Sporadically Fractal Basin Boundaries of Chaotic Systems
Hunt, B.R.; Ott, E.; Rosa, E. Jr.
1999-05-01
We demonstrate a new type of basin boundary for typical chaotic dynamical systems. For the case of a two dimensional map, this boundary has the character of the graph of a function that is smooth and differentiable except on a set of fractal dimensions less than one. In spite of the basin boundary being smooth {open_quotes}almost everywhere,{close_quotes} its fractal dimension exceeds one (implying degradation of one{close_quote}s ability to predict the attractor an orbit approaches in the presence of small initial condition uncertainty). We call such a boundary {ital sporadically fractal}. {copyright} {ital 1999} {ital The American Physical Society}
Fractal dimension in nonhyperbolic chaotic scattering
NASA Technical Reports Server (NTRS)
Lau, Yun-Tung; Finn, John M.; Ott, Edward
1991-01-01
In chaotic scattering there is a Cantor set of input-variable values of zero Lebesgue measure (i.e., zero total length) on which the scattering function is singular. For cases where the dynamics leading to chaotic scattering is nonhyperbolic (e.g., there are Kolmogorov-Arnol'd-Moser tori), the nature of this singular set is fundamentally different from that in the hyperbolic case. In particular, for the nonhyperbolic case, although the singular set has zero total length, strong evidence is presented to show that its fractal dimension is 1.
The chaotic atom model via a fractal approximation of motion
NASA Astrophysics Data System (ADS)
Agop, M.; Nica, P.; Gurlui, S.; Focsa, C.; Magop, D.; Borsos, Z.
2011-10-01
A new model of the atom is built based on a complete and detailed nonlinear dynamics analysis (complete time series, Poincaré sections, complete phase space, Lyapunov exponents, bifurcation diagrams and fractal analysis), through the correlation of the chaotic-stochastic model with a fractal one. Some specific mechanisms that ensure the atom functionality are proposed: gun, chaotic gun and multi-gun effects for the excited states (the classical analogue of quantum absorption) and the fractalization of the trajectories for the stationary states (a natural way of introducing the quantification).
NASA Astrophysics Data System (ADS)
Burkholder, Michael B.; Litster, Shawn
2016-05-01
In this study, we analyze the stability of two-phase flow regimes and their transitions using chaotic and fractal statistics, and we report new measurements of dynamic two-phase pressure drop hysteresis that is related to flow regime stability and channel water content. Two-phase flow dynamics are relevant to a variety of real-world systems, and quantifying transient two-phase flow phenomena is important for efficient design. We recorded two-phase (air and water) pressure drops and flow images in a microchannel under both steady and transient conditions. Using Lyapunov exponents and Hurst exponents to characterize the steady-state pressure fluctuations, we develop a new, measurable regime identification criteria based on the dynamic stability of the two-phase pressure signal. We also applied a new experimental technique by continuously cycling the air flow rate to study dynamic hysteresis in two-phase pressure drops, which is separate from steady-state hysteresis and can be used to understand two-phase flow development time scales. Using recorded images of the two-phase flow, we show that the capacitive dynamic hysteresis is related to channel water content and flow regime stability. The mixed-wettability microchannel and in-channel water introduction used in this study simulate a polymer electrolyte fuel cell cathode air flow channel.
Fractal boundaries in open hydrodynamical flows: Signatures of chaotic saddles
NASA Astrophysics Data System (ADS)
Péntek, Áron; Toroczkai, Zoltán; Tél, Tamás; Grebogi, Celso; Yorke, James A.
1995-05-01
We introduce the concept of fractal boundaries in open hydrodynamical flows based on two gedanken experiments carried out with passive tracer particles colored differently. It is shown that the signature for the presence of a chaotic saddle in the advection dynamics is a fractal boundary between regions of different colors. The fractal parts of the boundaries found in the two experiments contain either the stable or the unstable manifold of this chaotic set. We point out that these boundaries coincide with streak lines passing through appropriately chosen points. As an illustrative numerical experiment, we consider a model of the von Kármán vortex street, a time periodic two-dimensional flow of a viscous fluid around a cylinder.
