to an experimental system with known nonlinear forms. We chose a double pendulum, for which chaotic response were easily generated. Thus we confronted a two-degree-of-freedom system, which brought forth challenging issues; (3) A study of alternative reconstruction methods. The reconstruction of the phase space is necessary for the extraction of periodic orbits from the chaotic responses, which is needed in this work. Also, characterization of a nonlinear system is done in the reconstructed phase space. Such characterizations are needed to compare models with experiments. Finally, some nonlinear prediction methods can be applied in the reconstructed phase space. We developed two reconstruction methods that may be considered if the common method (method of delays) is not applicable.

Reconstruction of the dynamics of the climatic system from time-series data

PubMed Central

Nicolis, C.; Nicolis, G.

1986-01-01

The oxygen isotope record of the last million years, as provided by a deep sea core sediment, is analyzed by a method recently developed in the theory of dynamical systems. The analysis suggests that climatic variability is the manifestation of a chaotic dynamics described by an attractor of fractal dimensionality. A quantitative measure of the limited predictability of the climatic system is provided by the evaluation of the time-correlation function and the largest positive Lyapounov exponent of the system. PMID:16593650

Cross-entropy optimization for neuromodulation.

PubMed

Brar, Harleen K; Yunpeng Pan; Mahmoudi, Babak; Theodorou, Evangelos A

2016-08-01

This study presents a reinforcement learning approach for the optimization of the proportional-integral gains of the feedback controller represented in a computational model of epilepsy. The chaotic oscillator model provides a feedback control systems view of the dynamics of an epileptic brain with an internal feedback controller representative of the natural seizure suppression mechanism within the brain circuitry. Normal and pathological brain activity is simulated in this model by adjusting the feedback gain values of the internal controller. With insufficient gains, the internal controller cannot provide enough feedback to the brain dynamics causing an increase in correlation between different brain sites. This increase in synchronization results in the destabilization of the brain dynamics, which is representative of an epileptic seizure. To provide compensation for an insufficient internal controller an external controller is designed using proportional-integral feedback control strategy. A cross-entropy optimization algorithm is applied to the chaotic oscillator network model to learn the optimal feedback gains for the external controller instead of hand-tuning the gains to provide sufficient control to the pathological brain and prevent seizure generation. The correlation between the dynamics of neural activity within different brain sites is calculated for experimental data to show similar dynamics of epileptic neural activity as simulated by the network of chaotic oscillators.

Emergent patterns in interacting neuronal sub-populations

NASA Astrophysics Data System (ADS)

Kamal, Neeraj Kumar; Sinha, Sudeshna

2015-05-01

We investigate an ensemble of coupled model neurons, consisting of groups of varying sizes and intrinsic dynamics, ranging from periodic to chaotic, where the inter-group coupling interaction is effectively like a dynamic signal from a different sub-population. We observe that the minority group can significantly influence the majority group. For instance, when a small chaotic group is coupled to a large periodic group, the chaotic group de-synchronizes. However, counter-intuitively, when a small periodic group couples strongly to a large chaotic group, it leads to complete synchronization in the majority chaotic population, which also spikes at the frequency of the small periodic group. It then appears that the small group of periodic neurons can act like a pacemaker for the whole network. Further, we report the existence of varied clustering patterns, ranging from sets of synchronized clusters to anti-phase clusters, governed by the interplay of the relative sizes and dynamics of the sub-populations. So these results have relevance in understanding how a group can influence the synchrony of another group of dynamically different elements, reminiscent of event-related synchronization/de-synchronization in complex networks.

Modeling the Physics of Sliding Objects on Rotating Space Elevators and Other Non-relativistic Strings

NASA Astrophysics Data System (ADS)

Golubovic, Leonardo; Knudsen, Steven

2017-01-01

We consider general problem of modeling the dynamics of objects sliding on moving strings. We introduce a powerful computational algorithm that can be used to investigate the dynamics of objects sliding along non-relativistic strings. We use the algorithm to numerically explore fundamental physics of sliding climbers on a unique class of dynamical systems, Rotating Space Elevators (RSE). Objects sliding along RSE strings do not require internal engines or propulsion to be transported from the Earth's surface into outer space. By extensive numerical simulations, we find that sliding climbers may display interesting non-linear dynamics exhibiting both quasi-periodic and chaotic states of motion. While our main interest in this study is in the climber dynamics on RSEs, our results for the dynamics of sliding object are of more general interest. In particular, we designed tools capable of dealing with strongly nonlinear phenomena involving moving strings of any kind, such as the chaotic dynamics of sliding climbers observed in our simulations.

Generalized Gaussian wave packet dynamics: Integrable and chaotic systems.

PubMed

Pal, Harinder; Vyas, Manan; Tomsovic, Steven

2016-01-01

The ultimate semiclassical wave packet propagation technique is a complex, time-dependent Wentzel-Kramers-Brillouin method known as generalized Gaussian wave packet dynamics (GGWPD). It requires overcoming many technical difficulties in order to be carried out fully in practice. In its place roughly twenty years ago, linearized wave packet dynamics was generalized to methods that include sets of off-center, real trajectories for both classically integrable and chaotic dynamical systems that completely capture the dynamical transport. The connections between those methods and GGWPD are developed in a way that enables a far more practical implementation of GGWPD. The generally complex saddle-point trajectories at its foundation are found using a multidimensional Newton-Raphson root search method that begins with the set of off-center, real trajectories. This is possible because there is a one-to-one correspondence. The neighboring trajectories associated with each off-center, real trajectory form a path that crosses a unique saddle; there are exceptions that are straightforward to identify. The method is applied to the kicked rotor to demonstrate the accuracy improvement as a function of ℏ that comes with using the saddle-point trajectories.

A chaotic view of behavior change: a quantum leap for health promotion.

PubMed

Resnicow, Ken; Vaughan, Roger

2006-09-12

The study of health behavior change, including nutrition and physical activity behaviors, has been rooted in a cognitive-rational paradigm. Change is conceptualized as a linear, deterministic process where individuals weigh pros and cons, and at the point at which the benefits outweigh the cost change occurs. Consistent with this paradigm, the associated statistical models have almost exclusively assumed a linear relationship between psychosocial predictors and behavior. Such a perspective however, fails to account for non-linear, quantum influences on human thought and action. Consider why after years of false starts and failed attempts, a person succeeds at increasing their physical activity, eating healthier or losing weight. Or, why after years of success a person relapses. This paper discusses a competing view of health behavior change that was presented at the 2006 annual ISBNPA meeting in Boston. Rather than viewing behavior change from a linear perspective it can be viewed as a quantum event that can be understood through the lens of Chaos Theory and Complex Dynamic Systems. Key principles of Chaos Theory and Complex Dynamic Systems relevant to understanding health behavior change include: 1) Chaotic systems can be mathematically modeled but are nearly impossible to predict; 2) Chaotic systems are sensitive to initial conditions; 3) Complex Systems involve multiple component parts that interact in a nonlinear fashion; and 4) The results of Complex Systems are often greater than the sum of their parts. Accordingly, small changes in knowledge, attitude, efficacy, etc may dramatically alter motivation and behavioral outcomes. And the interaction of such variables can yield almost infinite potential patterns of motivation and behavior change. In the linear paradigm unaccounted for variance is generally relegated to the catch all "error" term, when in fact such "error" may represent the chaotic component of the process. The linear and chaotic paradigms are however, not mutually exclusive, as behavior change may include both chaotic and cognitive processes. Studies of addiction suggest that many decisions to change are quantum rather than planned events; motivation arrives as opposed to being planned. Moreover, changes made through quantum processes appear more enduring than those that involve more rational, planned processes. How such processes may apply to nutrition and physical activity behavior and related interventions merits examination.

Multi-piecewise quadratic nonlinearity memristor and its 2N-scroll and 2N + 1-scroll chaotic attractors system.

PubMed

Wang, Chunhua; Liu, Xiaoming; Xia, Hu

2017-03-01

In this paper, two kinds of novel ideal active flux-controlled smooth multi-piecewise quadratic nonlinearity memristors with multi-piecewise continuous memductance function are presented. The pinched hysteresis loop characteristics of the two memristor models are verified by building a memristor emulator circuit. Using the two memristor models establish a new memristive multi-scroll Chua's circuit, which can generate 2N-scroll and 2N+1-scroll chaotic attractors without any other ordinary nonlinear function. Furthermore, coexisting multi-scroll chaotic attractors are found in the proposed memristive multi-scroll Chua's circuit. Phase portraits, Lyapunov exponents, bifurcation diagrams, and equilibrium point analysis have been used to research the basic dynamics of the memristive multi-scroll Chua's circuit. The consistency of circuit implementation and numerical simulation verifies the effectiveness of the system design.

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

Dmitriev, Alexander S.; Yemelyanov, Ruslan Yu.; Moscow Institute of Physics and Technology

The paper deals with a new multi-element processor platform assigned for modelling the behaviour of interacting dynamical systems, i.e., active wireless network. Experimentally, this ensemble is implemented in an active network, the active nodes of which include direct chaotic transceivers and special actuator boards containing microcontrollers for modelling the dynamical systems and an information display unit (colored LEDs). The modelling technique and experimental results are described and analyzed.

Impact of an irregular friction formulation on dynamics of a minimal model for brake squeal

NASA Astrophysics Data System (ADS)

Stender, Merten; Tiedemann, Merten; Hoffmann, Norbert; Oberst, Sebastian

2018-07-01

Friction-induced vibrations are of major concern in the design of reliable, efficient and comfortable technical systems. Well-known examples for systems susceptible to self-excitation can be found in fluid structure interaction, disk brake squeal, rotor dynamics, hip implants noise and many more. While damping elements and amplitude reduction are well-understood in linear systems, nonlinear systems and especially self-excited dynamics still constitute a challenge for damping element design. Additionally, complex dynamical systems exhibit deterministic chaotic cores which add severe sensitivity to initial conditions to the system response. Especially the complex friction interface dynamics remain a challenging task for measurements and modeling. Today, mostly simple and regular friction models are investigated in the field of self-excited brake system vibrations. This work aims at investigating the effect of high-frequency irregular interface dynamics on the nonlinear dynamical response of a self-excited structure. Special focus is put on the characterization of the system response time series. A low-dimensional minimal model is studied which features self-excitation, gyroscopic effects and friction-induced damping. Additionally, the employed friction formulation exhibits temperature as inner variable and superposed chaotic fluctuations governed by a Lorenz attractor. The time scale of the irregular fluctuations is chosen one order smaller than the overall system dynamics. The influence of those fluctuations on the structural response is studied in various ways, i.e. in time domain and by means of recurrence analysis. The separate time scales are studied in detail and regimes of dynamic interactions are identified. The results of the irregular friction formulation indicate dynamic interactions on multiple time scales, which trigger larger vibration amplitudes as compared to regular friction formulations conventionally studied in the field of friction-induced vibrations.

Characterization of nonstationary chaotic systems

NASA Astrophysics Data System (ADS)

Serquina, Ruth; Lai, Ying-Cheng; Chen, Qingfei

2008-02-01

Nonstationary dynamical systems arise in applications, but little has been done in terms of the characterization of such systems, as most standard notions in nonlinear dynamics such as the Lyapunov exponents and fractal dimensions are developed for stationary dynamical systems. We propose a framework to characterize nonstationary dynamical systems. A natural way is to generate and examine ensemble snapshots using a large number of trajectories, which are capable of revealing the underlying fractal properties of the system. By defining the Lyapunov exponents and the fractal dimension based on a proper probability measure from the ensemble snapshots, we show that the Kaplan-Yorke formula, which is fundamental in nonlinear dynamics, remains valid most of the time even for nonstationary dynamical systems.

Qualitative and quantitative behaviour of planetary systems; Proceedings of the 3rd Alexander von Humboldt Colloquium on Celestial Mechanics, Ramsau, Austria, Mar. 29-Apr. 4, 1992

NASA Astrophysics Data System (ADS)

Dvorak, R.; Henrard, J.

1993-06-01

Topics addressed include planetary theories, the Sitnikov problem, asteroids, resonance, general dynamical systems, and chaos and stability. Particular attention is given to recent progress in the theory and application of symplectic integrators, a computer-aided analysis of the Sitnikov problem, the chaotic behavior of trajectories for the asteroidal resonances, and the resonant motion in the restricted three-body problem. Also discussed are the second order long-period motion of Hyperion, meteorites from the asteroid 6 Hebe, and least squares parameter estimation in chaotic differential equations.

Dynamic analysis and electronic circuit implementation of a novel 3D autonomous system without linear terms

NASA Astrophysics Data System (ADS)

Kengne, J.; Jafari, S.; Njitacke, Z. T.; Yousefi Azar Khanian, M.; Cheukem, A.

2017-11-01

Mathematical models (ODEs) describing the dynamics of almost all continuous time chaotic nonlinear systems (e.g. Lorenz, Rossler, Chua, or Chen system) involve at least a nonlinear term in addition to linear terms. In this contribution, a novel (and singular) 3D autonomous chaotic system without linear terms is introduced. This system has an especial feature of having two twin strange attractors: one ordinary and one symmetric strange attractor when the time is reversed. The complex behavior of the model is investigated in terms of equilibria and stability, bifurcation diagrams, Lyapunov exponent plots, time series and Poincaré sections. Some interesting phenomena are found including for instance, period-doubling bifurcation, antimonotonicity (i.e. the concurrent creation and annihilation of periodic orbits) and chaos while monitoring the system parameters. Compared to the (unique) case previously reported by Xu and Wang (2014) [31], the system considered in this work displays a more 'elegant' mathematical expression and experiences richer dynamical behaviors. A suitable electronic circuit (i.e. the analog simulator) is designed and used for the investigations. Pspice based simulation results show a very good agreement with the theoretical analysis.

The Six Fundamental Characteristics of Chaos and Their Clinical Relevance to Psychiatry: a New Hypothesis for the Origin of Psychosis

NASA Astrophysics Data System (ADS)

Schmid, Gary Bruno

Underlying idea: A new hypothesis about how the mental state of psychosis may arise in the brain as a "linear" information processing pathology is briefly introduced. This hypothesis is proposed in the context of a complementary approach to psychiatry founded in the logical paradigm of chaos theory. To best understand the relation between chaos theory and psychiatry, the semantic structure of chaos theory is analyzed with the help of six general, and six specific, fundamental characteristics which can be directly inferred from empirical observations on chaotic systems. This enables a mathematically and physically stringent perspective on psychological phenomena which until now could only be grasped intuitively: Chaotic systems are in a general sense dynamic, intrinsically coherent, deterministic, recursive, reactive and structured: in a specific sense, self-organizing, unpredictable, nonreproducible, triadic, unstable and self-similar. To a great extent, certain concepts of chaos theory can be associated with corresponding concepts in psychiatry, psychology and psychotherapy, thus enabling an understanding of the human psyche in general as a (fractal) chaotic system and an explanation of certain mental developments, such as the course of schizophrenia, the course of psychosis and psychotherapy as chaotic processes. General overview: A short comparison and contrast of classical and chaotic physical theory leads to four postulates and one hypothesis motivating a new, dynamic, nonlinear approach to classical, causal psychiatry: Process-Oriented PSYchiatry or "POPSY", for short. Four aspects of the relationship between chaos theory and POPSY are discussed: (1) The first of these, namely, Identification of Chaos / Picture of Illness involves a definition of Chaos / Psychosis and a discussion of the 6 logical characteristics of each. This leads to the concept of dynamical disease (definition, characteristics and examples) and to the idea of "psychological disturbance as dynamical illness". On the one hand, it is argued that the developmental course of psychosis is chaotic. On the other hand, we propose the hypothesis that the mental state of psychosis may be a linear information processing pathology. (2) The second aspect under discussion is the Assessment of Chaos / Diagnosis of Illness. In order to better understand how POPSY research treats this aspect, we take a look at the 3 different classes of (non-quantum) motion as models of 3 different possible courses of illness and outline present-day methods available for the quantitative assessment of chaotic (fractal) motion. (3) The third aspect, namely. Prediction of Chaos / Prognosis of Illness considers how each of these 3 classes of motion implies a different way of looking into the future: linear-causal, statistical and nonlinear-fractal, respectively (4) The fourth aspect of the relationship between chaos theory and POPSY, Control of Chaos / Treatment of Illness, is shown to have certain implications to complementary medicine. This paper completes with a short summary, conclusion and a closing remark.

Future missions studies: Combining Schatten's solar activity prediction model with a chaotic prediction model

NASA Technical Reports Server (NTRS)

Ashrafi, S.

1991-01-01

K. Schatten (1991) recently developed a method for combining his prediction model with our chaotic model. The philosophy behind this combined model and his method of combination is explained. Because the Schatten solar prediction model (KS) uses a dynamo to mimic solar dynamics, accurate prediction is limited to long-term solar behavior (10 to 20 years). The Chaotic prediction model (SA) uses the recently developed techniques of nonlinear dynamics to predict solar activity. It can be used to predict activity only up to the horizon. In theory, the chaotic prediction should be several orders of magnitude better than statistical predictions up to that horizon; beyond the horizon, chaotic predictions would theoretically be just as good as statistical predictions. Therefore, chaos theory puts a fundamental limit on predictability.

Deterministic chaotic dynamics of Raba River flow (Polish Carpathian Mountains)

NASA Astrophysics Data System (ADS)

Kędra, Mariola

2014-02-01

Is the underlying dynamics of river flow random or deterministic? If it is deterministic, is it deterministic chaotic? This issue is still controversial. The application of several independent methods, techniques and tools for studying daily river flow data gives consistent, reliable and clear-cut results to the question. The outcomes point out that the investigated discharge dynamics is not random but deterministic. Moreover, the results completely confirm the nonlinear deterministic chaotic nature of the studied process. The research was conducted on daily discharge from two selected gauging stations of the mountain river in southern Poland, the Raba River.

Conductance fluctuations in high mobility monolayer graphene: Nonergodicity, lack of determinism and chaotic behavior

PubMed Central

da Cunha, C. R.; Mineharu, M.; Matsunaga, M.; Matsumoto, N.; Chuang, C.; Ochiai, Y.; Kim, G.-H.; Watanabe, K.; Taniguchi, T.; Ferry, D. K.; Aoki, N.

2016-01-01

We have fabricated a high mobility device, composed of a monolayer graphene flake sandwiched between two sheets of hexagonal boron nitride. Conductance fluctuations as functions of a back gate voltage and magnetic field were obtained to check for ergodicity. Non-linear dynamics concepts were used to study the nature of these fluctuations. The distribution of eigenvalues was estimated from the conductance fluctuations with Gaussian kernels and it indicates that the carrier motion is chaotic at low temperatures. We argue that a two-phase dynamical fluid model best describes the transport in this system and can be used to explain the violation of the so-called ergodic hypothesis found in graphene. PMID:27609184

Conductance fluctuations in high mobility monolayer graphene: Nonergodicity, lack of determinism and chaotic behavior.

PubMed

da Cunha, C R; Mineharu, M; Matsunaga, M; Matsumoto, N; Chuang, C; Ochiai, Y; Kim, G-H; Watanabe, K; Taniguchi, T; Ferry, D K; Aoki, N

2016-09-09

We have fabricated a high mobility device, composed of a monolayer graphene flake sandwiched between two sheets of hexagonal boron nitride. Conductance fluctuations as functions of a back gate voltage and magnetic field were obtained to check for ergodicity. Non-linear dynamics concepts were used to study the nature of these fluctuations. The distribution of eigenvalues was estimated from the conductance fluctuations with Gaussian kernels and it indicates that the carrier motion is chaotic at low temperatures. We argue that a two-phase dynamical fluid model best describes the transport in this system and can be used to explain the violation of the so-called ergodic hypothesis found in graphene.

Chaos in the brain: imaging via chaoticity of EEG/MEG signals

NASA Astrophysics Data System (ADS)

Kowalik, Zbigniew J.; Elbert, Thomas; Rockstroh, Brigitte; Hoke, Manfried

1995-03-01

Brain electro- (EEG) or magnetoencephalogram (MEG) can be analyzed by using methods of the nonlinear system theory. We show that even for very short and nonstationary time series it is possible to functionally differentiate various brain activities. Usually the analysis assumes that the analyzed signals are both long and stationary, so that the classic spectral methods can be used. Even more convincing results can be obtained under these circumstances when the dimensional analysis or estimation of the Kolmogorov entropy or the Lyapunov exponent are performed. When measuring the spontaneous activity of a human brain the assumption of stationarity is questionable and `static' methods (correlation dimension, entropy, etc.) are then not adequate. In this case `dynamic' methods like pointwise-D2 dimension or chaoticity measures should be applied. Predictability measures in the form of local Lyapunov exponents are capable of revealing directly the chaoticity of a given process, and can practically be applied for functional differentiation of brain activity. We exemplify these in cases of apallic syndrome, tinnitus and schizophrenia. We show that: the average chaoticity in apallic syndrome differentiates brain states both in space and time, chaoticity changes temporally in case of schizophrenia (critical jumps of chaoticity), chaoticity changes locally in space, i.e., in the cortex plane in case of tinnitus.

Design and simulation of the micromixer with chaotic advection in twisted microchannels.

PubMed

Jen, Chun-Ping; Wu, Chung-Yi; Lin, Yu-Cheng; Wu, Ching-Yi

2003-05-01

Chaotic mixers with twisted microchannels were designed and simulated numerically in the present study. The phenomenon whereby a simple Eulerian velocity field may generate a chaotic response in the distribution of a Lagrangian marker is termed chaotic advection. Dynamic system theory indicates that chaotic particle motion can occur when a velocity field is either two-dimensional and time-dependent, or three-dimensional. In the present study, micromixers with three-dimensional structures of the twisted microchannel were designed in order to induce chaotic mixing. In addition to the basic T-mixer, three types of micromixers with inclined, oblique and wavelike microchannels were investigated. In the design of each twisted microchannel, the angle of the channels' bottoms alternates in each subsection. When the fluids enter the twisted microchannels, the flow sways around the varying structures within the microchannels. The designs of the twisted microchannels provide a third degree of freedom to the flow field in the microchannel. Therefore, chaotic regimes that lead to chaotic mixing may arise. The numerical results indicate that mixing occurs in the main channel and progressively larger mixing lengths are required as the Peclet number increased. The swaying of the flow in the twisted microchannel causes chaotic advection. Among the four micromixer designs, the micromixer with the inclined channel most improved mixing. Furthermore, using the inclined mixer with six subsections yielded optimum performance, decreasing the mixing length by up to 31% from that of the basic T-mixer.