Gallery of Chaotic Attractors Generated by Fractal Network
NASA Astrophysics Data System (ADS)
Bouallegue, Kais
During the last decade, fractal processes and chaotic systems were widely studied in many areas of research. Chaotic systems are highly dependent on initial conditions. Small changes in initial conditions can generate widely diverging or converging outcomes for both bifurcation or attraction in chaotic systems. In this work, we present a new method on how to generate a new family of chaotic attractors by combining these with a network of fractal processes. The proposed approach in this article is based upon the construction of a new system of fractal processes.
Is a chaotic multi-fractal approach for rainfall possible?
NASA Astrophysics Data System (ADS)
Sivakumar, Bellie
2001-04-01
Applications of the ideas gained from fractal theory to characterize rainfall have been one of the most exciting areas of research in recent times. The studies conducted thus far have nearly unanimously yielded positive evidence regarding the existence of fractal behaviour in rainfall. The studies also revealed the insufficiency of the mono-fractal approaches to characterizing the rainfall process in time and space and, hence, the necessity for multi-fractal approaches. The assumption behind multi-fractal approaches for rainfall is that the variability of the rainfall process could be directly modelled as a stochastic (or random) turbulent cascade process, since such stochastic cascade processes were found to generically yield multi-fractals. However, it has been observed recently that multi-fractal approaches might provide positive evidence of a multi-fractal nature not only in stochastic processes but also in, for example, chaotic processes. The purpose of the present study is to investigate the presence of both chaotic and fractal behaviours in the rainfall process to consider the possibility of using a chaotic multi-fractal approach for rainfall characterization. For this purpose, daily rainfall data observed at the Leaf River basin in Mississippi are studied, and only temporal analysis is carried out. The autocorrelation function, the power spectrum, the empirical probability distribution function, and the statistical moment scaling function are used as indicators to investigate the presence of fractal, whereas the presence of chaos is investigated by employing the correlation dimension method. The results from the fractal identification methods indicate that the rainfall data exhibit multi-fractal behaviour. The correlation dimension method yields a low dimension, suggesting the presence of chaotic behaviour. The existence of both multi-fractal and chaotic behaviours in the rainfall data suggests the possibility of a chaotic multi-fractal approach for
Experiments in chaotic dynamics
NASA Astrophysics Data System (ADS)
Moon, F. C.
Mathematical tools for the description of chaotic phenomena in physical systems are described and demonstrated, summarizing in part the principles presented in the author's book-length treatise on chaotic vibrations (Moon, 1987). Consideration is given to phase-plane and pseudo-phase-plane techniques, bifurcation diagrams, FFTs, autocorrelation functions, single and double Poincare maps, reduction to one-dimensional maps, Liapunov exponents, fractal dimensions, invariant distributions, chaos diagrams, and basin-boundary diagrams. The results obtained by application of these methods to data from typical mechanical and electronic oscillation experiments are presented graphically and discussed in detail.
Fractal dynamics of earthquakes
Bak, P.; Chen, K.
1995-05-01
Many objects in nature, from mountain landscapes to electrical breakdown and turbulence, have a self-similar fractal spatial structure. It seems obvious that to understand the origin of self-similar structures, one must understand the nature of the dynamical processes that created them: temporal and spatial properties must necessarily be completely interwoven. This is particularly true for earthquakes, which have a variety of fractal aspects. The distribution of energy released during earthquakes is given by the Gutenberg-Richter power law. The distribution of epicenters appears to be fractal with dimension D {approx} 1--1.3. The number of after shocks decay as a function of time according to the Omori power law. There have been several attempts to explain the Gutenberg-Richter law by starting from a fractal distribution of faults or stresses. But this is a hen-and-egg approach: to explain the Gutenberg-Richter law, one assumes the existence of another power-law--the fractal distribution. The authors present results of a simple stick slip model of earthquakes, which evolves to a self-organized critical state. Emphasis is on demonstrating that empirical power laws for earthquakes indicate that the Earth`s crust is at the critical state, with no typical time, space, or energy scale. Of course the model is tremendously oversimplified; however in analogy with equilibrium phenomena they do not expect criticality to depend on details of the model (universality).
Dynamical fractional chaotic inflation
NASA Astrophysics Data System (ADS)
Harigaya, Keisuke; Ibe, Masahiro; Schmitz, Kai; Yanagida, Tsutomu T.
2014-12-01
Chaotic inflation based on a simple monomial scalar potential, V (ϕ )∝ϕp, is an attractive large-field model of inflation capable of generating a sizable tensor-to-scalar ratio r . Therefore, assuming that future cosmic microwave background observations will confirm the large r value reported by BICEP2, it is important to determine what kind of dynamical mechanism could possibly endow the inflaton field with such a simple effective potential. In this paper, we answer this question in the context of field theory, i.e. in the framework of dynamical chaotic inflation, where strongly interacting supersymmetric gauge dynamics around the scale of grand unification dynamically generate a fractional power-law potential via the quantum effect of dimensional transmutation. In constructing explicit models, we significantly extend our previous work, as we now consider a large variety of possible underlying gauge dynamics and relax our conditions on the field content of the model. This allows us to realize almost arbitrary rational values for the power p in the inflaton potential. The present paper may hence be regarded as a first step toward a more complete theory of dynamical chaotic inflation.
Launching the chaotic realm of iso-fractals: A short remark
O'Schmidt, Nathan; Katebi, Reza; Corda, Christian
2015-03-10
In this brief note, we introduce the new, emerging sub-discipline of iso-fractals by highlighting and discussing the preliminary results of recent works. First, we note the abundance of fractal, chaotic, non-linear, and self-similar structures in nature while emphasizing the importance of studying such systems because fractal geometry is the language of chaos. Second, we outline the iso-fractal generalization of the Mandelbrot set to exemplify the newly generated Mandelbrot iso-sets. Third, we present the cutting-edge notion of dynamic iso-spaces and explain how a mathematical space can be iso-topically lifted with iso-unit functions that (continuously or discretely) change; in the discrete case examples, we mention that iteratively generated sequences like Fibonacci’s numbers and (the complex moduli of) Mandelbrot’s numbers can supply a deterministic chain of iso-units to construct an ordered series of (magnified and/or de-magnified) iso-spaces that are locally iso-morphic. Fourth, we consider the initiation of iso-fractals with Inopin’s holographic ring (IHR) topology and fractional statistics for 2D and 3D iso-spaces. In total, the reviewed iso-fractal results are a significant improvement over traditional fractals because the application of Santilli’s iso-mathematics arms us an extra degree of freedom for attacking problems in chaos. Finally, we conclude by proposing some questions and ideas for future research work.
Fractal scattering dynamics of the three-dimensional HOCl molecule
NASA Astrophysics Data System (ADS)
Lin, Yi-Der; Barr, Alex M.; Reichl, L. E.; Jung, Christof
2013-01-01
We compare the 2D and 3D classical fractal scattering dynamics of Cl and HO for energies just above dissociation of the HOCl molecule, using a realistic potential energy surface for the HOCl molecule and techniques developed to analyze 3D chaotic scattering processes. For parameter regimes where the HO dimer initially has small vibrational energy, only small intervals of initial conditions show fractal scattering behavior and the scattering process is well described by a 2D model. For parameter regimes where the HO dimer initially has large vibrational energy, the scattering process is fully 3D and is dominated by fractal behavior.
NASA Astrophysics Data System (ADS)
Freeman, Walter J.
2013-01-01
The first step of the sensory systems is to construct the meaning of the information they receive from the senses. They do this by generating random noise and then filtering the noise with adaptive filters. We simulate the operation with the solutions of matrices of ordinary differential equations that predict subcritical Hopf bifurcations between point and limit cycle attractors. The second step is integration of the outputs from the several sensory systems into a multisensory percept, called a gestalt, which in the third step is consolidated and stored as knowledge. Simulation of the second step requires use of landscapes of nonconvergent chaotic attractors. This is not deterministic chaos, which is much too brittle owing to the infinite sensitivity to initial conditions. It is a hybrid form we call stochastic chaos, which is stabilized by additive noise modeled on noise sources in the sensory systems. Thus bifurcation and chaos theory provides tools for succinct empirical models of cortical dynamics performing the most basic cognitive operations: generalization, abstraction, and categorization in constructing knowledge. The descriptions are in a form that is suitable for more advanced modeling using analog VLSI, neuropercolation from random graph theory, non-equilibrium dissipative thermodynamics, and macroscopic many-body physics. This review concludes with a summary of the applications of stochastic chaos in pattern classification and some prescriptions for neurobiologists on what to look for in large-scale anatomical formations.