Denoising of chaotic signal using independent component analysis and empirical mode decomposition with circulate translating

NASA Astrophysics Data System (ADS)

Wen-Bo, Wang; Xiao-Dong, Zhang; Yuchan, Chang; Xiang-Li, Wang; Zhao, Wang; Xi, Chen; Lei, Zheng

2016-01-01

In this paper, a new method to reduce noises within chaotic signals based on ICA (independent component analysis) and EMD (empirical mode decomposition) is proposed. The basic idea is decomposing chaotic signals and constructing multidimensional input vectors, firstly, on the base of EMD and its translation invariance. Secondly, it makes the independent component analysis on the input vectors, which means that a self adapting denoising is carried out for the intrinsic mode functions (IMFs) of chaotic signals. Finally, all IMFs compose the new denoised chaotic signal. Experiments on the Lorenz chaotic signal composed of different Gaussian noises and the monthly observed chaotic sequence on sunspots were put into practice. The results proved that the method proposed in this paper is effective in denoising of chaotic signals. Moreover, it can correct the center point in the phase space effectively, which makes it approach the real track of the chaotic attractor. Project supported by the National Science and Technology, China (Grant No. 2012BAJ15B04), the National Natural Science Foundation of China (Grant Nos. 41071270 and 61473213), the Natural Science Foundation of Hubei Province, China (Grant No. 2015CFB424), the State Key Laboratory Foundation of Satellite Ocean Environment Dynamics, China (Grant No. SOED1405), the Hubei Provincial Key Laboratory Foundation of Metallurgical Industry Process System Science, China (Grant No. Z201303), and the Hubei Key Laboratory Foundation of Transportation Internet of Things, Wuhan University of Technology, China (Grant No.2015III015-B02).

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

Osovski, Shmuel; Moiseyev, Nimrod

The recent pioneering experiments of the [Nature 412, 52 (2001)] and [Science, 293, 274 (2001)] groups have demonstrated the dynamical tunneling of cold atoms interacting with standing electromagnetic waves. It has been shown [Phys. Rev. Lett. 89, 253201 (2002)], that the tunneling oscillations observed in these experiments correspondingly stems from two- and three-Floquet quantum state mechanism and can be controlled by varying the experimental parameters. The question where are the fingerprints of the classical chaotic dynamics in a quantum dynamical process which is controlled by 2 or 3 quantum states remains open. Our calculations show that although the effective ({Dirac_h}/2{pi})more » associated with the two experiments is large, and the quantum system is far from its semiclassical limit, the quantum Floquet-Bloch quasienergy states still can be classified as regular and chaotic states. In both experiments the quantum and the classical phase-space entropies are quite similar, although the classical phase space is a mixed regular-chaotic space. It is also shown that as the wave packet which is initially localized at one of the two inner regular islands oscillates between them through the chaotic sea, it accumulates a random phase which causes the decay of the amplitude of the oscillating mean momentum, , as measured in both experiments. The extremely high sensitivity of the rate of decay of the oscillations of to the very small changes in the population of different Floquet-Bloch states, is another type of fingerprint of chaos in quantum dynamics that presumably was measured in the NIST and AUSTIN experiments for the first time.« less

ORIGIN OF THE CHAOTIC MOTION OF THE SATURNIAN SATELLITE ATLAS

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

Renner, S.; Vienne, A.; Cooper, N. J.

2016-05-01

We revisit the dynamics of Atlas. Using Cassini ISS astrometric observations spanning 2004 February to 2013 August, Cooper et al. found evidence that Atlas is currently perturbed by both a 54:53 corotation eccentricity resonance (CER) and a 54:53 Lindblad eccentricity resonance (LER) with Prometheus. They demonstrated that the orbit of Atlas is chaotic, with a Lyapunov time of order 10 years, as a direct consequence of the coupled resonant interaction (CER/LER) with Prometheus. Here we investigate the interactions between the two resonances using the CoraLin analytical model, showing that the chaotic zone fills almost all the corotation sites occupied bymore » the satellite's orbit. Four 70:67 apse-type mean motion resonances with Pandora are also overlapping, but these resonances have a much weaker effect. Frequency analysis allows us to highlight the coupling between the 54:53 resonances, and confirms that a simplified system including the perturbations due to Prometheus and Saturn's oblateness only captures the essential features of the dynamics.« less

Origin of the Chaotic Motion of the Saturnian Satellite Atlas

NASA Astrophysics Data System (ADS)

Renner, S.; Cooper, N. J.; El Moutamid, M.; Sicardy, B.; Vienne, A.; Murray, C. D.; Saillenfest, M.

2016-05-01

We revisit the dynamics of Atlas. Using Cassini ISS astrometric observations spanning 2004 February to 2013 August, Cooper et al. found evidence that Atlas is currently perturbed by both a 54:53 corotation eccentricity resonance (CER) and a 54:53 Lindblad eccentricity resonance (LER) with Prometheus. They demonstrated that the orbit of Atlas is chaotic, with a Lyapunov time of order 10 years, as a direct consequence of the coupled resonant interaction (CER/LER) with Prometheus. Here we investigate the interactions between the two resonances using the CoraLin analytical model, showing that the chaotic zone fills almost all the corotation sites occupied by the satellite's orbit. Four 70:67 apse-type mean motion resonances with Pandora are also overlapping, but these resonances have a much weaker effect. Frequency analysis allows us to highlight the coupling between the 54:53 resonances, and confirms that a simplified system including the perturbations due to Prometheus and Saturn's oblateness only captures the essential features of the dynamics.

Simultaneous trilateral communication based on three mutually coupled chaotic semiconductor lasers with optical feedback.

PubMed

Li, Qiliang; Lu, Shanshan; Bao, Qi; Chen, Dewang; Hu, Miao; Zeng, Ran; Yang, Guowei; Li, Shuqin

2018-01-10

In this paper, we propose a chaos-based scheme allowing for trilateral communication among three mutually coupled chaotic semiconductor lasers. The coupling through a partially transparent optical mirror between two lasers induces the chaotic dynamics. We numerically solve the delay rate equations of three lasers and demonstrate that the dynamics is completely synchronous. Herein, each laser is not only a transmitter but a receiver; three different messages are encoded by simultaneously modulating bias current of the three lasers. By monitoring the synchronization error between transmitter and receiver, and comparing the error with the message of the local laser, we can decipher the message of the sender. The investigation indicates that these messages introduced on the two ends of each link among three lasers can be simultaneously transmitted and restored, so the system can realize simultaneous trilateral communication. In this scheme, an eavesdropper can monitor the synchronization error, but one has no way to obtain the bits that are being sent, so the trilateral communication is secure.

Working Towards Führer: A Chaotic View

NASA Astrophysics Data System (ADS)

Cakar, Ulas

Leadership is a concept that has been discussed since the beginning of history. Even though there have been many theories in the field accepting leadership's role in bringing order, chaotic aspects of leadership are generally neglected. This chapter aims to examine the leadership beyond an orderly interpretation of universe. For this purpose, Third Reich period and leadership during this period will be examined. Ian Kershaw's "Working Towards Führer" concept provides a unique understanding of leadership concept. It goes beyond the dualist depiction of Third Reich, it does not state Adolf Hitler as an all powerful dictator, or a weak one. Rather, he expresses that due to the conditions in the Third Reich, Adolf Hitler was both of this. This complex situation can be understood deeper when it is examined through the lens of chaos theory. This study contributes to the field by being the first in using chaos theory for examining "Working Towards Führer" concept and its development. Seemingly orderly nature of synchronization process and its vortex will be shown. Adolf Hitler's storm spot position in the chaotic system and its dynamics are explained. War's entropic power and its effect on the downfall of the system is crucial in understanding this unique chaotic system. The chaotic pattern of "Working Towards Führer" offers an opportunity to analyze the complexities of the leadership concept.

A phase transition induces chaos in a predator-prey ecosystem with a dynamic fitness landscape.

PubMed

Gilpin, William; Feldman, Marcus W

2017-07-01

In many ecosystems, natural selection can occur quickly enough to influence the population dynamics and thus future selection. This suggests the importance of extending classical population dynamics models to include such eco-evolutionary processes. Here, we describe a predator-prey model in which the prey population growth depends on a prey density-dependent fitness landscape. We show that this two-species ecosystem is capable of exhibiting chaos even in the absence of external environmental variation or noise, and that the onset of chaotic dynamics is the result of the fitness landscape reversibly alternating between epochs of stabilizing and disruptive selection. We draw an analogy between the fitness function and the free energy in statistical mechanics, allowing us to use the physical theory of first-order phase transitions to understand the onset of rapid cycling in the chaotic predator-prey dynamics. We use quantitative techniques to study the relevance of our model to observational studies of complex ecosystems, finding that the evolution-driven chaotic dynamics confer community stability at the "edge of chaos" while creating a wide distribution of opportunities for speciation during epochs of disruptive selection-a potential observable signature of chaotic eco-evolutionary dynamics in experimental studies.

A new feedback image encryption scheme based on perturbation with dynamical compound chaotic sequence cipher generator

NASA Astrophysics Data System (ADS)

Tong, Xiaojun; Cui, Minggen; Wang, Zhu

2009-07-01

The design of the new compound two-dimensional chaotic function is presented by exploiting two one-dimensional chaotic functions which switch randomly, and the design is used as a chaotic sequence generator which is proved by Devaney's definition proof of chaos. The properties of compound chaotic functions are also proved rigorously. In order to improve the robustness against difference cryptanalysis and produce avalanche effect, a new feedback image encryption scheme is proposed using the new compound chaos by selecting one of the two one-dimensional chaotic functions randomly and a new image pixels method of permutation and substitution is designed in detail by array row and column random controlling based on the compound chaos. The results from entropy analysis, difference analysis, statistical analysis, sequence randomness analysis, cipher sensitivity analysis depending on key and plaintext have proven that the compound chaotic sequence cipher can resist cryptanalytic, statistical and brute-force attacks, and especially it accelerates encryption speed, and achieves higher level of security. By the dynamical compound chaos and perturbation technology, the paper solves the problem of computer low precision of one-dimensional chaotic function.

Forecasting fluctuating outbreaks in seasonally driven epidemics

NASA Astrophysics Data System (ADS)

Stone, Lewi

2009-03-01

Seasonality is a driving force that has major impact on the spatio-temporal dynamics of natural systems and their populations. This is especially true for the transmission of common infectious diseases such as influenza, measles, chickenpox, and pertussis. Here we gain new insights into the nonlinear dynamics of recurrent diseases through the analysis of the classical seasonally forced SIR epidemic model. Despite many efforts over the last decades, it has been difficult to gain general analytical insights because of the complex synchronization effects that can evolve between the external forcing and the model's natural oscillations. The analysis advanced here attempts to make progress in this direction by focusing on the dynamics of ``skips'' where we identify and predict years in which the epidemic is absent rather than outbreak years. Skipping events are intrinsic to the forced SIR model when parameterised in the chaotic regime. In fact, it is difficult if not impossible to locate realistic chaotic parameter regimes in which outbreaks occur regularly each year. This contrasts with the well known Rossler oscillator whose outbreaks recur regularly but whose amplitude vary chaotically in time (Uniform Phase Chaotic Amplitude oscillations). The goal of the present study is to develop a ``language of skips'' that makes it possible to predict under what conditions the next outbreak is likely to occur, and how many ``skips'' might be expected after any given outbreak. We identify a new threshold effect and give clear analytical conditions that allow accurate predictions. Moreover, the time of occurrence (i.e., phase) of an outbreak proves to be a useful new parameter that carries important epidemiological information. In forced systems, seasonal changes can prevent late-initiating outbreaks (i.e., having high phase) from running to completion. These principles yield forecasting tools that should have relevance for the study of newly emerging and reemerging diseases.

SIAM conference on applications of dynamical systems

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

Not Available

1992-01-01

A conference (Oct.15--19, 1992, Snowbird, Utah; sponsored by SIAM (Society for Industrial and Applied Mathematics) Activity Group on Dynamical Systems) was held that highlighted recent developments in applied dynamical systems. The main lectures and minisymposia covered theory about chaotic motion, applications in high energy physics and heart fibrillations, turbulent motion, Henon map and attractor, integrable problems in classical physics, pattern formation in chemical reactions, etc. The conference fostered an exchange between mathematicians working on theoretical issues of modern dynamical systems and applied scientists. This two-part document contains abstracts, conference program, and an author index.

SIAM conference on applications of dynamical systems. Abstracts and author index

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

Not Available

1992-12-31

A conference (Oct.15--19, 1992, Snowbird, Utah; sponsored by SIAM (Society for Industrial and Applied Mathematics) Activity Group on Dynamical Systems) was held that highlighted recent developments in applied dynamical systems. The main lectures and minisymposia covered theory about chaotic motion, applications in high energy physics and heart fibrillations, turbulent motion, Henon map and attractor, integrable problems in classical physics, pattern formation in chemical reactions, etc. The conference fostered an exchange between mathematicians working on theoretical issues of modern dynamical systems and applied scientists. This two-part document contains abstracts, conference program, and an author index.

Incomplete Thermalization from Trap-Induced Integrability Breaking: Lessons from Classical Hard Rods

NASA Astrophysics Data System (ADS)

Cao, Xiangyu; Bulchandani, Vir B.; Moore, Joel E.

2018-04-01

We study a one-dimensional gas of hard rods trapped in a harmonic potential, which breaks integrability of the hard-rod interaction in a nonuniform way. We explore the consequences of such broken integrability for the dynamics of a large number of particles and find three distinct regimes: initial, chaotic, and stationary. The initial regime is captured by an evolution equation for the phase-space distribution function. For any finite number of particles, this hydrodynamics breaks down and the dynamics becomes chaotic after a characteristic timescale determined by the interparticle distance and scattering length. The system fails to thermalize over the timescale studied (1 04 natural units), but the time-averaged ensemble is a stationary state of the hydrodynamic evolution. We close by discussing logical extensions of the results to similar systems of quantum particles.

Genome chaos: survival strategy during crisis.

PubMed

Liu, Guo; Stevens, Joshua B; Horne, Steven D; Abdallah, Batoul Y; Ye, Karen J; Bremer, Steven W; Ye, Christine J; Chen, David J; Heng, Henry H

2014-01-01

Genome chaos, a process of complex, rapid genome re-organization, results in the formation of chaotic genomes, which is followed by the potential to establish stable genomes. It was initially detected through cytogenetic analyses, and recently confirmed by whole-genome sequencing efforts which identified multiple subtypes including "chromothripsis", "chromoplexy", "chromoanasynthesis", and "chromoanagenesis". Although genome chaos occurs commonly in tumors, both the mechanism and detailed aspects of the process are unknown due to the inability of observing its evolution over time in clinical samples. Here, an experimental system to monitor the evolutionary process of genome chaos was developed to elucidate its mechanisms. Genome chaos occurs following exposure to chemotherapeutics with different mechanisms, which act collectively as stressors. Characterization of the karyotype and its dynamic changes prior to, during, and after induction of genome chaos demonstrates that chromosome fragmentation (C-Frag) occurs just prior to chaotic genome formation. Chaotic genomes seem to form by random rejoining of chromosomal fragments, in part through non-homologous end joining (NHEJ). Stress induced genome chaos results in increased karyotypic heterogeneity. Such increased evolutionary potential is demonstrated by the identification of increased transcriptome dynamics associated with high levels of karyotypic variance. In contrast to impacting on a limited number of cancer genes, re-organized genomes lead to new system dynamics essential for cancer evolution. Genome chaos acts as a mechanism of rapid, adaptive, genome-based evolution that plays an essential role in promoting rapid macroevolution of new genome-defined systems during crisis, which may explain some unwanted consequences of cancer treatment.

Chaotic dynamics of flexible beams driven by external white noise

NASA Astrophysics Data System (ADS)

Awrejcewicz, J.; Krysko, A. V.; Papkova, I. V.; Zakharov, V. M.; Erofeev, N. P.; Krylova, E. Yu.; Mrozowski, J.; Krysko, V. A.

2016-10-01

Mathematical models of continuous structural members (beams, plates and shells) subjected to an external additive white noise are studied. The structural members are considered as systems with infinite number of degrees of freedom. We show that in mechanical structural systems external noise can not only lead to quantitative changes in the system dynamics (that is obvious), but also cause the qualitative, and sometimes surprising changes in the vibration regimes. Furthermore, we show that scenarios of the transition from regular to chaotic regimes quantified by Fast Fourier Transform (FFT) can lead to erroneous conclusions, and a support of the wavelet analysis is needed. We have detected and illustrated the modifications of classical three scenarios of transition from regular vibrations to deterministic chaos. The carried out numerical experiment shows that the white noise lowers the threshold for transition into spatio-temporal chaotic dynamics. A transition into chaos via the proposed modified scenarios developed in this work is sensitive to small noise and significantly reduces occurrence of periodic vibrations. Increase of noise intensity yields decrease of the duration of the laminar signal range, i.e., time between two successive turbulent bursts decreases. Scenario of transition into chaos of the studied mechanical structures essentially depends on the control parameters, and it can be different in different zones of the constructed charts (control parameter planes). Furthermore, we found an interesting phenomenon, when increase of the noise intensity yields surprisingly the vibrational characteristics with a lack of noisy effect (chaos is destroyed by noise and windows of periodicity appear).

Chaotic itinerancy within the coupled dynamics between a physical body and neural oscillator networks

PubMed Central

Mori, Hiroki; Okuyama, Yuji; Asada, Minoru

2017-01-01

Chaotic itinerancy is a phenomenon in which the state of a nonlinear dynamical system spontaneously explores and attracts certain states in a state space. From this perspective, the diverse behavior of animals and its spontaneous transitions lead to a complex coupled dynamical system, including a physical body and a brain. Herein, a series of simulations using different types of non-linear oscillator networks (i.e., regular, small-world, scale-free, random) with a musculoskeletal model (i.e., a snake-like robot) as a physical body are conducted to understand how the chaotic itinerancy of bodily behavior emerges from the coupled dynamics between the body and the brain. A behavior analysis (behavior clustering) and network analysis for the classified behavior are then applied. The former consists of feature vector extraction from the motions and classification of the movement patterns that emerged from the coupled dynamics. The network structures behind the classified movement patterns are revealed by estimating the “information networks” different from the given non-linear oscillator networks based on the transfer entropy which finds the information flow among neurons. The experimental results show that: (1) the number of movement patterns and their duration depend on the sensor ratio to control the balance of strength between the body and the brain dynamics and on the type of the given non-linear oscillator networks; and (2) two kinds of information networks are found behind two kinds movement patterns with different durations by utilizing the complex network measures, clustering coefficient and the shortest path length with a negative and a positive relationship with the duration periods of movement patterns. The current results seem promising for a future extension of the method to a more complicated body and environment. Several requirements are also discussed. PMID:28796797

Chaotic itinerancy within the coupled dynamics between a physical body and neural oscillator networks.

PubMed

Park, Jihoon; Mori, Hiroki; Okuyama, Yuji; Asada, Minoru

2017-01-01

Chaotic itinerancy is a phenomenon in which the state of a nonlinear dynamical system spontaneously explores and attracts certain states in a state space. From this perspective, the diverse behavior of animals and its spontaneous transitions lead to a complex coupled dynamical system, including a physical body and a brain. Herein, a series of simulations using different types of non-linear oscillator networks (i.e., regular, small-world, scale-free, random) with a musculoskeletal model (i.e., a snake-like robot) as a physical body are conducted to understand how the chaotic itinerancy of bodily behavior emerges from the coupled dynamics between the body and the brain. A behavior analysis (behavior clustering) and network analysis for the classified behavior are then applied. The former consists of feature vector extraction from the motions and classification of the movement patterns that emerged from the coupled dynamics. The network structures behind the classified movement patterns are revealed by estimating the "information networks" different from the given non-linear oscillator networks based on the transfer entropy which finds the information flow among neurons. The experimental results show that: (1) the number of movement patterns and their duration depend on the sensor ratio to control the balance of strength between the body and the brain dynamics and on the type of the given non-linear oscillator networks; and (2) two kinds of information networks are found behind two kinds movement patterns with different durations by utilizing the complex network measures, clustering coefficient and the shortest path length with a negative and a positive relationship with the duration periods of movement patterns. The current results seem promising for a future extension of the method to a more complicated body and environment. Several requirements are also discussed.

Generating Spatiotemporal Joint Torque Patterns from Dynamical Synchronization of Distributed Pattern Generators

PubMed Central

Pitti, Alexandre; Lungarella, Max; Kuniyoshi, Yasuo

2009-01-01

Pattern generators found in the spinal cord are no more seen as simple rhythmic oscillators for motion control. Indeed, they achieve flexible and dynamical coordination in interaction with the body and the environment dynamics giving to rise motor synergies. Discovering the mechanisms underlying the control of motor synergies constitutes an important research question not only for neuroscience but also for robotics: the motors coordination of high dimensional robotic systems is still a drawback and new control methods based on biological solutions may reduce their overall complexity. We propose to model the flexible combination of motor synergies in embodied systems via partial phase synchronization of distributed chaotic systems; for specific coupling strength, chaotic systems are able to phase synchronize their dynamics to the resonant frequencies of one external force. We take advantage of this property to explore and exploit the intrinsic dynamics of one specified embodied system. In two experiments with bipedal walkers, we show how motor synergies emerge when the controllers phase synchronize to the body's dynamics, entraining it to its intrinsic behavioral patterns. This stage is characterized by directed information flow from the sensors to the motors exhibiting the optimal situation when the body dynamics drive the controllers (mutual entrainment). Based on our results, we discuss the relevance of our findings for modeling the modular control of distributed pattern generators exhibited in the spinal cord, and for exploring the motor synergies in robots. PMID:20011216

Chaotic universe model.

PubMed

Aydiner, Ekrem

2018-01-15

In this study, we consider nonlinear interactions between components such as dark energy, dark matter, matter and radiation in the framework of the Friedman-Robertson-Walker space-time and propose a simple interaction model based on the time evolution of the densities of these components. By using this model we show that these interactions can be given by Lotka-Volterra type equations. We numerically solve these coupling equations and show that interaction dynamics between dark energy-dark matter-matter or dark energy-dark matter-matter-radiation has a strange attractor for 0 > w de  >-1, w dm  ≥ 0, w m  ≥ 0 and w r  ≥ 0 values. These strange attractors with the positive Lyapunov exponent clearly show that chaotic dynamics appears in the time evolution of the densities. These results provide that the time evolution of the universe is chaotic. The present model may have potential to solve some of the cosmological problems such as the singularity, cosmic coincidence, big crunch, big rip, horizon, oscillation, the emergence of the galaxies, matter distribution and large-scale organization of the universe. The model also connects between dynamics of the competing species in biological systems and dynamics of the time evolution of the universe and offers a new perspective and a new different scenario for the universe evolution.

Origin of Complexity in Multicellular Organisms

NASA Astrophysics Data System (ADS)

Furusawa, Chikara; Kaneko, Kunihiko

2000-06-01

Through extensive studies of dynamical system modeling cellular growth and reproduction, we find evidence that complexity arises in multicellular organisms naturally through evolution. Without any elaborate control mechanism, these systems can exhibit complex pattern formation with spontaneous cell differentiation. Such systems employ a ``cooperative'' use of resources and maintain a larger growth speed than simple cell systems, which exist in a homogeneous state and behave ``selfishly.'' The relevance of the diversity of chemicals and reaction dynamics to the growth of a multicellular organism is demonstrated. Chaotic biochemical dynamics are found to provide the multipotency of stem cells.

Breaking time reversal in a simple smooth chaotic system.

PubMed

Tomsovic, Steven; Ullmo, Denis; Nagano, Tatsuro

2003-06-01

Within random matrix theory, the statistics of the eigensolutions depend fundamentally on the presence (or absence) of time reversal symmetry. Accepting the Bohigas-Giannoni-Schmit conjecture, this statement extends to quantum systems with chaotic classical analogs. For practical reasons, much of the supporting numerical studies of symmetry breaking have been done with billiards or maps, and little with simple, smooth systems. There are two main difficulties in attempting to break time reversal invariance in a continuous time system with a smooth potential. The first is avoiding false time reversal breaking. The second is locating a parameter regime in which the symmetry breaking is strong enough to transform the fluctuation properties fully to the broken symmetry case, and yet remain weak enough so as not to regularize the dynamics sufficiently that the system is no longer chaotic. We give an example of a system of two coupled quartic oscillators whose energy level statistics closely match with those of the Gaussian unitary ensemble, and which possesses only a minor proportion of regular motion in its phase space.

Study on Unified Chaotic System-Based Wind Turbine Blade Fault Diagnostic System

NASA Astrophysics Data System (ADS)

Kuo, Ying-Che; Hsieh, Chin-Tsung; Yau, Her-Terng; Li, Yu-Chung

At present, vibration signals are processed and analyzed mostly in the frequency domain. The spectrum clearly shows the signal structure and the specific characteristic frequency band is analyzed, but the number of calculations required is huge, resulting in delays. Therefore, this study uses the characteristics of a nonlinear system to load the complete vibration signal to the unified chaotic system, applying the dynamic error to analyze the wind turbine vibration signal, and adopting extenics theory for artificial intelligent fault diagnosis of the analysis signal. Hence, a fault diagnostor has been developed for wind turbine rotating blades. This study simulates three wind turbine blade states, namely stress rupture, screw loosening and blade loss, and validates the methods. The experimental results prove that the unified chaotic system used in this paper has a significant effect on vibration signal analysis. Thus, the operating conditions of wind turbines can be quickly known from this fault diagnostic system, and the maintenance schedule can be arranged before the faults worsen, making the management and implementation of wind turbines smoother, so as to reduce many unnecessary costs.

Discretization chaos - Feedback control and transition to chaos

NASA Technical Reports Server (NTRS)

Grantham, Walter J.; Athalye, Amit M.

1990-01-01

Problems in the design of feedback controllers for chaotic dynamical systems are considered theoretically, focusing on two cases where chaos arises only when a nonchaotic continuous-time system is discretized into a simpler discrete-time systems (exponential discretization and pseudo-Euler integration applied to Lotka-Volterra competition and prey-predator systems). Numerical simulation results are presented in extensive graphs and discussed in detail. It is concluded that care must be taken in applying standard dynamical-systems methods to control systems that may be discontinuous or nondifferentiable.

Asynchronous Rate Chaos in Spiking Neuronal Circuits

PubMed Central

Harish, Omri; Hansel, David

2015-01-01

The brain exhibits temporally complex patterns of activity with features similar to those of chaotic systems. Theoretical studies over the last twenty years have described various computational advantages for such regimes in neuronal systems. Nevertheless, it still remains unclear whether chaos requires specific cellular properties or network architectures, or whether it is a generic property of neuronal circuits. We investigate the dynamics of networks of excitatory-inhibitory (EI) spiking neurons with random sparse connectivity operating in the regime of balance of excitation and inhibition. Combining Dynamical Mean-Field Theory with numerical simulations, we show that chaotic, asynchronous firing rate fluctuations emerge generically for sufficiently strong synapses. Two different mechanisms can lead to these chaotic fluctuations. One mechanism relies on slow I-I inhibition which gives rise to slow subthreshold voltage and rate fluctuations. The decorrelation time of these fluctuations is proportional to the time constant of the inhibition. The second mechanism relies on the recurrent E-I-E feedback loop. It requires slow excitation but the inhibition can be fast. In the corresponding dynamical regime all neurons exhibit rate fluctuations on the time scale of the excitation. Another feature of this regime is that the population-averaged firing rate is substantially smaller in the excitatory population than in the inhibitory population. This is not necessarily the case in the I-I mechanism. Finally, we discuss the neurophysiological and computational significance of our results. PMID:26230679

Is the Filinov integral conditioning technique useful in semiclassical initial value representation methods?

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

Spanner, Michael; Batista, Victor S.; Brumer, Paul

2005-02-22

The utility of the Filinov integral conditioning technique, as implemented in semiclassical initial value representation (SC-IVR) methods, is analyzed for a number of regular and chaotic systems. For nonchaotic systems of low dimensionality, the Filinov technique is found to be quite ineffective at accelerating convergence of semiclassical calculations since, contrary to the conventional wisdom, the semiclassical integrands usually do not exhibit significant phase oscillations in regions of large integrand amplitude. In the case of chaotic dynamics, it is found that the regular component is accurately represented by the SC-IVR, even when using the Filinov integral conditioning technique, but that quantummore » manifestations of chaotic behavior was easily overdamped by the filtering technique. Finally, it is shown that the level of approximation introduced by the Filinov filter is, in general, comparable to the simpler ad hoc truncation procedure introduced by Kay [J. Chem. Phys. 101, 2250 (1994)].« less

A novel chaotic stream cipher and its application to palmprint template protection

NASA Astrophysics Data System (ADS)

Li, Heng-Jian; Zhang, Jia-Shu

2010-04-01

Based on a coupled nonlinear dynamic filter (NDF), a novel chaotic stream cipher is presented in this paper and employed to protect palmprint templates. The chaotic pseudorandom bit generator (PRBG) based on a coupled NDF, which is constructed in an inverse flow, can generate multiple bits at one iteration and satisfy the security requirement of cipher design. Then, the stream cipher is employed to generate cancelable competitive code palmprint biometrics for template protection. The proposed cancelable palmprint authentication system depends on two factors: the palmprint biometric and the password/token. Therefore, the system provides high-confidence and also protects the user's privacy. The experimental results of verification on the Hong Kong PolyU Palmprint Database show that the proposed approach has a large template re-issuance ability and the equal error rate can achieve 0.02%. The performance of the palmprint template protection scheme proves the good practicability and security of the proposed stream cipher.

A 2D chaotic path planning for mobile robots accomplishing boundary surveillance missions in adversarial conditions

NASA Astrophysics Data System (ADS)

Curiac, Daniel-Ioan; Volosencu, Constantin

2014-10-01

The path-planning algorithm represents a crucial issue for every autonomous mobile robot. In normal circumstances a patrol robot will compute an optimal path to ensure its task accomplishment, but in adversarial conditions the problem is getting more complicated. Here, the robot’s trajectory needs to be altered into a misleading and unpredictable path to cope with potential opponents. Chaotic systems provide the needed framework for obtaining unpredictable motion in all of the three basic robot surveillance missions: area, points of interests and boundary monitoring. Proficient approaches have been provided for the first two surveillance tasks, but for boundary patrol missions no method has been reported yet. This paper addresses the mentioned research gap by proposing an efficient method, based on chaotic dynamic of the Hénon system, to ensure unpredictable boundary patrol on any shape of chosen closed contour.

Analysis of chaos in high-dimensional wind power system.

PubMed

Wang, Cong; Zhang, Hongli; Fan, Wenhui; Ma, Ping

2018-01-01

A comprehensive analysis on the chaos of a high-dimensional wind power system is performed in this study. A high-dimensional wind power system is more complex than most power systems. An 11-dimensional wind power system proposed by Huang, which has not been analyzed in previous studies, is investigated. When the systems are affected by external disturbances including single parameter and periodic disturbance, or its parameters changed, chaotic dynamics of the wind power system is analyzed and chaotic parameters ranges are obtained. Chaos existence is confirmed by calculation and analysis of all state variables' Lyapunov exponents and the state variable sequence diagram. Theoretical analysis and numerical simulations show that the wind power system chaos will occur when parameter variations and external disturbances change to a certain degree.

Long-Range Correlations in Stride Intervals May Emerge from Non-Chaotic Walking Dynamics

PubMed Central

Ahn, Jooeun; Hogan, Neville

2013-01-01

Stride intervals of normal human walking exhibit long-range temporal correlations. Similar to the fractal-like behaviors observed in brain and heart activity, long-range correlations in walking have commonly been interpreted to result from chaotic dynamics and be a signature of health. Several mathematical models have reproduced this behavior by assuming a dominant role of neural central pattern generators (CPGs) and/or nonlinear biomechanics to evoke chaos. In this study, we show that a simple walking model without a CPG or biomechanics capable of chaos can reproduce long-range correlations. Stride intervals of the model revealed long-range correlations observed in human walking when the model had moderate orbital stability, which enabled the current stride to affect a future stride even after many steps. This provides a clear counterexample to the common hypothesis that a CPG and/or chaotic dynamics is required to explain the long-range correlations in healthy human walking. Instead, our results suggest that the long-range correlation may result from a combination of noise that is ubiquitous in biological systems and orbital stability that is essential in general rhythmic movements. PMID:24086274

An advanced analysis and modelling the air pollutant concentration temporal dynamics in atmosphere of the industrial cities: Odessa city

NASA Astrophysics Data System (ADS)

Buyadzhi, V. V.; Glushkov, A. V.; Khetselius, O. Yu; Ternovsky, V. B.; Serga, I. N.; Bykowszczenko, N.

2017-10-01

Results of analysis and modelling the air pollutant (dioxide of nitrogen) concentration temporal dynamics in atmosphere of the industrial city Odessa are presented for the first time and based on computing by nonlinear methods of the chaos and dynamical systems theories. A chaotic behaviour is discovered and investigated. To reconstruct the corresponding strange chaotic attractor, the time delay and embedding dimension are computed. The former is determined by the methods of autocorrelation function and average mutual information, and the latter is calculated by means of correlation dimension method and algorithm of false nearest neighbours. It is shown that low-dimensional chaos exists in the nitrogen dioxide concentration time series under investigation. Further, the Lyapunov’s exponents spectrum, Kaplan-Yorke dimension and Kolmogorov entropy are computed.

High-frequency chaotic dynamics enabled by optical phase-conjugation

PubMed Central

Mercier, Émeric; Wolfersberger, Delphine; Sciamanna, Marc

2016-01-01

Wideband chaos is of interest for applications such as random number generation or encrypted communications, which typically use optical feedback in a semiconductor laser. Here, we show that replacing conventional optical feedback with phase-conjugate feedback improves the chaos bandwidth. In the range of achievable phase-conjugate mirror reflectivities, the bandwidth increase reaches 27% when compared with feedback from a conventional mirror. Experimental measurements of the time-resolved frequency dynamics on nanosecond time-scales show that the bandwidth enhancement is related to the onset of self-pulsing solutions at harmonics of the external-cavity frequency. In the observed regime, the system follows a chaotic itinerancy among these destabilized high-frequency external-cavity modes. The recorded features are unique to phase-conjugate feedback and distinguish it from the long-standing problem of time-delayed feedback dynamics. PMID:26739806

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

Naoz, Smadar; Li, Gongjie; Zanardi, Macarena

The secular approximation of the hierarchical three body systems has been proven to be very useful in addressing many astrophysical systems, from planets to stars to black holes. In such a system, two objects are on a tight orbit and the tertiary is on a much wider orbit. Here, we study the dynamics of a system by taking the tertiary mass to zero and solve the hierarchical three body system up to the octupole level of approximation. We find a rich dynamics that the outer orbit undergoes due to gravitational perturbations from the inner binary. The nominal result of themore » precession of the nodes is mostly limited for the lowest order of approximation; however, when the octupole level of approximation is introduced, the system becomes chaotic, as expected, and the tertiary oscillates below and above 90°, similarly to the non-test particle flip behavior. We provide the Hamiltonian of the system and investigate the dynamics of the system from the quadrupole to the octupole level of approximations. We also analyze the chaotic and quasi-periodic orbital evolution by studying the surfaces of sections. Furthermore, including general relativity, we showcase the long-term evolution of individual debris disk particles under the influence of a far-away interior eccentric planet. We show that this dynamics can naturally result in retrograde objects and a puffy disk after a long timescale evolution (a few Gyr) for initially aligned configuration.« less

Force Analysis and Energy Operation of Chaotic System of Permanent-Magnet Synchronous Motor

NASA Astrophysics Data System (ADS)

Qi, Guoyuan; Hu, Jianbing

2017-12-01

The disadvantage of a nondimensionalized model of a permanent-magnet synchronous Motor (PMSM) is identified. The original PMSM model is transformed into a Kolmogorov system to aid dynamic force analysis. The vector field of the PMSM is analogous to the force field including four types of torque — inertial, internal, dissipative, and generalized external. Using the feedback thought, the error torque between external torque and dissipative torque is identified. The pitchfork bifurcation of the PMSM is performed. Four forms of energy are identified for the system — kinetic, potential, dissipative, and supplied. The physical interpretations of the decomposition of force and energy exchange are given. Casimir energy is stored energy, and its rate of change is the error power between the dissipative energy and the energy supplied to the motor. Error torque and error power influence the different types of dynamic modes. The Hamiltonian energy and Casimir energy are compared to find the function of each in producing the dynamic modes. A supremum bound for the chaotic attractor is proposed using the error power and Lagrange multiplier.

Blended particle filters for large-dimensional chaotic dynamical systems

PubMed Central

Majda, Andrew J.; Qi, Di; Sapsis, Themistoklis P.

2014-01-01

A major challenge in contemporary data science is the development of statistically accurate particle filters to capture non-Gaussian features in large-dimensional chaotic dynamical systems. Blended particle filters that capture non-Gaussian features in an adaptively evolving low-dimensional subspace through particles interacting with evolving Gaussian statistics on the remaining portion of phase space are introduced here. These blended particle filters are constructed in this paper through a mathematical formalism involving conditional Gaussian mixtures combined with statistically nonlinear forecast models compatible with this structure developed recently with high skill for uncertainty quantification. Stringent test cases for filtering involving the 40-dimensional Lorenz 96 model with a 5-dimensional adaptive subspace for nonlinear blended filtering in various turbulent regimes with at least nine positive Lyapunov exponents are used here. These cases demonstrate the high skill of the blended particle filter algorithms in capturing both highly non-Gaussian dynamical features as well as crucial nonlinear statistics for accurate filtering in extreme filtering regimes with sparse infrequent high-quality observations. The formalism developed here is also useful for multiscale filtering of turbulent systems and a simple application is sketched below. PMID:24825886

Ergodic properties of spiking neuronal networks with delayed interactions

NASA Astrophysics Data System (ADS)

Palmigiano, Agostina; Wolf, Fred

The dynamical stability of neuronal networks, and the possibility of chaotic dynamics in the brain pose profound questions to the mechanisms underlying perception. Here we advance on the tractability of large neuronal networks of exactly solvable neuronal models with delayed pulse-coupled interactions. Pulse coupled delayed systems with an infinite dimensional phase space can be studied in equivalent systems of fixed and finite degrees of freedom by introducing a delayer variable for each neuron. A Jacobian of the equivalent system can be analytically obtained, and numerically evaluated. We find that depending on the action potential onset rapidness and the level of heterogeneities, the asynchronous irregular regime characteristic of balanced state networks loses stability with increasing delays to either a slow synchronous irregular or a fast synchronous irregular state. In networks of neurons with slow action potential onset, the transition to collective oscillations leads to an increase of the exponential rate of divergence of nearby trajectories and of the entropy production rate of the chaotic dynamics. The attractor dimension, instead of increasing linearly with increasing delay as reported in many other studies, decreases until eventually the network reaches full synchrony

Alteration of chaotic advection in blood flow around partial blockage zone: Role of hematocrit concentration

NASA Astrophysics Data System (ADS)

Maiti, Soumyabrata; Chaudhury, Kaustav; DasGupta, Debabrata; Chakraborty, Suman

2013-01-01

Spatial distributions of particles carried by blood exhibit complex filamentary pattern under the combined effects of geometrical irregularities of the blood vessels and pulsating pumping by the heart. This signifies the existence of so called chaotic advection. In the present article, we argue that the understanding of such pathologically triggered chaotic advection is incomplete without giving due consideration to a major constituent of blood: abundant presence of red blood cells quantified by the hematocrit (HCT) concentration. We show that the hematocrit concentration in blood cells can alter the filamentary structures of the spatial distribution of advected particles in an intriguing manner. Our results reveal that there primarily are two major impacts of HCT concentrations towards dictating the chaotic dynamics of blood flow: changing the zone of influence of chaotic mixing and determining the enhancement of residence time of the advected particles away from the wall. This, in turn, may alter the extent of activation of platelets or other reactive biological entities, bearing immense consequence towards dictating the biophysical mechanisms behind possible life-threatening diseases originating in the circulatory system.

Computing with Chaos

NASA Astrophysics Data System (ADS)

Murali, K.; Sinah, Sudeshna; Ditto, William

2004-03-01

Recently there has been a new theoretical direction in harnessing the richness of spatially extended chaotic systems, namely the exploitation of coupled chaotic elements to do flexible computations [1]. The aim of this presentation is to demonstrate the use a single chaotic element to emulate different logic gates and perform different arithmetic tasks. Additionally we demonstrate that the elements can be controlled to switch easily between the different operational roles. Such a computing unit may then allow a more dynamic computer architecture and serve as ingredients of a general-purpose device more flexible than statically wired hardware. The theoretical scheme for flexible implementation of all these fundamental logical operations utilizing low dimensional chaos [1] will be reviewed along with a specific realization of the theory in a chaotic circuit [2]. Results will also be presented from experiments done on leech neurons. [1] Sinha, S., Munakata, T. and Ditto, W.L., Phys. Rev. E 65 036216 [2] "Experimental realization of the fundamental NOR Gate using a chaotic circuit," K. Murali, Sudeshna Sinha and William L. Ditto Phys. Rev. E 68, 016205 (2003).

Dynamics of Two Point Vortices in an External Compressible Shear Flow

NASA Astrophysics Data System (ADS)

Vetchanin, Evgeny V.; Mamaev, Ivan S.

2017-12-01

This paper is concerned with a system of equations that describes the motion of two point vortices in a flow possessing constant uniform vorticity and perturbed by an acoustic wave. The system is shown to have both regular and chaotic regimes of motion. In addition, simple and chaotic attractors are found in the system. Attention is given to bifurcations of fixed points of a Poincaré map which lead to the appearance of these regimes. It is shown that, in the case where the total vortex strength changes, the "reversible pitch-fork" bifurcation is a typical scenario of emergence of asymptotically stable fixed and periodic points. As a result of this bifurcation, a saddle point, a stable and an unstable point of the same period emerge from an elliptic point of some period. By constructing and analyzing charts of dynamical regimes and bifurcation diagrams we show that a cascade of period-doubling bifurcations is a typical scenario of transition to chaos in the system under consideration.

Effect of collision duration on the chaotic dynamics of a ball bouncing on a vertically vibrating plate

NASA Astrophysics Data System (ADS)

Jiang, Z. H.; Liang, Z. J.; Wu, A. C.; Zheng, R. H.

2018-03-01

Experiments have been performed to study the chaotic dynamics of a ball bouncing on a vertically vibrating plate. The velocity dependence of collision duration and coefficient of restitution is determined, and phase portraits of chaotic structures for the flight time and the relative collision velocities are obtained. Numerical calculations are carried out to examine the effects of velocity-dependent collision duration on the ball dynamics. It is revealed that when the collision is instantaneous, sticking solutions are always observed, whereas when the collision duration is taken into account, sticking solutions are destroyed and thereby chaos behaviors are induced.

Chaotic Ising-like dynamics in traffic signals

PubMed Central

Suzuki, Hideyuki; Imura, Jun-ichi; Aihara, Kazuyuki

2013-01-01

The green and red lights of a traffic signal can be viewed as the up and down states of an Ising spin. Moreover, traffic signals in a city interact with each other, if they are controlled in a decentralised way. In this paper, a simple model of such interacting signals on a finite-size two-dimensional lattice is shown to have Ising-like dynamics that undergoes a ferromagnetic phase transition. Probabilistic behaviour of the model is realised by chaotic billiard dynamics that arises from coupled non-chaotic elements. This purely deterministic model is expected to serve as a starting point for considering statistical mechanics of traffic signals. PMID:23350034

The chaotic "sculpting" of the Solar System

NASA Astrophysics Data System (ADS)

Tsiganis, K.

2006-01-01

The orbits of the large celestial bodies in our Solar System are stable for very long times, as can be shown by numerical simulation. This gives the erroneous impression of perpetual stability of the system. It is only when we study the orbital distribution of the numerous minor bodies in the Solar System that we discover the rich variety of complex dynamical processes that have in fact shaped our system. During the last decade, enormous progress has been made, in understanding the evolution of the system over the last ~3.9 Gy. However, it also became clear that, in order to unveil its behaviour during the first ~700 million years of its lifetime, we have to find convincing explanations for observations that appear as details of its dynamical architecture. In the following we are going to show how the two best known - and up to now unexplained - observations in the Solar System, namely (i) the heavily cratered surface of the Moon and (ii) the elliptic (and not circular) motion of the planets, lead us to the discovery of the chaotic sculpting of the Solar System [1]-[3].

On some dynamical chameleon systems

NASA Astrophysics Data System (ADS)

Burkin, I. M.; Kuznetsova, O. I.

2018-03-01

It is now well known that dynamical systems can be categorized into systems with self-excited attractors and systems with hidden attractors. A self-excited attractor has a basin of attraction that is associated with an unstable equilibrium, while a hidden attractor has a basin of attraction that does not intersect with small neighborhoods of any equilibrium points. Hidden attractors play the important role in engineering applications because they allow unexpected and potentially disastrous responses to perturbations in a structure like a bridge or an airplane wing. In addition, complex behaviors of chaotic systems have been applied in various areas from image watermarking, audio encryption scheme, asymmetric color pathological image encryption, chaotic masking communication to random number generator. Recently, researchers have discovered the so-called “chameleon systems”. These systems were so named because they demonstrate self-excited or hidden oscillations depending on the value of parameters. The present paper offers a simple algorithm of synthesizing one-parameter chameleon systems. The authors trace the evolution of Lyapunov exponents and the Kaplan-Yorke dimension of such systems which occur when parameters change.

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.