Bialek, J.M.
1988-01-01
Chaotic behavior may be observed in deterministic Hamiltonian Systems with as few as three dimensions, i.e, X, P, and t. The amount of chaotic behavior depends on the relative influence of the integrable and non-integrable parts of the Hamiltonian. The Standard Map is such a system and the amount of chaotic behavior may be varied by adjusting a single parameter. The global phase space portrait is a complicated mixture of quiescent and chaotic regions. First a new calculational method, characterized by a Fractual Diagram, is presented. This allows the quantitative prediction of the boundaries between regular and chaotic regions in phase space. Where these barriers are located gives qualitative insight into diffusion in phase space. The method is illustrated with the Standard Map but may be applied to any Hamiltonian System. The second phenomenon is the Universal Behavior predicted to occur for all area preserving maps. As a parameter is varied causing the mapping to become more chaotic a pattern is observed in the location and stability of the fixed points of the maps. The fixed points undergo an infinite sequence of period doubling bifurcations in a finite range of the parameter. The relative locations of the fixed point bifurcation and the parameter intervals between bifurcations both asymptotically approach constants which are Universal in that the same constants keep appearing in different problems.
Fractal dynamics of bioconvective patterns
NASA Technical Reports Server (NTRS)
Noever, David A.
1991-01-01
Biologically generated cellular patterns, sometimes called bioconvective patterns, are found to cluster into aggregates which follow fractal growth dynamics akin to diffusion-limited aggregation (DLA) models. The pattern formed is self-similar with fractal dimension of 1.66 +/-0.038. Bioconvective DLA branching results from thermal roughening which shifts the balance between ordering viscous forces and disordering cell motility and random diffusion. The phase diagram for pattern morphology includes DLA, boundary spokes, random clusters, and reverse clusters.
Cryptosystems based on chaotic dynamics
McNees, R.A.; Protopopescu, V.; Santoro, R.T.; Tolliver, J.S.
1993-08-01
An encryption scheme based on chaotic dynamics is presented. This scheme makes use of the efficient and reproducible generation of cryptographically secure pseudo random numbers from chaotic maps. The result is a system which encrypts quickly and possesses a large keyspace, even in small precision implementations. This system offers an excellent solution to several problems including the dissemination of key material, over the air rekeying, and other situations requiring the secure management of information.
Nonlinear dynamics in a microfluidic loop device: Chaos and Fractals
NASA Astrophysics Data System (ADS)
Maddala, Jeevan; Rengaswamy, Raghunathan
2012-11-01
Discrete decision making and resistive interactions between droplets in a microfluidic loop device induces fascinating nonlinear dynamics such as multi-stability and period doubling. Droplets entering the device at fixed time intervals can exit at different periods or chaotically. One of the periodic behaviors that is observed in a loop is the three-period behavior; this is consistent with the notion that three period behavior implies chaos. Switching between these different dynamical regimes is achieved by changing the inlet droplet feeding frequency. Chaotic behavior is observed between islands of periodic behavior. We show through simulations and experimental observations that the transitions between periods are indeed chaotic. Network model is used to study the dynamic behavior for different inlet feeding frequencies resulting in the development of a bifurcation map. The bifurcation map shows that the three period dynamics is preceded by chaos. A Lyapunov exponent is used to further validate these results. The exit droplet spacing shows several fascinating patterns when the model is simulated for a large number of droplets in the chaotic regime. One such chaotic regime produces a fractal that has a boundary of cardioid. The correlation dimension for a fractal pattern produced by this particular loop system is calculated to be 0.7.
Studies in Chaotic adiabatic dynamics
Jarzynski, C.