Bit-level plane image encryption based on coupled map lattice with time-varying delay

NASA Astrophysics Data System (ADS)

Lv, Xiupin; Liao, Xiaofeng; Yang, Bo

2018-04-01

Most of the existing image encryption algorithms had two basic properties: confusion and diffusion in a pixel-level plane based on various chaotic systems. Actually, permutation in a pixel-level plane could not change the statistical characteristics of an image, and many of the existing color image encryption schemes utilized the same method to encrypt R, G and B components, which means that the three color components of a color image are processed three times independently. Additionally, dynamical performance of a single chaotic system degrades greatly with finite precisions in computer simulations. In this paper, a novel coupled map lattice with time-varying delay therefore is applied in color images bit-level plane encryption to solve the above issues. Spatiotemporal chaotic system with both much longer period in digitalization and much excellent performances in cryptography is recommended. Time-varying delay embedded in coupled map lattice enhances dynamical behaviors of the system. Bit-level plane image encryption algorithm has greatly reduced the statistical characteristics of an image through the scrambling processing. The R, G and B components cross and mix with one another, which reduces the correlation among the three components. Finally, simulations are carried out and all the experimental results illustrate that the proposed image encryption algorithm is highly secure, and at the same time, also demonstrates superior performance.

Chaotic Excitation and Tidal Damping in the GJ 876 System

NASA Astrophysics Data System (ADS)

Puranam, Abhijit; Batygin, Konstantin

2018-04-01

The M-dwarf GJ 876 is the closest known star to harbor a multi-planetary system. With three outer planets locked in a chaotic Laplace-type resonance and an appreciably eccentric short-period super-Earth, this system represents a unique exposition of extrasolar planetary dynamics. A key question that concerns the long-term evolution of this system, and the fate of close-in planets in general, is how the significant eccentricity of the inner-most planet is maintained against tidal circularization on timescales comparable to the age of the universe. Here, we employ stochastic secular perturbation theory and N-body simulations to show that the orbit of the inner-most planet is shaped by a delicate balance between extrinsic chaotic forcing and tidal dissipation. As such, the planet’s orbital eccentricity represents an indirect measure of its tidal quality factor. Based on the system’s present-day architecture, we estimate that the extrasolar super-Earth GJ 876 d has a tidal Q ∼ 104–105, a value characteristic of solar system gas giants.

Brain-Inspired Photonic Signal Processor for Generating Periodic Patterns and Emulating Chaotic Systems

NASA Astrophysics Data System (ADS)

Antonik, Piotr; Haelterman, Marc; Massar, Serge

2017-05-01

Reservoir computing is a bioinspired computing paradigm for processing time-dependent signals. Its hardware implementations have received much attention because of their simplicity and remarkable performance on a series of benchmark tasks. In previous experiments, the output was uncoupled from the system and, in most cases, simply computed off-line on a postprocessing computer. However, numerical investigations have shown that feeding the output back into the reservoir opens the possibility of long-horizon time-series forecasting. Here, we present a photonic reservoir computer with output feedback, and we demonstrate its capacity to generate periodic time series and to emulate chaotic systems. We study in detail the effect of experimental noise on system performance. In the case of chaotic systems, we introduce several metrics, based on standard signal-processing techniques, to evaluate the quality of the emulation. Our work significantly enlarges the range of tasks that can be solved by hardware reservoir computers and, therefore, the range of applications they could potentially tackle. It also raises interesting questions in nonlinear dynamics and chaos theory.

Chaotic dynamics and diffusion in a piecewise linear equation

NASA Astrophysics Data System (ADS)

Shahrear, Pabel; Glass, Leon; Edwards, Rod

2015-03-01

Genetic interactions are often modeled by logical networks in which time is discrete and all gene activity states update simultaneously. However, there is no synchronizing clock in organisms. An alternative model assumes that the logical network is preserved and plays a key role in driving the dynamics in piecewise nonlinear differential equations. We examine dynamics in a particular 4-dimensional equation of this class. In the equation, two of the variables form a negative feedback loop that drives a second negative feedback loop. By modifying the original equations by eliminating exponential decay, we generate a modified system that is amenable to detailed analysis. In the modified system, we can determine in detail the Poincaré (return) map on a cross section to the flow. By analyzing the eigenvalues of the map for the different trajectories, we are able to show that except for a set of measure 0, the flow must necessarily have an eigenvalue greater than 1 and hence there is sensitive dependence on initial conditions. Further, there is an irregular oscillation whose amplitude is described by a diffusive process that is well-modeled by the Irwin-Hall distribution. There is a large class of other piecewise-linear networks that might be analyzed using similar methods. The analysis gives insight into possible origins of chaotic dynamics in periodically forced dynamical systems.

Ballistic and diffusive dynamics in a two-dimensional ideal gas of macroscopic chaotic Faraday waves.

PubMed

Welch, Kyle J; Hastings-Hauss, Isaac; Parthasarathy, Raghuveer; Corwin, Eric I

2014-04-01

We have constructed a macroscopic driven system of chaotic Faraday waves whose statistical mechanics, we find, are surprisingly simple, mimicking those of a thermal gas. We use real-time tracking of a single floating probe, energy equipartition, and the Stokes-Einstein relation to define and measure a pseudotemperature and diffusion constant and then self-consistently determine a coefficient of viscous friction for a test particle in this pseudothermal gas. Because of its simplicity, this system can serve as a model for direct experimental investigation of nonequilibrium statistical mechanics, much as the ideal gas epitomizes equilibrium statistical mechanics.

Transition to chaos of a vertical collapsible tube conveying air flow

NASA Astrophysics Data System (ADS)

Castillo Flores, F.; Cros, A.

2009-05-01

"Sky dancers", the large collapsible tubes used as advertising, are studied in this work through a simple experimental device. Our study is devoted to the nonlinear dynamics of this system and to its transition to chaos. Firstly, we have shown that after a collapse occurs, the air fills the tube at a different speed rate from the flow velocity. Secondly, the temporal intermittency is studied as the flow rate is increased. A statistical analysis shows that the chaotic times maintain roughly the same value by increasing air speed. On the other hand, laminar times become shorter, until the system reaches a completely chaotic state.

Local dynamics and spatiotemporal chaos. The Kuramoto- Sivashinsky equation: A case study

NASA Astrophysics Data System (ADS)

Wittenberg, Ralf Werner

The nature of spatiotemporal chaos in extended continuous systems is not yet well-understood. In this thesis, a model partial differential equation, the Kuramoto- Sivashinsky (KS) equation ut+uxxxx+uxx+uux =0 on a large one-dimensional periodic domain, is studied analytically, numerically, and through modeling to obtain a more detailed understanding of the observed spatiotemporally complex dynamics. In particular, with the aid of a wavelet decomposition, the relevant dynamical interactions are shown to be localized in space and scale. Motivated by these results, and by the idea that the attractor on a large domain may be understood via attractors on smaller domains, a spatially localized low- dimensional model for a minimal chaotic box is proposed. A (de)stabilized extension of the KS equation has recently attracted increased interest; for this situation, dissipativity and analyticity areproven, and an explicit shock-like solution is constructed which sheds light on the difficulties in obtaining optimal bounds for the KS equation. For the usual KS equation, the spatiotemporally chaotic state is carefully characterized in real, Fourier and wavelet space. The wavelet decomposition provides good scale separation which isolates the three characteristic regions of the dynamics: large scales of slow Gaussian fluctuations, active scales containing localized interactions of coherent structures, and small scales. Space localization is shown through a comparison of various correlation lengths and a numerical experiment in which different modes are uncoupled to estimate a dynamic interaction length. A detailed picture of the contributions of different scales to the spatiotemporally complex dynamics is obtained via a Galerkin projection of the KS equation onto the wavelet basis, and an extensive series of numerical experiments in which different combinations of wavelet levels are eliminated or forced. These results, and a formalism to derive an effective equation for periodized subsystems externally forced from a larger system, motivate various models for spatially localized forced systems. There is convincing evidence that short periodized systems, internally forced at the largest scales, form a minimal model for the observed extensively chaotic dynamics in larger domains.

Testing the FLI in the region of the Pallas asteroid family

NASA Astrophysics Data System (ADS)

Todorović, N.; Novaković, B.

2015-08-01

Computation of the fast Lyapunov indicator (FLI) is one of the most efficient numerical ways to characterize dynamical nature of motion and to detect phase-space structures in a large variety of dynamical models. Despite its effectiveness, FLI was mainly used for symplectic maps or simple Hamiltonians, but it has never been used to study dynamics of asteroids to a greater extent. This research shows that FLI could also be successfully applied to real (Solar system) dynamics. For this purpose, we focus on the main belt region where the Pallas asteroid family is located. By using the full Solar system model, different sets of initial conditions and different integration times, we managed not only to visualize a large multiplet of resonances located in the region, but also their structures, chaotic boundaries, stability islands therein and the positions of their mutual interaction. In the end, we have identified some of the most dominant resonances present in the region and established a link between these resonances and chaotic areas visible in our maps. We have illustrated that FLI once again has shown its efficiency to detect dynamical structures in the main belt, e.g. in the Pallas asteroid family, with a surprisingly good clarity.

Experimental analysis of the dynamical response of energy harvesting devices based on bistable laminated plates.

PubMed

Syta, A; Bowen, C R; Kim, H A; Rysak, A; Litak, G

The use of bistable laminates is a potential approach to realize broadband piezoelectric based energy harvesting systems. In this paper the dynamic response of a piezoelectric material attached to a bistable laminate plate is examined based on the experimental generated voltage time series. The system was subjected to harmonic excitations and exhibited single-well and snap-through vibrations of both periodic and chaotic character. To identify the dynamics of the system response we examined the frequency spectrum, bifurcation diagrams, phase portraits, and the 0-1 test.

The amazing evolutionary dynamics of non-linear optical systems with feedback

NASA Astrophysics Data System (ADS)

Yaroslavsky, Leonid

2013-09-01

Optical systems with feedback are, generally, non-linear dynamic systems. As such, they exhibit evolutionary behavior. In the paper we present results of experimental investigation of evolutionary dynamics of several models of such systems. The models are modifications of the famous mathematical "Game of Life". The modifications are two-fold: "Game of Life" rules are made stochastic and mutual influence of cells is made spatially non-uniform. A number of new phenomena in the evolutionary dynamics of the models are revealed: - "Ordering of chaos". Formation, from seed patterns, of stable maze-like patterns with chaotic "dislocations" that resemble natural patterns, such as skin patterns of some animals and fishes, see shell, fingerprints, magnetic domain patterns and alike, which one can frequently find in the nature. These patterns and their fragments exhibit a remarkable capability of unlimited growth. - "Self-controlled growth" of chaotic "live" formations into "communities" bounded, depending on the model, by a square, hexagon or octagon, until they reach a certain critical size, after which the growth stops. - "Eternal life in a bounded space" of "communities" after reaching a certain size and shape. - "Coherent shrinkage" of "mature", after reaching a certain size, "communities" into one of stable or oscillating patterns preserving in this process isomorphism of their bounding shapes until the very end.

A phase transition induces chaos in a predator-prey ecosystem with a dynamic fitness landscape

PubMed Central

2017-01-01

In many ecosystems, natural selection can occur quickly enough to influence the population dynamics and thus future selection. This suggests the importance of extending classical population dynamics models to include such eco-evolutionary processes. Here, we describe a predator-prey model in which the prey population growth depends on a prey density-dependent fitness landscape. We show that this two-species ecosystem is capable of exhibiting chaos even in the absence of external environmental variation or noise, and that the onset of chaotic dynamics is the result of the fitness landscape reversibly alternating between epochs of stabilizing and disruptive selection. We draw an analogy between the fitness function and the free energy in statistical mechanics, allowing us to use the physical theory of first-order phase transitions to understand the onset of rapid cycling in the chaotic predator-prey dynamics. We use quantitative techniques to study the relevance of our model to observational studies of complex ecosystems, finding that the evolution-driven chaotic dynamics confer community stability at the “edge of chaos” while creating a wide distribution of opportunities for speciation during epochs of disruptive selection—a potential observable signature of chaotic eco-evolutionary dynamics in experimental studies. PMID:28678792

Geometrical analysis of the LiCN vibrational dynamics: a stability geometrical indicator.

PubMed

Vergel, A; Benito, R M; Losada, J C; Borondo, F

2014-02-01

The vibrational dynamics of the LiNC/LiCN molecular system is examined making use of the Riemannian geometry. Stability and chaoticity are analyzed, in this context, by means of the Jacobi-Levi-Civita equations, derived from the Jacobi metric, and its solutions. A dynamical indicator, called stability geometrical indicator, is introduced in order to ascertain the dynamical characteristics of stability and chaos in the molecule under study.

Complexity and network dynamics in physiological adaptation: an integrated view.

PubMed

Baffy, György; Loscalzo, Joseph

2014-05-28

Living organisms constantly interact with their surroundings and sustain internal stability against perturbations. This dynamic process follows three fundamental strategies (restore, explore, and abandon) articulated in historical concepts of physiological adaptation such as homeostasis, allostasis, and the general adaptation syndrome. These strategies correspond to elementary forms of behavior (ordered, chaotic, and static) in complex adaptive systems and invite a network-based analysis of the operational characteristics, allowing us to propose an integrated framework of physiological adaptation from a complex network perspective. Applicability of this concept is illustrated by analyzing molecular and cellular mechanisms of adaptation in response to the pervasive challenge of obesity, a chronic condition resulting from sustained nutrient excess that prompts chaotic exploration for system stability associated with tradeoffs and a risk of adverse outcomes such as diabetes, cardiovascular disease, and cancer. Deconstruction of this complexity holds the promise of gaining novel insights into physiological adaptation in health and disease. Published by Elsevier Inc.

Multistability in Chua's circuit with two stable node-foci

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

Bao, B. C.; Wang, N.; Xu, Q.

2016-04-15

Only using one-stage op-amp based negative impedance converter realization, a simplified Chua's diode with positive outer segment slope is introduced, based on which an improved Chua's circuit realization with more simpler circuit structure is designed. The improved Chua's circuit has identical mathematical model but completely different nonlinearity to the classical Chua's circuit, from which multiple attractors including coexisting point attractors, limit cycle, double-scroll chaotic attractor, or coexisting chaotic spiral attractors are numerically simulated and experimentally captured. Furthermore, with dimensionless Chua's equations, the dynamical properties of the Chua's system are studied including equilibrium and stability, phase portrait, bifurcation diagram, Lyapunov exponentmore » spectrum, and attraction basin. The results indicate that the system has two symmetric stable nonzero node-foci in global adjusting parameter regions and exhibits the unusual and striking dynamical behavior of multiple attractors with multistability.« less

A case study in bifurcation theory

NASA Astrophysics Data System (ADS)

Khmou, Youssef

This short paper is focused on the bifurcation theory found in map functions called evolution functions that are used in dynamical systems. The most well-known example of discrete iterative function is the logistic map that puts into evidence bifurcation and chaotic behavior of the topology of the logistic function. We propose a new iterative function based on Lorentizan function and its generalized versions, based on numerical study, it is found that the bifurcation of the Lorentzian function is of second-order where it is characterized by the absence of chaotic region.

Deterministic representation of chaos with application to turbulence

NASA Technical Reports Server (NTRS)

Zak, M.

1987-01-01

Chaotic motions of nonlinear dynamical systems are decomposed into mean components and fluctuations. The approach is based upon the concept that the fluctuations driven by the instability of the original (unperturbed) motion grow until a new stable state is approached. The Reynolds-type equations written for continuous as well as for finite-degrees-of-freedom dynamical systems are closed by using this stabilization principle. The theory is applied to conservative systems, to strange attractors and to turbulent motions.

Chaotic behavior of renal sympathetic nerve activity: effect of baroreceptor denervation and cardiac failure.

PubMed

DiBona, G F; Jones, S Y; Sawin, L L

2000-09-01

Nonlinear dynamic analysis was used to examine the chaotic behavior of renal sympathetic nerve activity in conscious rats subjected to either complete baroreceptor denervation (sinoaortic and cardiac baroreceptor denervation) or induction of congestive heart failure (CHF). The peak interval sequence of synchronized renal sympathetic nerve discharge was extracted and used for analysis. In control rats, this yielded a system whose correlation dimension converged to a low value over the embedding dimension range of 10-15 and whose greatest Lyapunov exponent was positive. Complete baroreceptor denervation was associated with a decrease in the correlation dimension of the system (before 2.65 +/- 0.27, after 1.64 +/- 0.17; P < 0.01) and a reduction in chaotic behavior (greatest Lyapunov exponent: 0.201 +/- 0.008 bits/data point before, 0.177 +/- 0.004 bits/data point after, P < 0.02). CHF, a state characterized by impaired sinoaortic and cardiac baroreceptor regulation of renal sympathetic nerve activity, was associated with a similar decrease in the correlation dimension (control 3.41 +/- 0.23, CHF 2.62 +/- 0.26; P < 0.01) and a reduction in chaotic behavior (greatest Lyapunov exponent: 0.205 +/- 0.048 bits/data point control, 0.136 +/- 0.033 bits/data point CHF, P < 0.02). These results indicate that removal of sinoaortic and cardiac baroreceptor regulation of renal sympathetic nerve activity, occurring either physiologically or pathophysiologically, is associated with a decrease in the correlation dimensions of the system and a reduction in chaotic behavior.

Chaotic dynamics of a microswimmer in Poiseuille flow.

PubMed

Chacón, Ricardo

2013-11-01

The chaotic dynamics of pointlike, spherical particles in cylindrical Poiseuille flow is theoretically characterized and numerically confirmed when their own intrinsic swimming velocity undergoes temporal fluctuations around an average value. Two dimensionless ratios associated with the three significant temporal scales of the problem are identified that fully determine the chaos scenario. In particular, small but finite periodic fluctuations of swimming speed result in chaotic or regular motion depending on the position and orientation of the microswimmer with respect to the flow center line. Remarkably, the spatial extension of chaotic microswimmers is found to depend crucially on the fluctuations' period and amplitude and to be highly sensitive to the Fourier spectrum of the fluctuations. This has implications for the design of artificial microswimmers.

Experimental demonstration of chaotic scattering of microwaves

NASA Astrophysics Data System (ADS)

Doron, E.; Smilansky, U.; Frenkel, A.

1990-12-01

Reflection of microwaves from a cavity is measured in a frequency domain where the underlying classical chaotic scattering leaves a clear mark on the wave dynamics. We check the hypothesis that the fluctuations of the S matrix can be described in terms of parameters characterizing the chaotic classical scatteirng. Absorption of energy in the cavity walls is shown to significantly affect the results, and is linked to time-domain properties of the scattering in a general way. We also show that features whose origin is entirely due to wave dynamics (e.g., the enhancement of the Wigner time delay due to time-reversal symmetry) coexist with other features which characterize the underlying classical dynamics.

Crisis of the chaotic attractor of a climate model: a transfer operator approach

NASA Astrophysics Data System (ADS)

Tantet, Alexis; Lucarini, Valerio; Lunkeit, Frank; Dijkstra, Henk A.

2018-05-01

The destruction of a chaotic attractor leading to rough changes in the dynamics of a dynamical system is studied. Local bifurcations are known to be characterised by a single or a pair of characteristic exponents crossing the imaginary axis. As a result, the approach of such bifurcations in the presence of noise can be inferred from the slowing down of the decay of correlations (Held and Kleinen 2004 Geophys. Res. Lett. 31 1–4). On the other hand, little is known about global bifurcations involving high-dimensional attractors with several positive Lyapunov exponents. It is known that the global stability of chaotic attractors may be characterised by the spectral properties of the Koopman (Mauroy and Mezić 2016 IEEE Trans. Autom. Control 61 3356–69) or the transfer operators governing the evolution of statistical ensembles. Accordingly, it has recently been shown (Tantet 2017 J. Stat. Phys. 1–33) that a boundary crisis in the Lorenz flow coincides with the approach to the unit circle of the eigenvalues of these operators associated with motions about the attractor, the stable resonances. A second class of resonances, the unstable resonances, are responsible for the decay of correlations and mixing on the attractor. In the deterministic case, these cannot be expected to be affected by general boundary crises. Here, however, we give an example of a chaotic system in which slowing down of the decay of correlations of some observables does occur at the approach of a boundary crisis. The system considered is a high-dimensional, chaotic climate model of physical relevance. Moreover, coarse-grained approximations of the transfer operators on a reduced space, constructed from a long time series of the system, give evidence that this behaviour is due to the approach of unstable resonances to the unit circle. That the unstable resonances are affected by the crisis can be physically understood from the fact that the process responsible for the instability, the ice-albedo feedback, is also active on the attractor. Finally, we discuss implications regarding response theory and the design of early-warning signals.

Consistency properties of chaotic systems driven by time-delayed feedback

NASA Astrophysics Data System (ADS)

Jüngling, T.; Soriano, M. C.; Oliver, N.; Porte, X.; Fischer, I.

2018-04-01

Consistency refers to the property of an externally driven dynamical system to respond in similar ways to similar inputs. In a delay system, the delayed feedback can be considered as an external drive to the undelayed subsystem. We analyze the degree of consistency in a generic chaotic system with delayed feedback by means of the auxiliary system approach. In this scheme an identical copy of the nonlinear node is driven by exactly the same signal as the original, allowing us to verify complete consistency via complete synchronization. In the past, the phenomenon of synchronization in delay-coupled chaotic systems has been widely studied using correlation functions. Here, we analytically derive relationships between characteristic signatures of the correlation functions in such systems and unequivocally relate them to the degree of consistency. The analytical framework is illustrated and supported by numerical calculations of the logistic map with delayed feedback for different replica configurations. We further apply the formalism to time series from an experiment based on a semiconductor laser with a double fiber-optical feedback loop. The experiment constitutes a high-quality replica scheme for studying consistency of the delay-driven laser and confirms the general theoretical results.

Breathers and thermal relaxation as a temporal process: A possible way to detect breathers in experimental situations

NASA Astrophysics Data System (ADS)

Castrejón Pita, A. A.; Castrejón Pita, J. R.; Sarmiento G., A.

2005-06-01

Breather stability and longevity in thermally relaxing nonlinear arrays is investigated under the scrutiny of the analysis and tools employed for time series and state reconstruction of a dynamical system. We briefly review the methods used in the analysis and characterize a breather in terms of the results obtained with such methods. Our present work focuses on spontaneously appearing breathers in thermal Fermi-Pasta-Ulam arrays but we believe that the conclusions are general enough to describe many other related situations; the particular case described in detail is presented as another example of systems where three incommensurable frequencies dominate their chaotic dynamics (reminiscent of the Ruelle-Takens scenario for the appearance of chaotic behavior in nonlinear systems). This characterization may also be of great help for the discovery of breathers in experimental situations where the temporal evolution of a local variable (like the site energy) is the only available/measured data.

Hybrid electronic/optical synchronized chaos communication system.

PubMed

Toomey, J P; Kane, D M; Davidović, A; Huntington, E H

2009-04-27

A hybrid electronic/optical system for synchronizing a chaotic receiver to a chaotic transmitter has been demonstrated. The chaotic signal is generated electronically and injected, in addition to a constant bias current, to a semiconductor laser to produce an optical carrier for transmission. The optical chaotic carrier is photodetected to regenerate an electronic signal for synchronization in a matched electronic receiver The system has been successfully used for the transmission and recovery of a chaos masked message that is added to the chaotic optical carrier. Past demonstrations of synchronized chaos based, secure communication systems have used either an electronic chaotic carrier or an optical chaotic carrier (such as the chaotic output of various nonlinear laser systems). This is the first electronic/optical hybrid system to be demonstrated. We call this generation of a chaotic optical carrier by electronic injection.

Chaotic attractors of relaxation oscillators

NASA Astrophysics Data System (ADS)

Guckenheimer, John; Wechselberger, Martin; Young, Lai-Sang

2006-03-01

We develop a general technique for proving the existence of chaotic attractors for three-dimensional vector fields with two time scales. Our results connect two important areas of dynamical systems: the theory of chaotic attractors for discrete two-dimensional Henon-like maps and geometric singular perturbation theory. Two-dimensional Henon-like maps are diffeomorphisms that limit on non-invertible one-dimensional maps. Wang and Young formulated hypotheses that suffice to prove the existence of chaotic attractors in these families. Three-dimensional singularly perturbed vector fields have return maps that are also two-dimensional diffeomorphisms limiting on one-dimensional maps. We describe a generic mechanism that produces folds in these return maps and demonstrate that the Wang-Young hypotheses are satisfied. Our analysis requires a careful study of the convergence of the return maps to their singular limits in the Ck topology for k >= 3. The theoretical results are illustrated with a numerical study of a variant of the forced van der Pol oscillator.

Trojan resonant dynamics, stability, and chaotic diffusion, for parameters relevant to exoplanetary systems

NASA Astrophysics Data System (ADS)

Páez, Rocío Isabel; Efthymiopoulos, Christos

2015-02-01

The possibility that giant extrasolar planets could have small Trojan co-orbital companions has been examined in the literature from both viewpoints of the origin and dynamical stability of such a configuration. Here we aim to investigate the dynamics of hypothetical small Trojan exoplanets in domains of secondary resonances embedded within the tadpole domain of motion. To this end, we consider the limit of a massless Trojan companion of a giant planet. Without other planets, this is a case of the elliptic restricted three body problem (ERTBP). The presence of additional planets (hereafter referred to as the restricted multi-planet problem, RMPP) induces new direct and indirect secular effects on the dynamics of the Trojan body. The paper contains a theoretical and a numerical part. In the theoretical part, we develop a Hamiltonian formalism in action-angle variables, which allows us to treat in a unified way resonant dynamics and secular effects on the Trojan body in both the ERTBP or the RMPP. In both cases, our formalism leads to a decomposition of the Hamiltonian in two parts, . , called the basic model, describes resonant dynamics in the short-period (epicyclic) and synodic (libration) degrees of freedom, while contains only terms depending trigonometrically on slow (secular) angles. is formally identical in the ERTBP and the RMPP, apart from a re-definition of some angular variables. An important physical consequence of this analysis is that the slow chaotic diffusion along resonances proceeds in both the ERTBP and the RMPP by a qualitatively similar dynamical mechanism. We found that this is best approximated by the paradigm of `modulational diffusion'. In the paper's numerical part, we then focus on the ERTBP in order to make a detailed numerical demonstration of the chaotic diffusion process along resonances. Using color stability maps, we first provide a survey of the resonant web for characteristic mass parameter values of the primary, in which the secondary resonances from 1:5 to 1:12 (ratio of the short over the synodic period), as well as their transverse resonant multiplets, appear. We give numerical examples of diffusion of weakly chaotic orbits in the resonant web. We finally make a statistics of the escaping times in the resonant domain, and find power-law tails of the distribution of the escaping times for the slowly diffusing chaotic orbits. Implications of resonant dynamics in the search for Trojan exoplanets are discussed.

Nonlinear analysis of the occurrence of hurricanes in the Gulf of Mexico and the Caribbean Sea

NASA Astrophysics Data System (ADS)

Rojo-Garibaldi, Berenice; Salas-de-León, David Alberto; Adela Monreal-Gómez, María; Sánchez-Santillán, Norma Leticia; Salas-Monreal, David

2018-04-01

Hurricanes are complex systems that carry large amounts of energy. Their impact often produces natural disasters involving the loss of human lives and materials, such as infrastructure, valued at billions of US dollars. However, not everything about hurricanes is negative, as hurricanes are the main source of rainwater for the regions where they develop. This study shows a nonlinear analysis of the time series of the occurrence of hurricanes in the Gulf of Mexico and the Caribbean Sea obtained from 1749 to 2012. The construction of the hurricane time series was carried out based on the hurricane database of the North Atlantic basin hurricane database (HURDAT) and the published historical information. The hurricane time series provides a unique historical record on information about ocean-atmosphere interactions. The Lyapunov exponent indicated that the system presented chaotic dynamics, and the spectral analysis and nonlinear analyses of the time series of the hurricanes showed chaotic edge behavior. One possible explanation for this chaotic edge is the individual chaotic behavior of hurricanes, either by category or individually regardless of their category and their behavior on a regular basis.

Chaos-based partial image encryption scheme based on linear fractional and lifting wavelet transforms

NASA Astrophysics Data System (ADS)

Belazi, Akram; Abd El-Latif, Ahmed A.; Diaconu, Adrian-Viorel; Rhouma, Rhouma; Belghith, Safya

2017-01-01

In this paper, a new chaos-based partial image encryption scheme based on Substitution-boxes (S-box) constructed by chaotic system and Linear Fractional Transform (LFT) is proposed. It encrypts only the requisite parts of the sensitive information in Lifting-Wavelet Transform (LWT) frequency domain based on hybrid of chaotic maps and a new S-box. In the proposed encryption scheme, the characteristics of confusion and diffusion are accomplished in three phases: block permutation, substitution, and diffusion. Then, we used dynamic keys instead of fixed keys used in other approaches, to control the encryption process and make any attack impossible. The new S-box was constructed by mixing of chaotic map and LFT to insure the high confidentiality in the inner encryption of the proposed approach. In addition, the hybrid compound of S-box and chaotic systems strengthened the whole encryption performance and enlarged the key space required to resist the brute force attacks. Extensive experiments were conducted to evaluate the security and efficiency of the proposed approach. In comparison with previous schemes, the proposed cryptosystem scheme showed high performances and great potential for prominent prevalence in cryptographic applications.

Non-Stationary Effects and Cross Correlations in Solar Activity

NASA Astrophysics Data System (ADS)

Nefedyev, Yuri; Panischev, Oleg; Demin, Sergey

2016-07-01

In this paper within the framework of the Flicker-Noise Spectroscopy (FNS) we consider the dynamic properties of the solar activity by analyzing the Zurich sunspot numbers. As is well-known astrophysics objects are the non-stationary open systems, whose evolution are the quite individual and have the alternation effects. The main difference of FNS compared to other related methods is the separation of the original signal reflecting the dynamics of solar activity into three frequency bands: system-specific "resonances" and their interferential contributions at lower frequencies, chaotic "random walk" ("irregularity-jump") components at larger frequencies, and chaotic "irregularity-spike" (inertial) components in the highest frequency range. Specific parameters corresponding to each of the bands are introduced and calculated. These irregularities as well as specific resonance frequencies are considered as the information carriers on every hierarchical level of the evolution of a complex natural system with intermittent behavior, consecutive alternation of rapid chaotic changes in the values of dynamic variables on small time intervals with small variations of the values on longer time intervals ("laminar" phases). The jump and spike irregularities are described by power spectra and difference moments (transient structural functions) of the second order. FNS allows revealing the most crucial points of the solar activity dynamics by means of "spikiness" factor. It is shown that this variable behaves as the predictor of crucial changes of the sunspot number dynamics, particularly when the number comes up to maximum value. The change of averaging interval allows revealing the non-stationary effects depending by 11-year cycle and by inside processes in a cycle. To consider the cross correlations between the different variables of solar activity we use the Zurich sunspot numbers and the sequence of corona's radiation energy. The FNS-approach allows extracting the information about cross correlation dynamics between the signals from separate points of the studied system. The 3D cross correlators and their plain projections allow revealing the periodic laws of solar evolution. Work was supported by grants RFBR 15-02-01638-a and 16-02-00496-a.

Chaotic Motions in the Dynamics of Space Tethered Systems. 1. Analysis of the Problem

NASA Astrophysics Data System (ADS)

Pirozhenko, A. V.

The determined-chaos phenomenon in the dynamics of space tethered systems is analyzed. A model problem, the essence of stochastic regimes of motion in the oscillation of masses in the internal degrees of freedom is formulated. A number of calculus approaches to the phenomenon is considered and the supposition is made that it is impossible to define the essence of the phenomenon by the mathematical methods traditional for mechanics.

Quasibound states in a triple Gaussian potential

NASA Astrophysics Data System (ADS)

Reichl, L. E.; Porter, Max D.

2018-04-01

We derive the transmission probabilities and delay times, and identify quasibound state structures in an open quantum system consisting of three Gaussian potential energy peaks, a system whose classical scattering dynamics we show to be chaotic. Such open quantum systems can serve as models for nanoscale quantum devices and their wave dynamics are similar to electromagnetic wave dynamics in optical microcavities. We use a quantum web to determine energy regimes for which the system exhibits the quantum manifestations of chaos, and we show that the classical scattering dynamics contains a significant amount of chaos. We also derive an exact expression for the non-Hermitian Hamiltonian whose eigenvalues give quasibound state energies and lifetimes of the system.

Universality and chaotic dynamics in reactive scattering of ultracold KRb molecules with K atoms

NASA Astrophysics Data System (ADS)

Li, Ming; Makrides, Constantinos; Petrov, Alexander; Kotochigova, Svetlana; Croft, James F. E.; Balakrishnan, Naduvalath; Kendrick, Brian K.

2017-04-01

We study the benchmark reaction between the most-celebrated ultracold polar molecule, KRb, with an ultracold K atom. For the first time we map out an accurate ab initio ground potential energy surface of the K2Rb complex in full dimensionality and performed a numerically exact quantum-mechanical calculation of reaction dynamics based on coupled-channels approach in hyperspherical coordinates. An analysis of the adiabatic hyperspherical potentials reveals a chaotic distribution for the short-range complex that plays a key role in governing the reaction outcome. The equivalent distribution for a lighter collisional system with a smaller density of states (here the Li2Yb trimer) only shows random behavior. We find an extreme sensitivity of our chaotic system to a small perturbation associated with the weak non-additive three-body potential contribution that does not affect the total reaction rate coefficient but leads to a significant change in the rotational distribution in the product molecule. In both cases the distribution of these rates is random or Poissonian. This work was supported in part by NSF Grant PHY-1505557 (N.B.) and PHY-1619788 (S.K.), ARO MURI Grant No. W911NF-12-1-0476 (N.B. & S.K.), and DOE LDRD Grant No. 20170221ER (B.K.).

Revealing the Character of Orbits in a Binary System Consisting of a Primary Galaxy and a Satellite Companion

NASA Astrophysics Data System (ADS)

Zotos, Euaggelos E.

2013-02-01

In this article, we present a galactic gravitational model of three degrees of freedom (3D), in order to study and reveal the character of the orbits of the stars, in a binary stellar system composed of a primary quiet or active galaxy and a small satellite companion galaxy. Our main dynamical analysis will be focused on the behaviour of the primary galaxy. We investigate in detail the regular or chaotic nature of motion, in two different cases: (i) the time-independent model in both 2D and 3D dynamical systems and (ii) the time-evolving 3D model. For the description of the structure of the 2D system, we use the classical method of the Poincaré (x, px ), y = 0, py < 0 phase plane. In order to study the structure of the phase space of the 3D system, we take sections in the plane y = 0 of the 3D orbits, whose initial conditions differ from the plane parent periodic orbits, only by the z component. The set of the four-dimensional points in the (x, px , z, pz ) phase space is projected on the (z, pz ) plane. The maximum Lyapunov characteristic exponent is used in order to make an estimation of the chaoticity of our galactic system, in both 2D and 3D dynamical models. Our numerical calculations indicate that the percentage of the chaotic orbits increases when the primary galaxy has a dense and massive nucleus. The presence of the dense galactic core also increases the stellar velocities near the center of the galaxy. Moreover, for small values of the distance R between the two bodies, low-energy stars display chaotic motion, near the central region of the galaxy, while for larger values of the distance R, the motion in active galaxies is entirely regular for low-energy stars. Our simulations suggest that in galaxies with a satellite companion, the chaotic nature of motion is not only a result of the galactic interaction between the primary galaxy and its companion, but also a result caused by the presence of the dense nucleus in the core of the primary galaxy. Theoretical arguments are presented in order to support and interpret the numerically derived outcomes. Furthermore, we follow the 3D evolution of the primary galaxy, when mass is transported adiabatically from the disk to the nucleus. Our numerical results are in satisfactory agreement with observational data obtained from the M51-type binary stellar systems. A comparison between the present research and similar and earlier work is also made.

Symmetry restoring bifurcations and quasiperiodic chaos induced by a new intermittency in a vibro-impact system.

PubMed

Yue, Yuan; Miao, Pengcheng; Xie, Jianhua; Celso, Grebogi

2016-11-01

Quasiperiodic chaos (QC), which is a combination of quasiperiodic sets and a chaotic set, is uncovered in the six dimensional Poincaré map of a symmetric three-degree of freedom vibro-impact system. Accompanied by symmetry restoring bifurcation, this QC is the consequence of a novel intermittency that occurs between two conjugate quasiperiodic sets and a chaotic set. The six dimensional Poincaré map P is the 2-fold composition of another virtual implicit map Q, yielding the symmetry of the system. Map Q can capture two conjugate attractors, which is at the core of the dynamics of the vibro-impact system. Three types of symmetry restoring bifurcations are analyzed in detail. First, if two conjugate chaotic attractors join together, the chaos-chaos intermittency induced by attractor-merging crisis takes place. Second, if two conjugate quasiperiodic sets are suddenly embedded in a chaotic one, QC is induced by a new intermittency between the three attractors. Third, if two conjugate quasiperiodic attractors connect with each other directly, they merge to form a single symmetric quasiperiodic one. For the second case, the new intermittency is caused by the collision of two conjugate quasiperiodic attractors with an unstable symmetric limit set. As the iteration number is increased, the largest finite-time Lyapunov exponent of the QC does not converge to a constant, but fluctuates in the positive region.

Markov and non-Markov processes in complex systems by the dynamical information entropy

NASA Astrophysics Data System (ADS)

Yulmetyev, R. M.; Gafarov, F. M.

1999-12-01

We consider the Markov and non-Markov processes in complex systems by the dynamical information Shannon entropy (DISE) method. The influence and important role of the two mutually dependent channels of entropy alternation (creation or generation of correlation) and anti-correlation (destroying or annihilation of correlation) have been discussed. The developed method has been used for the analysis of the complex systems of various natures: slow neutron scattering in liquid cesium, psychology (short-time numeral and pattern human memory and effect of stress on the dynamical taping-test), random dynamics of RR-intervals in human ECG (problem of diagnosis of various disease of the human cardio-vascular systems), chaotic dynamics of the parameters of financial markets and ecological systems.

Mixed-mode oscillations and chaos in a prey-predator system with dormancy of predators.

PubMed

Kuwamura, Masataka; Chiba, Hayato

2009-12-01

It is shown that the dormancy of predators induces mixed-mode oscillations and chaos in the population dynamics of a prey-predator system under certain conditions. The mixed-mode oscillations and chaos are shown to bifurcate from a coexisting equilibrium by means of the theory of fast-slow systems. These results may help to find experimental conditions under which one can demonstrate chaotic population dynamics in a simple phytoplankton-zooplankton (-resting eggs) community in a microcosm with a short duration.

A novel high-resolution chaotic lidar with optical injection to chaotic laser diode

NASA Astrophysics Data System (ADS)

Wang, Yun-cai; Wang, An-bang

2008-03-01

A novel chaotic lidar with high resolution is proposed and studied theoretically. In chaotic lidar system, the chaotic laser emitted from chaotic laser diode is split into two beams: the probe and the reference light. The ranging is achieved by correlating the reference waveform with the delayed probe waveform backscattered from the target. In chaotic lidar systems presented previously, the chaotic signal source is laser diode with optical feedback or with optical injection by another one. The ranging resolution is limited by the bandwidth of chaotic laser which determined by the configuration of chaotic signal source. We proposed a novel chaotic lidar which ranging resolution is enhanced significantly by external optical injected chaotic laser diode. With the bandwidth-enhanced chaotic laser, the range resolution of the chaotic lidar system with optical injection is roughly two times compared with that of without optical injection. The resolution increases with injection strength increasing in a certain frequency detuning range.

Emergence and transitions of dynamic patterns of thickness oscillation of the plasmodium of the true slime mold Physarum polycephalum

NASA Astrophysics Data System (ADS)

Takagi, Seiji; Ueda, Tetsuo

2008-03-01

The emergence and transitions of various spatiotemporal patterns of thickness oscillation were studied in the freshly isolated protoplasm of the Physarum plasmodium. New patterns, such as standing waves, and chaotic and rotating spirals, developed successively before the well-documented synchronous pattern appeared. There was also a spontaneous opposite transition from synchrony to chaotic and rotating spirals. Rotating spiral waves were observed in the large migrating plasmodium, where the vein structures were being destroyed. Thus, the Physarum plasmodium exhibits versatile patterns, which are generally expected in coupled oscillator systems. This paper discusses the physiological roles of spatiotemporal patterns, comparing them with other biological systems.

Scale relativity: from quantum mechanics to chaotic dynamics.

NASA Astrophysics Data System (ADS)

Nottale, L.

Scale relativity is a new approach to the problem of the origin of fundamental scales and of scaling laws in physics, which consists in generalizing Einstein's principle of relativity to the case of scale transformations of resolutions. We recall here how it leads one to the concept of fractal space-time, and to introduce a new complex time derivative operator which allows to recover the Schrödinger equation, then to generalize it. In high energy quantum physics, it leads to the introduction of a Lorentzian renormalization group, in which the Planck length is reinterpreted as a lowest, unpassable scale, invariant under dilatations. These methods are successively applied to two problems: in quantum mechanics, that of the mass spectrum of elementary particles; in chaotic dynamics, that of the distribution of planets in the Solar System.

Quantum Tunneling and Chaos in Classical Scale Walkers

NASA Astrophysics Data System (ADS)

Su, Jenny; Dijksman, Joshua; Ward, Jeremy; Behringer, Robert

2014-03-01

We study the behavior of `walkers' small droplets bouncing on a fluid layer vibrated at amplitudes just below the onset of Faraday instability. It was shown recently that despite their macroscopic size, the droplet dynamics are stochastic in nature and reminiscent of the dual particle-wave dynamics in the realm of quantum mechanics (Couder PRL 2006). We use these walkers to study how chaos, which is macroscopically unpredictable, will manifest in a quantum setting. Pecora showed in 2011 that tunneling for particles that have a chaotic ground state is different from tunneling for particles with a regular ground state (PRE 2011). In the experiment we gather data that illustrates the particle trajectory and tunneling behavior as particles transition across the barrier in the double well system with both integrable and chaotic shapes.

Noise, chaos, and (ɛ, τ)-entropy per unit time

NASA Astrophysics Data System (ADS)

Gaspard, Pierre; Wang, Xiao-Jing

1993-12-01

The degree of dynamical randomness of different time processes is characterized in terms of the (ε, τ)-entropy per unit time. The (ε, τ)-entropy is the amount of information generated per unit time, at different scales τ of time and ε of the observables. This quantity generalizes the Kolmogorov-Sinai entropy per unit time from deterministic chaotic processes, to stochastic processes such as fluctuations in mesoscopic physico-chemical phenomena or strong turbulence in macroscopic spacetime dynamics. The random processes that are characterized include chaotic systems, Bernoulli and Markov chains, Poisson and birth-and-death processes, Ornstein-Uhlenbeck and Yaglom noises, fractional Brownian motions, different regimes of hydrodynamical turbulence, and the Lorentz-Boltzmann process of nonequilibrium statistical mechanics. We also extend the (ε, τ)-entropy to spacetime processes like cellular automata, Conway's game of life, lattice gas automata, coupled maps, spacetime chaos in partial differential equations, as well as the ideal, the Lorentz, and the hard sphere gases. Through these examples it is demonstrated that the (ε, τ)-entropy provides a unified quantitative measure of dynamical randomness to both chaos and noises, and a method to detect transitions between dynamical states of different degrees of randomness as a parameter of the system is varied.

From Weakly Chaotic Dynamics to Deterministic Subdiffusion via Copula Modeling

NASA Astrophysics Data System (ADS)

Nazé, Pierre

2018-03-01

Copula modeling consists in finding a probabilistic distribution, called copula, whereby its coupling with the marginal distributions of a set of random variables produces their joint distribution. The present work aims to use this technique to connect the statistical distributions of weakly chaotic dynamics and deterministic subdiffusion. More precisely, we decompose the jumps distribution of Geisel-Thomae map into a bivariate one and determine the marginal and copula distributions respectively by infinite ergodic theory and statistical inference techniques. We verify therefore that the characteristic tail distribution of subdiffusion is an extreme value copula coupling Mittag-Leffler distributions. We also present a method to calculate the exact copula and joint distributions in the case where weakly chaotic dynamics and deterministic subdiffusion statistical distributions are already known. Numerical simulations and consistency with the dynamical aspects of the map support our results.

Introduction

NASA Astrophysics Data System (ADS)

Cohen, E. G. D.