1994-01-01
Chaotic adiabatic dynamics refers to the study of systems exhibiting chaotic evolution under slowly time-dependent equations of motion. In this dissertation the author restricts his attention to Hamiltonian chaotic adiabatic systems. The results presented are organized around a central theme, namely, that the energies of such systems evolve diffusively. He begins with a general analysis, in which he motivates and derives a Fokker-Planck equation governing this process of energy diffusion. He applies this equation to study the {open_quotes}goodness{close_quotes} of an adiabatic invariant associated with chaotic motion. This formalism is then applied to two specific examples. The first is that of a gas of noninteracting point particles inside a hard container that deforms slowly with time. Both the two- and three-dimensional cases are considered. The results are discussed in the context of the Wall Formula for one-body dissipation in nuclear physics, and it is shown that such a gas approaches, asymptotically with time, an exponential velocity distribution. The second example involves the Fermi mechanism for the acceleration of cosmic rays. Explicit evolution equations are obtained for the distribution of cosmic ray energies within this model, and the steady-state energy distribution that arises when this equation is modified to account for the injection and removal of cosmic rays is discussed. Finally, the author re-examines the multiple-time-scale approach as applied to the study of phase space evolution under a chaotic adiabatic Hamiltonian. This leads to a more rigorous derivation of the above-mentioned Fokker-Planck equation, and also to a new term which has relevance to the problem of chaotic adiabatic reaction forces (the forces acting on slow, heavy degrees of freedom due to their coupling to light, fast chaotic degrees).
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 practical test for noisy chaotic dynamics
NASA Astrophysics Data System (ADS)
BenSaïda, Ahmed
2015-12-01
This code computes the largest Lyapunov exponent and tests for the presence of a chaotic dynamics, as opposed to stochastic dynamics, in a noisy scalar series. The program runs under MATLAB® programming language.
Regular transport dynamics produce chaotic travel times.
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. PMID:25019866
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.
Fractal boundaries in magnetotail particle dynamics
NASA Technical Reports Server (NTRS)
Chen, J.; Rexford, J. L.; Lee, Y. C.
1990-01-01
It has been recently established that particle dynamics in the magnetotail geometry can be described as a nonintegrable Hamiltonian system with well-defined entry and exit regions through which stochastic orbits can enter and exit the system after repeatedly crossing the equatorial plane. It is shown that the phase space regions occupied by orbits of different numbers of equatorial crossings or different exit modes are separated by fractal boundaries. The fractal boundaries in an entry region for stochastic orbits are examined and the capacity dimension is determined.
Fractal boundaries in magnetotail particle dynamics
NASA Astrophysics Data System (ADS)
Chen, J.; Rexford, J. L.; Lee, Y. C.
1990-07-01
It has been recently established that particle dynamics in the magnetotail geometry can be described as a nonintegrable Hamiltonian system with well-defined entry and exit regions through which stochastic orbits can enter and exit the system after repeatedly crossing the equatorial plane. It is shown that the phase space regions occupied by orbits of different numbers of equatorial crossings or different exit modes are separated by fractal boundaries. The fractal boundaries in an entry region for stochastic orbits are examined and the capacity dimension is determined.
Dynamic visual cryptography based on chaotic oscillations
NASA Astrophysics Data System (ADS)
Petrauskiene, Vilma; Palivonaite, Rita; Aleksa, Algiment; Ragulskis, Minvydas
2014-01-01
Dynamic visual cryptography scheme based on chaotic oscillations is proposed in this paper. Special computational algorithms are required for hiding the secret image in the cover moiré grating, but the decryption of the secret is completely visual. The secret image is leaked in the form of time-averaged geometric moiré fringes when the cover image is oscillated by a chaotic law. The relationship among the standard deviation of the stochastic time variable, the pitch of the moiré grating and the pixel size ensuring visual decryption of the secret is derived. The parameters of these chaotic oscillations must be carefully preselected before the secret image is leaked from the cover image. Several computational experiments are used to illustrate the functionality and the applicability of the proposed image hiding technique.