Lecture notes are organized around the key word dissipation, while focusing on a presentation of modern theoretical developments in the study of irreversible phenomena. A broad cross-disciplinary perspective towards non-equilibrium statistical mechanics is backed by the general theory of nonlinear and complex dynamical systems. The classical-quantum intertwine and semiclassical dissipative borderline issue (decoherence, "classical out of quantum") are here included . Special emphasis is put on links between the theory of classical and quantum dynamical systems (temporal disorder, dynamical chaos and transport processes) with central problems of non-equilibrium statistical mechanics like e.g. the connection between dynamics and thermodynamics, relaxation towards equilibrium states and mechanisms capable to drive and next maintain the physical system far from equilibrium, in a non-equilibrium steady (stationary) state. The notion of an equilibrium state - towards which a system naturally evolves if left undisturbed - is a fundamental concept of equilibrium statistical mechanics. Taken as a primitive point of reference that allows to give an unambiguous status to near equilibrium and far from equilibrium systems, together with the dynamical notion of a relaxation (decay) towards a prescribed asymptotic invariant measure or probability distribution (properties of ergodicity and mixing are implicit). A related issue is to keep under control the process of driving a physical system away from an initial state of equilibrium and either keeping it in another (non-equilibrium) steady state or allowing to restore the initial data (return back, relax). To this end various models of environment (heat bath, reservoir, thermostat, measuring instrument etc.), and the environment - system coupling are analyzed. The central theme of the book is the dynamics of dissipation and various mechanisms responsible for the irreversible behaviour (transport properties) of open systems on classical and quantum levels of description. A distinguishing feature of these lecture notes is that microscopic foundations of irreversibility are investigated basically in terms of "small" systems, when the "system" and/or "environment" may have a finite (and small) number of degrees of freedom and may be bounded. This is to be contrasted with the casual understanding of statistical mechanics which is regarded to refer to systems with a very large number of degrees of freedom. In fact, it is commonly accepted that the accumulation of effects due to many (range of the Avogadro number) particles is required for statistical mechanics reasoning. Albeit those large numbers are not at all sufficient for transport properties. A helpful hint towards this conceptual turnover comes from the observation that for chaotic dynamical systems the random time evolution proves to be compatible with the underlying purely deterministic laws of motion. Chaotic features of the classical dynamics already appear in systems with two degrees of freedom and such systems need to be described in statistical terms, if we wish to quantify the dynamics of relaxation towards an invariant ergodic measure. The relaxation towards equilibrium finds a statistical description through an analysis of statistical ensembles. This entails an extension of the range of validity of statistical mechanics to small classical systems. On the other hand, the dynamics of fluctuations in macroscopic dissipative systems (due to their molecular composition and thermal mobility) may render a characterization of such systems as being chaotic. That motivates attempts of understanding the role of microscopic chaos and various "chaotic hypotheses" - dynamical systems approach is being pushed down to the level of atoms, molecules and complex matter constituents, whose natural substitute are low-dimensional model subsystems (encompassing as well the mesoscopic "quantum chaos") - in non-equilibrium transport phenomena. On the way a number of questions is addressed like e.g.: is there, or what is the nature of a connection between chaos (modern theory of dynamical systems) and irreversible thermodynamics; can really quantum chaos explain some peculiar features of quantum transport? The answer in both cases is positive, modulo a careful discrimination between viewing the dynamical chaos as a necessary or sufficient basis for irreversibility. In those dynamical contexts, another key term dynamical semigroups refers to major technical tools appropriate for the "dissipative mathematics", modelling irreversible behaviour on the classical and quantum levels of description. Dynamical systems theory and "quantum chaos" research involve both a high level of mathematical sophistication and heavy computer "experimentation". One of the present volume specific flavors is a tutorial access to quite advanced mathematical tools. They gradually penetrate the classical and quantum dynamical semigroup description, while culminating in the noncommutative Brillouin zone construction as a prerequisite to understand transport in aperiodic solids. Lecture notes are structured into chapters to give a better insight into major conceptual streamlines. Chapter I is devoted to a discussion of non-equilibrium steady states and, through so-called chaotic hypothesis combined with suitable fluctuation theorems, elucidates the role of Sinai-Ruelle-Bowen distribution in both equilibrium and non-equilibrium statistical physics frameworks (E. G. D. Cohen). Links between dynamics and statistics (Boltzmann versus Tsallis) are also discussed. Fluctuation relations and a survey of deterministic thermostats are given in the context of non-equilibrium steady states of fluids (L. Rondoni). Response of systems driven far from equilibrium is analyzed on the basis of a central assertion about the existence of the statistical representation in terms of an ensemble of dynamical realizations of the driving process. Non-equilibrium work relation is deduced for irreversible processes (C. Jarzynski). The survey of non-equilibrium steady states in statistical mechanics of classical and quantum systems employs heat bath models and the random matrix theory input. The quantum heat bath analysis and derivation of fluctuation-dissipation theorems is performed by means of the influence functional technique adopted to solve quantum master equations (D. Kusnezov). Chapter II deals with an issue of relaxation and its dynamical theory in both classical and quantum contexts. Pollicott-Ruelle resonance background for the exponential decay scenario is discussed for irreversible processes of diffusion in the Lorentz gas and multibaker models (P. Gaspard). The Pollicott-Ruelle theory reappears as a major inspiration in the survey of the behaviour of ensembles of chaotic systems, with a focus on model systems for which no rigorous results concerning the exponential decay of correlations in time is available (S. Fishman). The observation, that non-equilibrium transport processes in simple classical chaotic systems can be described in terms of fractal structures developing in the system phase space, links their formation and properties with the entropy production in the course of diffusion processes displaying a low dimensional deterministic (chaotic) origin (J. R. Dorfman). Chapter III offers an introduction to the theory of dynamical semigroups. Asymptotic properties of Markov operators and Markov semigroups acting in the set of probability densities (statistical ensemble notion is implicit) are analyzed. Ergodicity, mixing, strong (complete) mixing and sweeping are discussed in the familiar setting of "noise, chaos and fractals" (R. Rudnicki). The next step comprises a passage to quantum dynamical semigroups and completely positive dynamical maps, with an ultimate goal to introduce a consistent framework for the analysis of irreversible phenomena in open quantum systems, where dissipation and decoherence are crucial concepts (R. Alicki). Friction and damping in classical and quantum mechanics of finite dissipative systems is analyzed by means of Markovian quantum semigroups with special emphasis on the issue of complete positivity (M. Fannes). Specific two-level model systems of elementary particle physics (kaons) and rudiments of neutron interferometry are employed to elucidate a distinction between positivity and complete positivity (F. Benatti). Quantization of dynamics of stochastic models related to equilibrium Gibbs states results in dynamical maps which form quantum stochastic dynamical semigroups (W. A. Majewski). Chapter IV addresses diverse but deeply interrelated features of driven chaotic (mesoscopic) classical and quantum systems, their dissipative properties, notions of quantum irreversibility, entanglement, dephasing and decoherence. A survey of non-perturbative quantum effects for open quantum systems is concluded by outlining the discrepancies between random matrix theory and non-perturbative semiclassical predictions (D. Cohen). As a useful supplement to the subject of bounded open systems, methods of quantum state control in a cavity (coherent versus incoherent dynamics and dissipation) are described for low dimensional quantum systems (A. Buchleitner). The dynamics of open quantum systems can be alternatively described by means of non-Markovian stochastic Schrödinger equation, jointly for an open system and its environment, which moves us beyond the Linblad evolution scenario of Markovian dynamical semigroups. The quantum Brownian motion is considered (W. Strunz) . Chapter V enforces a conceptual transition 'from "small" to "large" systems with emphasis on irreversible thermodynamics of quantum transport. Typical features of the statistical mechanics of infinitely extended systems and the dynamical (small) systems approach are described by means of representative examples of relaxation towards asymptotic steady states: quantum one-dimensional lattice conductor and an open multibaker map (S. Tasaki). Dissipative transport in aperiodic solids is reviewed by invoking methods on noncommutative geometry. The anomalous Drude formula is derived. The occurence of quantum chaos is discussed together with its main consequences (J. Bellissard). The chapter is concluded by a survey of scaling limits of the N-body Schrödinger quantum dynamics, where classical evolution equations of irreversible statistical mechanics (linear Boltzmann, Hartree, Vlasov) emerge "out of quantum". In particular, a scaling limit of one body quantum dynamics with impurities (static random potential) and that of quantum dynamics with weakly coupled phonons are shown to yield the linear Boltzmann equation (L. Erdös). Various interrelations between chapters and individual lectures, plus a detailed fine-tuned information about the subject matter coverage of the volume, can be recovered by examining an extensive index.

Chaotic non-planar vibrations of the thin elastica. Part I: Experimental observation of planar instability

NASA Astrophysics Data System (ADS)

Cusumano, J. P.; Moon, F. C.

1995-01-01

In this two-part paper, the results of an investigation into the non-linear dynamics of a flexible cantilevered rod (the elastica) with a thin rectangular cross-section are presented. An experimental examination of the dynamics of the elastica over a broad parameter range forms the core of Part I. In Part II, the experimental work is related to a theoretical study of the mechanics of the elastica, and the study of a two-degree-of-freedom model obtained by modal projection. The experimental system used in this investigation is a rod with clamped-free boundary conditions, forced by sinusoidally displacing the clamped end. Planar periodic motions of the driven elastica are shown to lose stability at distinct resonant wedges, and the resulting motions are shown in general to be non-planar, chaotic, bending-torsion oscillations. Non-planar motions in all resonances exhibit energy cascading and dynamic two-well phenomena, and a family of asymmetric, bending-torsion non-linear modes is discovered. Correlation dimension calculations are used to estimate the number of active degrees of freedom in the system.

Chaos for cardiac arrhythmias through a one-dimensional modulation equation for alternans

PubMed Central

Dai, Shu; Schaeffer, David G.

2010-01-01

Instabilities in cardiac dynamics have been widely investigated in recent years. One facet of this work has studied chaotic behavior, especially possible correlations with fatal arrhythmias. Previously chaotic behavior was observed in various models, specifically in the breakup of spiral and scroll waves. In this paper we study cardiac dynamics and find spatiotemporal chaotic behavior through the Echebarria–Karma modulation equation for alternans in one dimension. Although extreme parameter values are required to produce chaos in this model, it seems significant mathematically that chaos may occur by a different mechanism from previous observations. PMID:20590327

Dynamical noise filter and conditional entropy analysis in chaos synchronization.

PubMed

Wang, Jiao; Lai, C-H

2006-06-01

It is shown that, in a chaotic synchronization system whose driving signal is exposed to channel noise, the estimation of the drive system states can be greatly improved by applying the dynamical noise filtering to the response system states. If the noise is bounded in a certain range, the estimation errors, i.e., the difference between the filtered responding states and the driving states, can be made arbitrarily small. This property can be used in designing an alternative digital communication scheme. An analysis based on the conditional entropy justifies the application of dynamical noise filtering in generating quality synchronization.

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.

Nonlinear Time-Reversal in a Wave Chaotic System

NASA Astrophysics Data System (ADS)

Frazier, Matthew; Taddese, Biniyam; Ott, Edward; Antonsen, Thomas; Anlage, Steven

2012-02-01

Time reversal mirrors are particularly simple to implement in wave chaotic systems and form the basis for a new class of sensors [1-3]. These sensors work by applying the quantum mechanical concepts of Loschmidt echo and fidelity decay to classical waves. The sensors make explicit use of time-reversal invariance and spatial reciprocity in a wave chaotic system to remotely measure the presence of small perturbations to the system. The underlying ray chaos increases the sensitivity to small perturbations throughout the volume explored by the waves. We extend our time-reversal mirror to include a discrete element with a nonlinear dynamical response. The initially injected pulse interacts with the nonlinear element, generating new frequency components originating at the element. By selectively filtering for and applying the time-reversal mirror to the new frequency components, we focus a pulse only onto the element, without knowledge of its location. Furthermore, we demonstrate transmission of arbitrary patterns of pulses to the element, creating a targeted communication channel to the exclusion of 'eavesdroppers' at other locations in the system. [1] Appl. Phys. Lett. 95, 114103 (2009) [2] J. Appl. Phys. 108, 1 (2010) [3] Acta Physica Polonica A 112, 569 (2007)

Amplitude-Phase Modulation, Topological Horseshoe and Scaling Attractor of a Dynamical System

NASA Astrophysics Data System (ADS)

Li, Chun-Lai; Li, Wen; Zhang, Jing; Xie, Yuan-Xi; Zhao, Yi-Bo

2016-09-01

A three-dimensional autonomous chaotic system is discussed in this paper. Some basic dynamical properties of the system, including phase portrait, Poincaré map, power spectrum, Kaplan-Yorke dimension, Lyapunov exponent spectra, signal amplitude and topological horseshoe are studied theoretically and numerically. The main finding by analysis is that the signal amplitude can be modulated via controlling the coefficients of the linear term, cross-product term and squared term simultaneously or respectively, and the phase of x3 can be modulated by the product of the coefficients of the linear term and cross-product term. Furthermore, scaling chaotic attractors of this system are achieved by modified projective synchronization with an optimization-based linear coupling method, which is safer for secure communications than the existed synchronization scheme since the scaling factors can be regarded as the security encoding key. Supported by Hunan Provincial Natural Science Foundation of China under Grant No. 2016JJ4036, University Natural Science Foundation of Jiangsu Province under Grant No. 14KJB120007 and the National Natural Science Foundation of China under Grant Nos. 11504176 and 11602084

On the chaotic diffusion in multidimensional Hamiltonian systems

NASA Astrophysics Data System (ADS)

Cincotta, P. M.; Giordano, C. M.; Martí, J. G.; Beaugé, C.

2018-01-01

We present numerical evidence that diffusion in the herein studied multidimensional near-integrable Hamiltonian systems departs from a normal process, at least for realistic timescales. Therefore, the derivation of a diffusion coefficient from a linear fit on the variance evolution of the unperturbed integrals fails. We review some topics on diffusion in the Arnold Hamiltonian and yield numerical and theoretical arguments to show that in the examples we considered, a standard coefficient would not provide a good estimation of the speed of diffusion. However, numerical experiments concerning diffusion would provide reliable information about the stability of the motion within chaotic regions of the phase space. In this direction, we present an extension of previous results concerning the dynamical structure of the Laplace resonance in Gliese-876 planetary system considering variations of the orbital parameters accordingly to the error introduced by the radial velocity determination. We found that a slight variation of the eccentricity of planet c would destabilize the inner region of the resonance that, though chaotic, shows stable when adopting the best fit values for the parameters.

Hierarchical structure in sharply divided phase space for the piecewise linear map

NASA Astrophysics Data System (ADS)

Akaishi, Akira; Aoki, Kazuki; Shudo, Akira

2017-05-01

We have studied a two-dimensional piecewise linear map to examine how the hierarchical structure of stable regions affects the slow dynamics in Hamiltonian systems. In the phase space there are infinitely many stable regions, each of which is polygonal-shaped, and the rest is occupied by chaotic orbits. By using symbolic representation of stable regions, a procedure to compute the edges of the polygons is presented. The stable regions are hierarchically distributed in phase space and the edges of the stable regions show the marginal instability. The cumulative distribution of the recurrence time obeys a power law as ˜t-2 , the same as the one for the system with phase space, which is composed of a single stable region and chaotic components. By studying the symbol sequence of recurrence trajectories, we show that the hierarchical structure of stable regions has no significant effect on the power-law exponent and that only the marginal instability on the boundary of stable regions is responsible for determining the exponent. We also discuss the relevance of the hierarchical structure to those in more generic chaotic systems.

Chaos, Fractals and Their Applications

NASA Astrophysics Data System (ADS)

Thompson, J. Michael T.

2016-12-01

This paper gives an up-to-date account of chaos and fractals, in a popular pictorial style for the general scientific reader. A brief historical account covers the development of the subject from Newton’s laws of motion to the astronomy of Poincaré and the weather forecasting of Lorenz. Emphasis is given to the important underlying concepts, embracing the fractal properties of coastlines and the logistics of population dynamics. A wide variety of applications include: NASA’s discovery and use of zero-fuel chaotic “superhighways” between the planets; erratic chaotic solutions generated by Euler’s method in mathematics; atomic force microscopy; spontaneous pattern formation in chemical and biological systems; impact mechanics in offshore engineering and the chatter of cutting tools; controlling chaotic heartbeats. Reference is made to a number of interactive simulations and movies accessible on the web.

Designing attractive models via automated identification of chaotic and oscillatory dynamical regimes.

PubMed

Silk, Daniel; Kirk, Paul D W; Barnes, Chris P; Toni, Tina; Rose, Anna; Moon, Simon; Dallman, Margaret J; Stumpf, Michael P H

2011-10-04

Chaos and oscillations continue to capture the interest of both the scientific and public domains. Yet despite the importance of these qualitative features, most attempts at constructing mathematical models of such phenomena have taken an indirect, quantitative approach, for example, by fitting models to a finite number of data points. Here we develop a qualitative inference framework that allows us to both reverse-engineer and design systems exhibiting these and other dynamical behaviours by directly specifying the desired characteristics of the underlying dynamical attractor. This change in perspective from quantitative to qualitative dynamics, provides fundamental and new insights into the properties of dynamical systems.

Synchronization of unidirectionally delay-coupled chaotic oscillators with memory

NASA Astrophysics Data System (ADS)

Jaimes-Reátegui, Rider; Vera-Ávila, Victor P.; Sevilla-Escoboza, Ricardo; Huerta-Cuéllar, Guillermo; Castañeda-Hernández, Carlos E.; Chiu-Zarate, Roger; Pisarchik, Alexander N.

2016-11-01

We study synchronization of two chaotic oscillators coupled with time delay in a master-slave configuration and with delayed positive feedback in the slave oscillator which acts as memory. The dynamics of the slave oscillator is analyzed with bifurcation diagrams of the peak value of the system variable with respect to the coupling and feedback strengths and two delay times. For small coupling, when the oscillators' phases synchronize, memory can induce bistability and stabilize periodic orbits, whereas for stronger coupling it is not possible. The delayed feedback signal impairs synchronization, simultaneously enhancing coherence of the slave oscillator.

Impact of inelastic processes on the chaotic dynamics of a Bose-Einstein condensate trapped into a moving optical lattice

NASA Astrophysics Data System (ADS)

Tchatchueng, Sylvin; Siewe Siewe, Martin; Marie Moukam Kakmeni, François; Tchawoua, Clément

2017-03-01

We investigate the dynamics of a Bose-Einstein condensate with attractive two-body and repulsive three-body interactions between atoms trapped into a moving optical lattice and subjected to some inelastic processes (a linear atomic feeding and two dissipative terms related to dipolar relaxation and three-body recombination). We are interested in finding out how the nonconservative terms mentioned above act on the dynamical behaviour of the condensate, and how they can be used in the control of possible chaotic dynamics. Seeking the wave function of condensate on the form of Bloch waves, we notice that the real amplitude of the condensate is governed by an integro-differential equation. As theoretical tool of prediction of homoclinic and heteroclinic chaos, we use the Melnikov method, which provides two Melnikov functions related to homoclinic and heteroclinic bifurcations. Applying the Melnikov criterion, some regions of instability are plotted in the parameter space and reveal complex dynamics (solitonic stable solutions, weak and strong instabilities leading to collapse, growth-collapse cycles and finally to chaotic oscillations). It comes from some parameter space that coupling the optical intensity and parameters related to atomic feeding and atomic losses (dissipations) as control parameters can help to reduce or annihilate chaotic behaviours of the condensate. Moreover, the theoretical study reveals that there is a certain ratio between the atomic feeding parameter and the parameters related to the dissipation for the occurrence of chaotic oscillations in the dynamics of condensate. The theoretical predictions are verified by numerical simulations (Poincaré sections), and there is a certain reliability of our analytical treatment.

Hydrodynamically induced oscillations and traffic dynamics in 1D microfludic networks

NASA Astrophysics Data System (ADS)

Bartolo, Denis; Jeanneret, Raphael

2011-03-01

We report on the traffic dynamics of particles driven through a minimal microfluidic network. Even in the minimal network consisting in a single loop, the traffic dynamics has proven to yield complex temporal patterns, including periodic, multi-periodic or chaotic sequences. This complex dynamics arises from the strongly nonlinear hydrodynamic interactions between the particles, that takes place at a junction. To better understand the consequences of this nontrivial coupling, we combined theoretical, numerical and experimental efforts and solved the 3-body problem in a 1D loop network. This apparently simple dynamical system revealed a rich and unexpected dynamics, including coherent spontaneous oscillations along closed orbits. Striking similarities between Hamiltonian systems and this driven dissipative system will be explained.

A bifurcation giving birth to order in an impulsively driven complex system

NASA Astrophysics Data System (ADS)

Seshadri, Akshay; Sujith, R. I.

2016-08-01

Nonlinear oscillations lie at the heart of numerous complex systems. Impulsive forcing arises naturally in many scenarios, and we endeavour to study nonlinear oscillators subject to such forcing. We model these kicked oscillatory systems as a piecewise smooth dynamical system, whereby their dynamics can be investigated. We investigate the problem of pattern formation in a turbulent combustion system and apply this formalism with the aim of explaining the observed dynamics. We identify that the transition of this system from low amplitude chaotic oscillations to large amplitude periodic oscillations is the result of a discontinuity induced bifurcation. Further, we provide an explanation for the occurrence of intermittent oscillations in the system.

A bifurcation giving birth to order in an impulsively driven complex system

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

Seshadri, Akshay, E-mail: akshayseshadri@gmail.com; Sujith, R. I., E-mail: sujith@iitm.ac.in

Nonlinear oscillations lie at the heart of numerous complex systems. Impulsive forcing arises naturally in many scenarios, and we endeavour to study nonlinear oscillators subject to such forcing. We model these kicked oscillatory systems as a piecewise smooth dynamical system, whereby their dynamics can be investigated. We investigate the problem of pattern formation in a turbulent combustion system and apply this formalism with the aim of explaining the observed dynamics. We identify that the transition of this system from low amplitude chaotic oscillations to large amplitude periodic oscillations is the result of a discontinuity induced bifurcation. Further, we provide anmore » explanation for the occurrence of intermittent oscillations in the system.« less

Flatness-based adaptive fuzzy control of chaotic finance dynamics

NASA Astrophysics Data System (ADS)

Rigatos, G.; Siano, P.; Loia, V.; Tommasetti, A.; Troisi, O.

2017-11-01

A flatness-based adaptive fuzzy control is applied to the problem of stabilization of the dynamics of a chaotic finance system, describing interaction between the interest rate, the investment demand and the price exponent. By proving that the system is differentially flat and by applying differential flatness diffeomorphisms, its transformation to the linear canonical (Brunovsky) is performed. For the latter description of the system, the design of a stabilizing state feedback controller becomes possible. A first problem in the design of such a controller is that the dynamic model of the finance system is unknown and thus it has to be identified with the use neurofuzzy approximators. The estimated dynamics provided by the approximators is used in the computation of the control input, thus establishing an indirect adaptive control scheme. The learning rate of the approximators is chosen from the requirement the system's Lyapunov function to have always a negative first-order derivative. Another problem that has to be dealt with is that the control loop is implemented only with the use of output feedback. To estimate the non-measurable state vector elements of the finance system, a state observer is implemented in the control loop. The computation of the feedback control signal requires the solution of two algebraic Riccati equations at each iteration of the control algorithm. Lyapunov stability analysis demonstrates first that an H-infinity tracking performance criterion is satisfied. This signifies elevated robustness against modelling errors and external perturbations. Moreover, the global asymptotic stability is proven for the control loop.

The chaotic dynamical aperture

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

Lee, S.Y.; Tepikian, S.

1985-10-01

Nonlinear magnetic forces become more important for particles in the modern large accelerators. These nonlinear elements are introduced either intentionally to control beam dynamics or by uncontrollable random errors. Equations of motion in the nonlinear Hamiltonian are usually non-integrable. Because of the nonlinear part of the Hamiltonian, the tune diagram of accelerators is a jungle. Nonlinear magnet multipoles are important in keeping the accelerator operation point in the safe quarter of the hostile jungle of resonant tunes. Indeed, all the modern accelerator design have taken advantages of nonlinear mechanics. On the other hand, the effect of the uncontrollable random multipolesmore » should be evaluated carefully. A powerful method of studying the effect of these nonlinear multipoles is using a particle tracking calculation, where a group of test particles are tracing through these magnetic multipoles in the accelerator hundreds to millions of turns in order to test the dynamical aperture of the machine. These methods are extremely useful in the design of a large accelerator such as SSC, LEP, HERA and RHIC. These calculations unfortunately take tremendous amount of computing time. In this paper, we try to apply the existing method in the nonlinear dynamics to study the possible alternative solution. When the Hamiltonian motion becomes chaotic, the tune of the machine becomes undefined. The aperture related to the chaotic orbit can be identified as chaotic dynamical aperture. We review the method of determining chaotic orbit and apply the method to nonlinear problems in accelerator physics. We then discuss the scaling properties and effect of random sextupoles.« less

Synthetic Modeling of Autonomous Learning with a Chaotic Neural Network

NASA Astrophysics Data System (ADS)

Funabashi, Masatoshi

We investigate the possible role of intermittent chaotic dynamics called chaotic itinerancy, in interaction with nonsupervised learnings that reinforce and weaken the neural connection depending on the dynamics itself. We first performed hierarchical stability analysis of the Chaotic Neural Network model (CNN) according to the structure of invariant subspaces. Irregular transition between two attractor ruins with positive maximum Lyapunov exponent was triggered by the blowout bifurcation of the attractor spaces, and was associated with riddled basins structure. We secondly modeled two autonomous learnings, Hebbian learning and spike-timing-dependent plasticity (STDP) rule, and simulated the effect on the chaotic itinerancy state of CNN. Hebbian learning increased the residence time on attractor ruins, and produced novel attractors in the minimum higher-dimensional subspace. It also augmented the neuronal synchrony and established the uniform modularity in chaotic itinerancy. STDP rule reduced the residence time on attractor ruins, and brought a wide range of periodicity in emerged attractors, possibly including strange attractors. Both learning rules selectively destroyed and preserved the specific invariant subspaces, depending on the neuron synchrony of the subspace where the orbits are situated. Computational rationale of the autonomous learning is discussed in connectionist perspective.

Child allowances, fertility, and chaotic dynamics.

PubMed

Chen, Hung-Ju; Li, Ming-Chia

2013-06-01

This paper analyzes the dynamics in an overlapping generations model with the provision of child allowances. Fertility is an increasing function of child allowances and there exists a threshold effect of the marginal effect of child allowances on fertility. We show that if the effectiveness of child allowances is sufficiently high, an intermediate-sized tax rate will be enough to generate chaotic dynamics. Besides, a decrease in the inter-temporal elasticity of substitution will prevent the occurrence of irregular cycles.

Macro- and micro-chaotic structures in the Hindmarsh-Rose model of bursting neurons

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

Barrio, Roberto, E-mail: rbarrio@unizar.es; Serrano, Sergio; Angeles Martínez, M.

2014-06-01

We study a plethora of chaotic phenomena in the Hindmarsh-Rose neuron model with the use of several computational techniques including the bifurcation parameter continuation, spike-quantification, and evaluation of Lyapunov exponents in bi-parameter diagrams. Such an aggregated approach allows for detecting regions of simple and chaotic dynamics, and demarcating borderlines—exact bifurcation curves. We demonstrate how the organizing centers—points corresponding to codimension-two homoclinic bifurcations—along with fold and period-doubling bifurcation curves structure the biparametric plane, thus forming macro-chaotic regions of onion bulb shapes and revealing spike-adding cascades that generate micro-chaotic structures due to the hysteresis.

Quorum Sensing in Populations of Spatially Extended Chaotic Oscillators Coupled Indirectly via a Heterogeneous Environment

NASA Astrophysics Data System (ADS)

Li, Bing-Wei; Cao, Xiao-Zhi; Fu, Chenbo

2017-12-01

Many biological and chemical systems could be modeled by a population of oscillators coupled indirectly via a dynamical environment. Essentially, the environment by which the individual element communicates with each other is heterogeneous. Nevertheless, most of previous works considered the homogeneous case only. Here we investigated the dynamical behaviors in a population of spatially distributed chaotic oscillators immersed in a heterogeneous environment. Various dynamical synchronization states (such as oscillation death, phase synchronization, and complete synchronized oscillation) as well as their transitions were explored. In particular, we uncovered a non-traditional quorum sensing transition: increasing the population density leaded to a transition from oscillation death to synchronized oscillation at first, but further increasing the density resulted in degeneration from complete synchronization to phase synchronization or even from phase synchronization to desynchronization. The underlying mechanism of this finding was attributed to the dual roles played by the population density. What's more, by treating the environment as another component of the oscillator, the full system was then effectively equivalent to a locally coupled system. This fact allowed us to utilize the master stability functions approach to predict the occurrence of complete synchronization oscillation, which agreed with that from the direct numerical integration of the system. The potential candidates for the experimental realization of our model were also discussed.

Quantum and classical chaos in kicked coupled Jaynes-Cummings cavities

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

Hayward, A. L. C.; Greentree, Andrew D.

2010-06-15

We consider two Jaynes-Cummings cavities coupled periodically with a photon hopping term. The semiclassical phase space is chaotic, with regions of stability over some ranges of the parameters. The quantum case exhibits dynamic localization and dynamic tunneling between classically forbidden regions. We explore the correspondence between the classical and quantum phase space and propose an implementation in a circuit QED system.

On the control of the chaotic attractors of the 2-d Navier-Stokes equations.

PubMed

Smaoui, Nejib; Zribi, Mohamed

2017-03-01

The control problem of the chaotic attractors of the two dimensional (2-d) Navier-Stokes (N-S) equations is addressed in this paper. First, the Fourier Galerkin method based on a reduced-order modelling approach developed by Chen and Price is applied to the 2-d N-S equations to construct a fifth-order system of nonlinear ordinary differential equations (ODEs). The dynamics of the fifth-order system was studied by analyzing the system's attractor for different values of Reynolds number, R e . Then, control laws are proposed to drive the states of the ODE system to a desired attractor. Finally, an adaptive controller is designed to synchronize two reduced order ODE models having different Reynolds numbers and starting from different initial conditions. Simulation results indicate that the proposed control schemes work well.

Synchronization of strange non-chaotic attractors via unidirectional coupling of quasiperiodically-forced systems

NASA Astrophysics Data System (ADS)

Sivaganesh, G.; Daniel Sweetlin, M.; Arulgnanam, A.

2016-07-01

In this paper, we present a numerical investigation on the robust synchronization phenomenon observed in a unidirectionally-coupled quasiperiodically-forced simple nonlinear electronic circuit system exhibiting strange non-chaotic attractors (SNAs) in its dynamics. The SNA obtained in the simple quasiperiodic system is characterized for its SNA behavior. Then, we studied the nature of the synchronized state in unidirectionally coupled SNAs by using the Master-Slave approach. The stability of the synchronized state is studied through the master stability functions (MSF) obtained for coupling different state variables of the drive and response system. The property of robust synchronization is analyzed for one type of coupling of the state variables through phase portraits, conditional lyapunov exponents and the Kaplan-Yorke dimension. The phenomenon of complete synchronization of SNAs via a unidirectional coupling scheme is reported for the first time.

On the control of the chaotic attractors of the 2-d Navier-Stokes equations

NASA Astrophysics Data System (ADS)

Smaoui, Nejib; Zribi, Mohamed

2017-03-01

The control problem of the chaotic attractors of the two dimensional (2-d) Navier-Stokes (N-S) equations is addressed in this paper. First, the Fourier Galerkin method based on a reduced-order modelling approach developed by Chen and Price is applied to the 2-d N-S equations to construct a fifth-order system of nonlinear ordinary differential equations (ODEs). The dynamics of the fifth-order system was studied by analyzing the system's attractor for different values of Reynolds number, Re. Then, control laws are proposed to drive the states of the ODE system to a desired attractor. Finally, an adaptive controller is designed to synchronize two reduced order ODE models having different Reynolds numbers and starting from different initial conditions. Simulation results indicate that the proposed control schemes work well.

A nonlinear controller design for permanent magnet motors using a synchronization-based technique inspired from the Lorenz system.

PubMed

Zaher, Ashraf A

2008-03-01

The dynamic behavior of a permanent magnet synchronous machine (PMSM) is analyzed. Nominal and special operating conditions are explored to show that the PMSM can experience chaos. A nonlinear controller is introduced to control these unwanted chaotic oscillations and to bring the PMSM to a stable steady state. The designed controller uses a pole-placement approach to force the closed-loop system to follow the performance of a simple first-order linear system with zero steady-state error to a desired set point. The similarity between the mathematical model of the PMSM and the famous chaotic Lorenz system is utilized to design a synchronization-based state observer using only the angular speed for feedback. Simulation results verify the effectiveness of the proposed controller in eliminating the chaotic oscillations while using a single feedback signal. The superiority of the proposed controller is further demonstrated by comparing it with a conventional PID controller. Finally, a laboratory-based experiment was conducted using the MCK2812 C Pro-MS(BL) motion control kit to confirm the theoretical results and to verify both the causality and versatility of the proposed controller.

Order or chaos in Boolean gene networks depends on the mean fraction of canalizing functions

NASA Astrophysics Data System (ADS)

Karlsson, Fredrik; Hörnquist, Michael

2007-10-01

We explore the connection between order/chaos in Boolean networks and the naturally occurring fraction of canalizing functions in such systems. This fraction turns out to give a very clear indication of whether the system possesses ordered or chaotic dynamics, as measured by Derrida plots, and also the degree of order when we compare different networks with the same number of vertices and edges. By studying also a wide distribution of indegrees in a network, we show that the mean probability of canalizing functions is a more reliable indicator of the type of dynamics for a finite network than the classical result on stability relating the bias to the mean indegree. Finally, we compare by direct simulations two biologically derived networks with networks of similar sizes but with power-law and Poisson distributions of indegrees, respectively. The biologically motivated networks are not more ordered than the latter, and in one case the biological network is even chaotic while the others are not.

A PRELIMINARY STUDY ON THE FRACTAL PHENOMENON: “DISCONNECTED+ DISCONNECTED=CONNECTED”

NASA Astrophysics Data System (ADS)

Wang, Da; Liu, Shutang; Zhao, Yang

The well-known Parrondo’s paradox: “losing+losing=winning” [G. P. Harmer and D. Abbott, Parrondo’s paradox, Stat. Sci. 14 (2009) 206-213.] indicated that two games with negative gains can generate a new game with positive gain. By extending the Parrondo’s philosophy into chaos research, it was shown that the periodic alteration of two chaotic dynamics results in an ordered dynamics, that is the phenomenon: “chaos+chaos=order” [J. Almeida, D. Peralta-Salas and M. Romera, Can two chaotic systems give rise to order, Physica D 200 124-132 (2005)]. This paper further extends these researches into fractal research by proposing that two disconnected Julia sets can originate a new connected Julia set via alternating order. This new parrondian paradoxical phenomenon can be stated in the Parrondo’s terms as “disconnected+disconnected=connected”.

Genetic training of network using chaos concept: application to QSAR studies of vibration modes of tetrahedral halides.

PubMed

Lu, Qingzhang; Shen, Guoli; Yu, Ruqin

2002-11-15

The chaotic dynamical system is introduced in genetic algorithm to train ANN to formulate the CGANN algorithm. Logistic mapping as one of the most important chaotic dynamic mappings provides each new generation a high chance to hold GA's population diversity. This enhances the ability to overcome overfitting in training an ANN. The proposed CGANN has been used for QSAR studies to predict the tetrahedral modes (nu(1)(A1) and nu(2)(E)) of halides [MX(4)](epsilon). The frequencies predicted by QSAR were compared with those calculated by quantum chemistry methods including PM3, AM1, and MNDO/d. The possibility of improving the predictive ability of QSAR by including quantum chemistry parameters as feature variables has been investigated using tetrahedral tetrahalide examples. Copyright 2002 Wiley Periodicals, Inc.

Chaotic operation and chaos control of travelling wave ultrasonic motor.

PubMed

Shi, Jingzhuo; Zhao, Fujie; Shen, Xiaoxi; Wang, Xiaojie

2013-08-01

The travelling wave ultrasonic motor, which is a nonlinear dynamic system, has complex chaotic phenomenon with some certain choices of system parameters and external inputs, and its chaotic characteristics have not been studied until now. In this paper, the preliminary study of the chaos phenomenon in ultrasonic motor driving system has been done. The experiment of speed closed-loop control is designed to obtain several groups of time sampling data sequence of the amplitude of driving voltage, and phase-space reconstruction is used to analyze the chaos characteristics of these time sequences. The largest Lyapunov index is calculated and the result is positive, which shows that the travelling wave ultrasonic motor has chaotic characteristics in a certain working condition Then, the nonlinear characteristics of travelling wave ultrasonic motor are analyzed which includes Lyapunov exponent map, the bifurcation diagram and the locus of voltage relative to speed based on the nonlinear chaos model of a travelling wave ultrasonic motor. After that, two kinds of adaptive delay feedback controllers are designed in this paper to control and suppress chaos in USM speed control system. Simulation results show that the method can control unstable periodic orbits, suppress chaos in USM control system. Proportion-delayed feedback controller was designed following and arithmetic of fuzzy logic was used to adaptively adjust the delay time online. Simulation results show that this method could fast and effectively change the chaos movement into periodic or fixed-point movement and make the system enter into stable state from chaos state. Finally the chaos behavior was controlled. Copyright © 2013 Elsevier B.V. All rights reserved.

Resonances in a Chaotic Attractor Crisis of the Lorenz Flow

NASA Astrophysics Data System (ADS)

Tantet, Alexis; Lucarini, Valerio; Dijkstra, Henk A.

2018-02-01

Local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions. Lyapunov exponents and their associated covariant Lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems. However, it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle. This behavior is investigated here for a boundary crisis in the Lorenz flow, for which neither the Lyapunov exponents nor the covariant Lyapunov vectors provide a criterion for the crisis. Instead, the convergence of the time evolution of probability densities to the invariant measure, governed by the semigroup of transfer operators, is expected to slow down at the approach of the crisis. Such convergence is described by the eigenvalues of the generator of this semigroup, which can be divided into two families, referred to as the stable and unstable Ruelle-Pollicott resonances, respectively. The former describes the convergence of densities to the attractor (or escape from a repeller) and is estimated from many short time series sampling the state space. The latter is responsible for the decay of correlations, or mixing, and can be estimated from a long times series, invoking ergodicity. It is found numerically for the Lorenz flow that the stable resonances do approach the imaginary axis during the crisis, as is indicative of the loss of global stability of the attractor. On the other hand, the unstable resonances, and a fortiori the decay of correlations, do not flag the proximity of the crisis, thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises.

Resonances in a Chaotic Attractor Crisis of the Lorenz Flow

NASA Astrophysics Data System (ADS)

Tantet, Alexis; Lucarini, Valerio; Dijkstra, Henk A.

2017-12-01

Local bifurcations of stationary points and limit cycles have successfully been characterized in terms of the critical exponents of these solutions. Lyapunov exponents and their associated covariant Lyapunov vectors have been proposed as tools for supporting the understanding of critical transitions in chaotic dynamical systems. However, it is in general not clear how the statistical properties of dynamical systems change across a boundary crisis during which a chaotic attractor collides with a saddle. This behavior is investigated here for a boundary crisis in the Lorenz flow, for which neither the Lyapunov exponents nor the covariant Lyapunov vectors provide a criterion for the crisis. Instead, the convergence of the time evolution of probability densities to the invariant measure, governed by the semigroup of transfer operators, is expected to slow down at the approach of the crisis. Such convergence is described by the eigenvalues of the generator of this semigroup, which can be divided into two families, referred to as the stable and unstable Ruelle-Pollicott resonances, respectively. The former describes the convergence of densities to the attractor (or escape from a repeller) and is estimated from many short time series sampling the state space. The latter is responsible for the decay of correlations, or mixing, and can be estimated from a long times series, invoking ergodicity. It is found numerically for the Lorenz flow that the stable resonances do approach the imaginary axis during the crisis, as is indicative of the loss of global stability of the attractor. On the other hand, the unstable resonances, and a fortiori the decay of correlations, do not flag the proximity of the crisis, thus questioning the usual design of early warning indicators of boundary crises of chaotic attractors and the applicability of response theory close to such crises.

A Secure Dynamic Identity and Chaotic Maps Based User Authentication and Key Agreement Scheme for e-Healthcare Systems.

PubMed

Li, Chun-Ta; Lee, Cheng-Chi; Weng, Chi-Yao; Chen, Song-Jhih

2016-11-01

Secure user authentication schemes in many e-Healthcare applications try to prevent unauthorized users from intruding the e-Healthcare systems and a remote user and a medical server can establish session keys for securing the subsequent communications. However, many schemes does not mask the users' identity information while constructing a login session between two or more parties, even though personal privacy of users is a significant topic for e-Healthcare systems. In order to preserve personal privacy of users, dynamic identity based authentication schemes are hiding user's real identity during the process of network communications and only the medical server knows login user's identity. In addition, most of the existing dynamic identity based authentication schemes ignore the inputs verification during login condition and this flaw may subject to inefficiency in the case of incorrect inputs in the login phase. Regarding the use of secure authentication mechanisms for e-Healthcare systems, this paper presents a new dynamic identity and chaotic maps based authentication scheme and a secure data protection approach is employed in every session to prevent illegal intrusions. The proposed scheme can not only quickly detect incorrect inputs during the phases of login and password change but also can invalidate the future use of a lost/stolen smart card. Compared the functionality and efficiency with other authentication schemes recently, the proposed scheme satisfies desirable security attributes and maintains acceptable efficiency in terms of the computational overheads for e-Healthcare systems.

Chaotic Dynamics in a Low-Energy Transfer Strategy to the Equilateral Equilibrium Points in the Earth-Moon System

NASA Astrophysics Data System (ADS)

Salazar, F. J. T.; Macau, E. E. N.; Winter, O. C.

In the frame of the equilateral equilibrium points exploration, numerous future space missions will require maximization of payload mass, simultaneously achieving reasonable transfer times. To fulfill this request, low-energy non-Keplerian orbits could be used to reach L4 and L5 in the Earth-Moon system instead of high energetic transfers. Previous studies have shown that chaos in physical systems like the restricted three-body Earth-Moon-particle problem can be used to direct a chaotic trajectory to a target that has been previously considered. In this work, we propose to transfer a spacecraft from a circular Earth Orbit in the chaotic region to the equilateral equilibrium points L4 and L5 in the Earth-Moon system, exploiting the chaotic region that connects the Earth with the Moon and changing the trajectory of the spacecraft (relative to the Earth) by using a gravity assist maneuver with the Moon. Choosing a sequence of small perturbations, the time of flight is reduced and the spacecraft is guided to a proper trajectory so that it uses the Moon's gravitational force to finally arrive at a desired target. In this study, the desired target will be an orbit about the Lagrangian equilibrium points L4 or L5. This strategy is not only more efficient with respect to thrust requirement, but also its time transfer is comparable to other known transfer techniques based on time optimization.

Scaling of chaos in strongly nonlinear lattices.

PubMed

Mulansky, Mario

2014-06-01

Although it is now understood that chaos in complex classical systems is the foundation of thermodynamic behavior, the detailed relations between the microscopic properties of the chaotic dynamics and the macroscopic thermodynamic observations still remain mostly in the dark. In this work, we numerically analyze the probability of chaos in strongly nonlinear Hamiltonian systems and find different scaling properties depending on the nonlinear structure of the model. We argue that these different scaling laws of chaos have definite consequences for the macroscopic diffusive behavior, as chaos is the microscopic mechanism of diffusion. This is compared with previous results on chaotic diffusion [M. Mulansky and A. Pikovsky, New J. Phys. 15, 053015 (2013)], and a relation between microscopic chaos and macroscopic diffusion is established.

Chaos in Atomic Force Microscopy

NASA Astrophysics Data System (ADS)

Hu, Shuiqing; Raman, Arvind

2006-01-01

Chaotic oscillations of microcantilever tips in dynamic atomic force microscopy (AFM) are reported and characterized. Systematic experiments performed using a variety of microcantilevers under a wide range of operating conditions indicate that softer AFM microcantilevers bifurcate from periodic to chaotic oscillations near the transition from the noncontact to the tapping regimes. Careful Lyapunov exponent and noise titration calculations of the tip oscillation data confirm their chaotic nature. AFM images taken by scanning the chaotically oscillating tips over the sample show small, but significant metrology errors at the nanoscale due to this “deterministic” uncertainty.

Chaos enhancing tunneling in a coupled Bose-Einstein condensate with a double driving.

PubMed

Rong, Shiguang; Hai, Wenhua; Xie, Qiongtao; Zhu, Qianquan

2009-09-01

We study the effects of chaotic dynamics on atomic tunneling between two weakly coupled Bose-Einstein condensates driven by a double-frequency periodic field. Under the Melnikov's chaos criterion, we divide the parameter space into three parts of different types, regular region, low-chaoticity region, and high-chaoticity region, and give the accurate boundaries between the different regions. It is found that the atomic tunneling can be enhanced in the presence of chaos. Particularly, in the high-chaoticity regions, the chaos-induced inversion of the population imbalance is observed numerically.

Scilab software package for the study of dynamical systems

NASA Astrophysics Data System (ADS)

Bordeianu, C. C.; Beşliu, C.; Jipa, Al.; Felea, D.; Grossu, I. V.

2008-05-01

This work presents a new software package for the study of chaotic flows and maps. The codes were written using Scilab, a software package for numerical computations providing a powerful open computing environment for engineering and scientific applications. It was found that Scilab provides various functions for ordinary differential equation solving, Fast Fourier Transform, autocorrelation, and excellent 2D and 3D graphical capabilities. The chaotic behaviors of the nonlinear dynamics systems were analyzed using phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov-Sinai entropy. Various well known examples are implemented, with the capability of the users inserting their own ODE. Program summaryProgram title: Chaos Catalogue identifier: AEAP_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEAP_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.html No. of lines in distributed program, including test data, etc.: 885 No. of bytes in distributed program, including test data, etc.: 5925 Distribution format: tar.gz Programming language: Scilab 3.1.1 Computer: PC-compatible running Scilab on MS Windows or Linux Operating system: Windows XP, Linux RAM: below 100 Megabytes Classification: 6.2 Nature of problem: Any physical model containing linear or nonlinear ordinary differential equations (ODE). Solution method: Numerical solving of ordinary differential equations. The chaotic behavior of the nonlinear dynamical system is analyzed using Poincaré sections, phase-space maps, autocorrelation functions, power spectra, Lyapunov exponents and Kolmogorov-Sinai entropies. Restrictions: The package routines are normally able to handle ODE systems of high orders (up to order twelve and possibly higher), depending on the nature of the problem. Running time: 10 to 20 seconds for problems that do not involve Lyapunov exponents calculation; 60 to 1000 seconds for problems that involve high orders ODE and Lyapunov exponents calculation.

Numerical computation of orbits and rigorous verification of existence of snapback repellers.

PubMed

Peng, Chen-Chang

2007-03-01

In this paper we show how analysis from numerical computation of orbits can be applied to prove the existence of snapback repellers in discrete dynamical systems. That is, we present a computer-assisted method to prove the existence of a snapback repeller of a specific map. The existence of a snapback repeller of a dynamical system implies that it has chaotic behavior [F. R. Marotto, J. Math. Anal. Appl. 63, 199 (1978)]. The method is applied to the logistic map and the discrete predator-prey system.

Communication scheme using a hyperchaotic semiconductor laser model: Chaos shift key revisited

NASA Astrophysics Data System (ADS)

Fataf, N. A. A.; Palit, Sanjay Kumar; Mukherjee, Sayan; Said, M. R. M.; Son, Doan Hoai; Banerjee, Santo

2017-11-01

Based on the Maxwell-Bloch equations, we considered a five-dimensional ODE system, describing the dynamics of a semiconductor laser. The system has rich dynamics with multi-periodic, chaotic and hyperchaotic states. In this analysis, we have investigated the hyperchaotic nature of the aforesaid model and proposed a communication scheme, the generalized form of chaos shift keys, where the coupled systems do not need to be in the synchronized state. The results are implemented with the hyperchaotic laser model followed by a comprehensive security analysis.

Mathematics of Failures in Complex Systems: Characterization and Mitigation of Service Failures in Complex Dynamic Systems

DTIC Science & Technology

2007-06-30

fractal dimensions and Lyapunov exponents . Fractal dimensions characterize geometri- cal complexity of dynamics (e.g., spatial distribution of points along...ant classi3ers (e.g., Lyapunov exponents , and fractal dimensions). The 3rst three steps show how chaotic systems may be separated from stochastic...correlated random walk in which a ¼ 2H, where H is the Hurst exponen interval 0pHp1 with the case H ¼ 0:5 corresponding to a simple rando This model has been

Autonomous choices among deterministic evolution-laws as source of uncertainty

NASA Astrophysics Data System (ADS)

Trujillo, Leonardo; Meyroneinc, Arnaud; Campos, Kilver; Rendón, Otto; Sigalotti, Leonardo Di G.

2018-03-01

We provide evidence of an extreme form of sensitivity to initial conditions in a family of one-dimensional self-ruling dynamical systems. We prove that some hyperchaotic sequences are closed-form expressions of the orbits of these pseudo-random dynamical systems. Each chaotic system in this family exhibits a sensitivity to initial conditions that encompasses the sequence of choices of the evolution rule in some collection of maps. This opens a possibility to extend current theories of complex behaviors on the basis of intrinsic uncertainty in deterministic chaos.

Optimizing homogenization by chaotic unmixing?

NASA Astrophysics Data System (ADS)

Weijs, Joost; Bartolo, Denis

2016-11-01

A number of industrial processes rely on the homogeneous dispersion of non-brownian particles in a viscous fluid. An ideal mixing would yield a so-called hyperuniform particle distribution. Such configurations are characterized by density fluctuations that grow slower than the standard √{ N}-fluctuations. Even though such distributions have been found in several natural structures, e.g. retina receptors in birds, they have remained out of experimental reach until very recently. Over the last 5 years independent experiments and numerical simulations have shown that periodically driven suspensions can self-assemble hyperuniformally. Simple as the recipe may be, it has one important disadvantage. The emergence of hyperuniform states co-occurs with a critical phase transition from reversible to non reversible particle dynamics. As a consequence the homogenization dynamics occurs over a time that diverges with the system size (critical slowing down). Here, we discuss how this process can be sped up by exploiting the stirring properties of chaotic advection. Among the questions that we answer are: What are the physical mechanisms in a chaotic flow that are relevant for hyperuniformity? How can we tune the flow parameters such to obtain optimal hyperuniformity in the fastest way? JW acknowledges funding by NWO (Netherlands Organisation for Scientific Research) through a Rubicon Grant.

Algorthms and prigrams complex for chaotic dynamics investigation of the Earth artificial sat-ellites. (Russian Title: Комплекс алгоритмов и программ для исследования хаотичности в динамике искусственных спутников Земли )

NASA Astrophysics Data System (ADS)

Bordovitsyna, T. V.; Aleksandrova, A. G.; Chuvashov, I. N.

2010-12-01

In this paper complex of algorithms and programs for revelation and investigation of dynamical chaotic state in the motion of the Earth artificial satellites by parallel computing is presented. Complex has been based on the program "Numerical model of the system artificial satellites motion" for cluster "Skiff Cyberia". Factor MEGNO as main indicator of chaotic state has been used. The factor is computed by combined numerical integration of equations of the motion, equations in variation and equations of MEGNO parameters. The results of program complex testing in the problem of MEGNO parameters calculation for different types of geostationary orbits are presented.

Using Chaotic System in Encryption

NASA Astrophysics Data System (ADS)

Findik, Oğuz; Kahramanli, Şirzat

In this paper chaotic systems and RSA encryption algorithm are combined in order to develop an encryption algorithm which accomplishes the modern standards. E.Lorenz's weather forecast' equations which are used to simulate non-linear systems are utilized to create chaotic map. This equation can be used to generate random numbers. In order to achieve up-to-date standards and use online and offline status, a new encryption technique that combines chaotic systems and RSA encryption algorithm has been developed. The combination of RSA algorithm and chaotic systems makes encryption system.

Resonant responses and chaotic dynamics of composite laminated circular cylindrical shell with membranes

NASA Astrophysics Data System (ADS)

Zhang, W.; Liu, T.; Xi, A.; Wang, Y. N.

2018-06-01

This paper is focused on the resonant responses and chaotic dynamics of a composite laminated circular cylindrical shell with radially pre-stretched membranes at both ends and clamped along a generatrix. Based on the two-degree-of-freedom non-autonomous nonlinear equations of this system, the method of multiple scales is employed to obtain the four-dimensional nonlinear averaged equation. The resonant case considered here is the primary parametric resonance-1/2 subharmonic resonance and 1:1 internal resonance. Corresponding to several selected parameters, the frequency-response curves are obtained. From the numerical results, we find that the hardening-spring-type behaviors and jump phenomena are exhibited. The jump phenomena also occur in the amplitude curves of the temperature parameter excitation. Moreover, it is found that the temperature parameter excitation, the coupling degree of two order modes and the detuning parameters can effect the nonlinear oscillations of this system. The periodic and chaotic motions of the composite laminated circular cylindrical shell clamped along a generatrix are demonstrated by the bifurcation diagrams, the maximum Lyapunov exponents, the phase portraits, the waveforms, the power spectrums and the Poincaré map. The temperature parameter excitation shows that the Pomeau-Manneville type intermittent chaos occur under the certain initial conditions. It is also found that there exist the twin phenomena between the Pomeau-Manneville type intermittent chaos and the period-doubling bifurcation.

On the conservation of the Jacobi integral in the post-Newtonian circular restricted three-body problem

NASA Astrophysics Data System (ADS)

Dubeibe, F. L.; Lora-Clavijo, F. D.; González, Guillermo A.

2017-05-01

In the present paper, using the first-order approximation of the n-body Lagrangian (derived on the basis of the post-Newtonian gravitational theory of Einstein, Infeld, and Hoffman), we explicitly write down the equations of motion for the planar circular restricted three-body problem in the Solar system. Additionally, with some simplified assumptions, we obtain two formulas for estimating the values of the mass-distance and velocity-speed of light ratios appropriate for a given post-Newtonian approximation. We show that the formulas derived in the present study, lead to good numerical accuracy in the conservation of the Jacobi constant and almost allow for an equivalence between the Lagrangian and Hamiltonian approaches at the same post-Newtonian order. Accordingly, the dynamics of the system is analyzed in terms of the Poincaré sections method and Lyapunov exponents, finding that for specific values of the Jacobi constant the dynamics can be either chaotic or regular. Our results suggest that the chaoticity of the post-Newtonian system is slightly increased in comparison with its Newtonian counterpart.

Quantum signature of chaos and thermalization in the kicked Dicke model

NASA Astrophysics Data System (ADS)

Ray, S.; Ghosh, A.; Sinha, S.

2016-09-01

We study the quantum dynamics of the kicked Dicke model (KDM) in terms of the Floquet operator, and we analyze the connection between chaos and thermalization in this context. The Hamiltonian map is constructed by suitably taking the classical limit of the Heisenberg equation of motion to study the corresponding phase-space dynamics, which shows a crossover from regular to chaotic motion by tuning the kicking strength. The fixed-point analysis and calculation of the Lyapunov exponent (LE) provide us with a complete picture of the onset of chaos in phase-space dynamics. We carry out a spectral analysis of the Floquet operator, which includes a calculation of the quasienergy spacing distribution and structural entropy to show the correspondence to the random matrix theory in the chaotic regime. Finally, we analyze the thermodynamics and statistical properties of the bosonic sector as well as the spin sector, and we discuss how such a periodically kicked system relaxes to a thermalized state in accordance with the laws of statistical mechanics.

Regular and chaotic motions of the Chaplygin sleigh with periodically switched location of nonholonomic constraint

NASA Astrophysics Data System (ADS)

Kuznetsov, Sergey P.

2017-04-01

We consider motions of the Chaplygin sleigh on a plane supposing that the nonholonomic constraint is located periodically turn by turn at each of three legs supporting the sleigh. We assume that at switching on the constraint the respective element (“knife-edge”) is directed along the local velocity vector and becomes fixed relatively to the sleigh for a certain time interval till the next switch. Differential equations of the mathematical model are formulated and analytical derivation of a 2D map for the state transformation on the switching period is provided. The dynamics takes place with conservation of the mechanical energy. Numerical simulations show phenomena characteristic to nonholonomic systems with complex dynamics. In particular, on the energy surface attractors may occur responsible for regular sustained motions settling in domains of prevalent area compression by the map. In addition, chaotic and quasi-periodic regimes take place similar to those observed in conservative nonlinear dynamics.

Quantum signature of chaos and thermalization in the kicked Dicke model.

PubMed

Ray, S; Ghosh, A; Sinha, S

2016-09-01

We study the quantum dynamics of the kicked Dicke model (KDM) in terms of the Floquet operator, and we analyze the connection between chaos and thermalization in this context. The Hamiltonian map is constructed by suitably taking the classical limit of the Heisenberg equation of motion to study the corresponding phase-space dynamics, which shows a crossover from regular to chaotic motion by tuning the kicking strength. The fixed-point analysis and calculation of the Lyapunov exponent (LE) provide us with a complete picture of the onset of chaos in phase-space dynamics. We carry out a spectral analysis of the Floquet operator, which includes a calculation of the quasienergy spacing distribution and structural entropy to show the correspondence to the random matrix theory in the chaotic regime. Finally, we analyze the thermodynamics and statistical properties of the bosonic sector as well as the spin sector, and we discuss how such a periodically kicked system relaxes to a thermalized state in accordance with the laws of statistical mechanics.

Global dynamics and diffusion in triaxial galactic models

NASA Astrophysics Data System (ADS)

Papaphilippou, Y.

We apply the Frequency Map Analysis method to the 3--dimensional logarithmic galactic potential in order to clarify the dynamical behaviour of triaxial power--law galactic models. All the fine dynamical details are displayed in the complete frequency map, a direct representation of the system's Arnol'd web. The influence of resonant lines and the extent of the chaotic zones are directly associated with the physical space of the system. Some new results related with the diffusion of galactic orbits are also discussed. This approach reveals many unknown dynamical features of triaxial galactic potentials and provides strong indications that chaos should be an innate characteristic of triaxial configurations.

Adaptative synchronization in multi-output fractional-order complex dynamical networks and secure communications

NASA Astrophysics Data System (ADS)

Mata-Machuca, Juan L.; Aguilar-López, Ricardo

2018-01-01

This work deals with the adaptative synchronization of complex dynamical networks with fractional-order nodes and its application in secure communications employing chaotic parameter modulation. The complex network is composed of multiple fractional-order systems with mismatch parameters and the coupling functions are given to realize the network synchronization. We introduce a fractional algebraic synchronizability condition (FASC) and a fractional algebraic identifiability condition (FAIC) which are used to know if the synchronization and parameters estimation problems can be solved. To overcome these problems, an adaptative synchronization methodology is designed; the strategy consists in proposing multiple receiver systems which tend to follow asymptotically the uncertain transmitters systems. The coupling functions and parameters of the receiver systems are adjusted continually according to a convenient sigmoid-like adaptative controller (SLAC), until the measurable output errors converge to zero, hence, synchronization between transmitter and receivers is achieved and message signals are recovered. Indeed, the stability analysis of the synchronization error is based on the fractional Lyapunov direct method. Finally, numerical results corroborate the satisfactory performance of the proposed scheme by means of the synchronization of a complex network consisting of several fractional-order unified chaotic systems.

Deterministic chaos and fractal complexity in the dynamics of cardiovascular behavior: perspectives on a new frontier.

PubMed

Sharma, Vijay

2009-09-10

Physiological systems such as the cardiovascular system are capable of five kinds of behavior: equilibrium, periodicity, quasi-periodicity, deterministic chaos and random behavior. Systems adopt one or more these behaviors depending on the function they have evolved to perform. The emerging mathematical concepts of fractal mathematics and chaos theory are extending our ability to study physiological behavior. Fractal geometry is observed in the physical structure of pathways, networks and macroscopic structures such the vasculature and the His-Purkinje network of the heart. Fractal structure is also observed in processes in time, such as heart rate variability. Chaos theory describes the underlying dynamics of the system, and chaotic behavior is also observed at many levels, from effector molecules in the cell to heart function and blood pressure. This review discusses the role of fractal structure and chaos in the cardiovascular system at the level of the heart and blood vessels, and at the cellular level. Key functional consequences of these phenomena are highlighted, and a perspective provided on the possible evolutionary origins of chaotic behavior and fractal structure. The discussion is non-mathematical with an emphasis on the key underlying concepts.

Deterministic Chaos and Fractal Complexity in the Dynamics of Cardiovascular Behavior: Perspectives on a New Frontier

PubMed Central

Sharma, Vijay

2009-01-01

Physiological systems such as the cardiovascular system are capable of five kinds of behavior: equilibrium, periodicity, quasi-periodicity, deterministic chaos and random behavior. Systems adopt one or more these behaviors depending on the function they have evolved to perform. The emerging mathematical concepts of fractal mathematics and chaos theory are extending our ability to study physiological behavior. Fractal geometry is observed in the physical structure of pathways, networks and macroscopic structures such the vasculature and the His-Purkinje network of the heart. Fractal structure is also observed in processes in time, such as heart rate variability. Chaos theory describes the underlying dynamics of the system, and chaotic behavior is also observed at many levels, from effector molecules in the cell to heart function and blood pressure. This review discusses the role of fractal structure and chaos in the cardiovascular system at the level of the heart and blood vessels, and at the cellular level. Key functional consequences of these phenomena are highlighted, and a perspective provided on the possible evolutionary origins of chaotic behavior and fractal structure. The discussion is non-mathematical with an emphasis on the key underlying concepts. PMID:19812706

Chaotic dynamics of Comet 1P/Halley: Lyapunov exponent and survival time expectancy

NASA Astrophysics Data System (ADS)

Muñoz-Gutiérrez, M. A.; Reyes-Ruiz, M.; Pichardo, B.

2015-03-01

The orbital elements of Comet Halley are known to a very high precision, suggesting that the calculation of its future dynamical evolution is straightforward. In this paper we seek to characterize the chaotic nature of the present day orbit of Comet Halley and to quantify the time-scale over which its motion can be predicted confidently. In addition, we attempt to determine the time-scale over which its present day orbit will remain stable. Numerical simulations of the dynamics of test particles in orbits similar to that of Comet Halley are carried out with the MERCURY 6.2 code. On the basis of these we construct survival time maps to assess the absolute stability of Halley's orbit, frequency analysis maps to study the variability of the orbit, and we calculate the Lyapunov exponent for the orbit for variations in initial conditions at the level of the present day uncertainties in our knowledge of its orbital parameters. On the basis of our calculations of the Lyapunov exponent for Comet Halley, the chaotic nature of its motion is demonstrated. The e-folding time-scale for the divergence of initially very similar orbits is approximately 70 yr. The sensitivity of the dynamics on initial conditions is also evident in the self-similarity character of the survival time and frequency analysis maps in the vicinity of Halley's orbit, which indicates that, on average, it is unstable on a time-scale of hundreds of thousands of years. The chaotic nature of Halley's present day orbit implies that a precise determination of its motion, at the level of the present-day observational uncertainty, is difficult to predict on a time-scale of approximately 100 yr. Furthermore, we also find that the ejection of Halley from the Solar system or its collision with another body could occur on a time-scale as short as 10 000 yr.

Homoclinic behaviors and chaotic motions of double layered viscoelastic nanoplates based on nonlocal theory and extended Melnikov method.

PubMed

Wang, Yu; Li, Feng-Ming; Wang, Yi-Ze

2015-06-01

The nonlinear dynamical equations are established for the double layered viscoelastic nanoplates (DLNP) subjected to in-plane excitation based on the nonlocal theory and von Kármán large deformation theory. The extended high dimensional homoclinic Melnikov method is employed to study the homoclinic phenomena and chaotic motions for the parametrically excited DLNP system. The criteria for the homoclinic transverse intersection for both the asynchronous and synchronous buckling cases are proposed. Lyapunov exponents and phase portraits are obtained to verify the Melnikov-type analysis. The influences of structural parameters on the transverse homoclinic orbits and homoclinic bifurcation sets are discussed for the two buckling cases. Some novel phenomena are observed in the investigation. It should be noticed that the nonlocal effect on the homoclinic behaviors and chaotic motions is quite remarkable. Hence, the small scale effect should be taken into account for homoclinic and chaotic analysis for nanostructures. It is significant that the nonlocal effect on the homoclinic phenomena for the asynchronous buckling case is quite different from that for the synchronous buckling case. Moreover, due to the van der Walls interaction between the layers, the nonlocal effect on the homoclinic behaviors and chaotic motions for high order mode is rather tiny under the asynchronous buckling condition.

Application of chaotic attractor analysis in crack assessment of plates

NASA Astrophysics Data System (ADS)

Jalili, Sina; Daneshmehr, A. R.

2018-03-01

Part-through crack presence with limited length is one of the prevalent defects in plate structures. However, this type of damage has only a slight effect on the dynamic response of the structures. In this paper the modified line spring method (MLSM) is used to develop a nonlinear multi-degree of freedom model of part through cracked rectangular plate and chaotic interrogation is implemented to assess crack-induced degradation in the nonlinear model. After a convergence study of the proposed model in time series domain in which the plate subjected to Lorenz-type chaotic excitation, the tuning of interrogation is conducted by crossing the Lyapunov exponents' spectrums of the nonlinear model of the plate and chaotic signal. In this research nonlinear prediction error (NPE) is proposed as a damage sensitive feature which deals with the chaotic attractor of the excited system response. It is found that there are ranges of tuning parameter that result in higher damage sensitivity of the NPE. Damage characteristics such as: length, angle, location and depth of crack are considered as parameters to be varied to scrutinize the response of the plates. Results show that NPE generally has significantly higher sensitivity in comparison with conventional frequency-based methods; however this property has different levels for various boundary conditions.

Homoclinic behaviors and chaotic motions of double layered viscoelastic nanoplates based on nonlocal theory and extended Melnikov method

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

Wang, Yu; Wang, Yi-Ze; Li, Feng-Ming, E-mail: fmli@bjut.edu.cn

2015-06-15

The nonlinear dynamical equations are established for the double layered viscoelastic nanoplates (DLNP) subjected to in-plane excitation based on the nonlocal theory and von Kármán large deformation theory. The extended high dimensional homoclinic Melnikov method is employed to study the homoclinic phenomena and chaotic motions for the parametrically excited DLNP system. The criteria for the homoclinic transverse intersection for both the asynchronous and synchronous buckling cases are proposed. Lyapunov exponents and phase portraits are obtained to verify the Melnikov-type analysis. The influences of structural parameters on the transverse homoclinic orbits and homoclinic bifurcation sets are discussed for the two bucklingmore » cases. Some novel phenomena are observed in the investigation. It should be noticed that the nonlocal effect on the homoclinic behaviors and chaotic motions is quite remarkable. Hence, the small scale effect should be taken into account for homoclinic and chaotic analysis for nanostructures. It is significant that the nonlocal effect on the homoclinic phenomena for the asynchronous buckling case is quite different from that for the synchronous buckling case. Moreover, due to the van der Walls interaction between the layers, the nonlocal effect on the homoclinic behaviors and chaotic motions for high order mode is rather tiny under the asynchronous buckling condition.« less

Paradigms of Complexity: Fractals and Structures in the Sciences

NASA Astrophysics Data System (ADS)

Novak, Miroslav M.

The Table of Contents for the book is as follows: * Preface * The Origin of Complexity (invited talk) * On the Existence of Spatially Uniform Scaling Laws in the Climate System * Multispectral Backscattering: A Fractal-Structure Probe * Small-Angle Multiple Scattering on a Fractal System of Point Scatterers * Symmetric Fractals Generated by Cellular Automata * Bispectra and Phase Correlations for Chaotic Dynamical Systems * Self-Organized Criticality Models of Neural Development * Altered Fractal and Irregular Heart Rate Behavior in Sick Fetuses * Extract Multiple Scaling in Long-Term Heart Rate Variability * A Semi-Continous Box Counting Method for Fractal Dimension Measurement of Short Single Dimension Temporal Signals - Preliminary Study * A Fractional Brownian Motion Model of Cracking * Self-Affine Scaling Studies on Fractography * Coarsening of Fractal Interfaces * A Fractal Model of Ocean Surface Superdiffusion * Stochastic Subsurface Flow and Transport in Fractal Fractal Conductivity Fields * Rendering Through Iterated Function Systems * The σ-Hull - The Hull Where Fractals Live - Calculating a Hull Bounded by Log Spirals to Solve the Inverse IFS-Problem by the Detected Orbits * On the Multifractal Properties of Passively Convected Scalar Fields * New Statistical Textural Transforms for Non-Stationary Signals: Application to Generalized Mutlifractal Analysis * Laplacian Growth of Parallel Needles: Their Mullins-Sekerka Instability * Entropy Dynamics Associated with Self-Organization * Fractal Properties in Economics (invited talk) * Fractal Approach to the Regional Seismic Event Discrimination Problem * Fractal and Topological Complexity of Radioactive Contamination * Pattern Selection: Nonsingular Saffman-Taylor Finger and Its Dynamic Evolution with Zero Surface Tension * A Family of Complex Wavelets for the Characterization of Singularities * Stabilization of Chaotic Amplitude Fluctuations in Multimode, Intracavity-Doubled Solid-State Lasers * Chaotic Dynamics of Elastic-Plastic Beams * The Riemann Non-Differentiable Function and Identities for the Gaussian Sums * Revealing the Multifractal Nature of Failure Sequence * The Fractal Nature of wood Revealed by Drying * Squaring the Circle: Diffusion Volume and Acoustic Behaviour of a Fractal Structure * Relationship Between Acupuncture Holographic Units and Fetus Development; Fractal Features of Two Acupuncture Holographic Unit Systems * The Fractal Properties of the Large-Scale Magnetic Fields on the Sun * Fractal Analysis of Tide Gauge Data * Author Index