Quantifying chaotic dynamics from interspike intervals
NASA Astrophysics Data System (ADS)
Pavlov, A. N.; Pavlova, O. N.; Mohammad, Y. K.; Shihalov, G. M.
2015-03-01
We address the problem of characterization of chaotic dynamics at the input of a threshold device described by an integrate-and-fire (IF) or a threshold crossing (TC) model from the output sequences of interspike intervals (ISIs). We consider the conditions under which quite short sequences of spiking events provide correct identification of the dynamical regime characterized by the single positive Lyapunov exponent (LE). We discuss features of detecting the second LE for both types of the considered models of events generation.
Virtual Libraries: Interactive Support Software and an Application in Chaotic Models.
ERIC Educational Resources Information Center
Katsirikou, Anthi; Skiadas, Christos; Apostolou, Apostolos; Rompogiannakis, Giannis
This paper begins with a discussion of the characteristics and the singularity of chaotic systems, including dynamic systems theory, chaotic orbit, fractals, chaotic attractors, and characteristics of chaotic systems. The second section addresses the digital libraries (DL) concept and the appropriateness of chaotic models, including definition and…
Chaotic dynamics of a magnetic nanoparticle.
Bragard, J; Pleiner, H; Suarez, O J; Vargas, P; Gallas, J A C; Laroze, D
2011-09-01
We study the deterministic spin dynamics of an anisotropic magnetic particle in the presence of a magnetic field with a constant longitudinal and a time-dependent transverse component using the Landau-Lifshitz-Gilbert equation. We characterize the dynamical behavior of the system through calculation of the Lyapunov exponents, Poincaré sections, bifurcation diagrams, and Fourier power spectra. In particular we explore the positivity of the largest Lyapunov exponent as a function of the magnitude and frequency of the applied magnetic field and its direction with respect to the main anisotropy axis of the magnetic particle. We find that the system presents multiple transitions between regular and chaotic behaviors. We show that the dynamical phases display a very complicated structure of intricately intermingled chaotic and regular phases. PMID:22060537
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
Urey Prize Lecture - Chaotic dynamics in the solar system
NASA Astrophysics Data System (ADS)
Wisdom, J.
1987-11-01
Newton's equations have chaotic solutions as well as regular solutions. There are several physical situations in the solar system where chaotic solutions of Newton's equations play an important role. There are examples of both chaotic rotation and chaotic orbital evolution. Hyperion is currently tumbling chaotically. Many of the other irregularly shaped satellites in the solar system have had chaotic rotations in the past. This episode of chaotic tumbling could have had a significant effect on the orbital histories of these satellites. Chaotic orbital evolution seems to be an essential ingredient in the explanation of the Kirkwood gaps in the distribution of asteroids. Chaotic trajectories at the 3/1 commensurability have the correct properties to provide a dynamical route for the transport of meteoritic material from the asteroid belt to Earth.
Chaotic spectra: How to extract dynamic information
Taylor, H.S.; Gomez Llorente, J.M.; Zakrzewski, J.; Kulander, K.C.
1988-10-01
Nonlinear dynamics is applied to chaotic unassignable atomic and molecular spectra with the aim of extracting detailed information about regular dynamic motions that exist over short intervals of time. It is shown how this motion can be extracted from high resolution spectra by doing low resolution studies or by Fourier transforming limited regions of the spectrum. These motions mimic those of periodic orbits (PO) and are inserts into the dominant chaotic motion. Considering these inserts and the PO as a dynamically decoupled region of space, resonant scattering theory and stabilization methods enable us to compute ladders of resonant states which interact with the chaotic quasi-continuum computed in principle from basis sets placed off the PO. The interaction of the resonances with the quasicontinuum explains the low resolution spectra seen in such experiments. It also allows one to associate low resolution features with a particular PO. The motion on the PO thereby supplies the molecular movements whose quantization causes the low resolution spectra. Characteristic properties of the periodic orbit based resonances are discussed. The method is illustrated on the photoabsorption spectrum of the hydrogen atom in a strong magnetic field and on the photodissociation spectrum of H/sub 3//sup +/. Other molecular systems which are currently under investigation using this formalism are also mentioned. 53 refs., 10 figs., 2 tabs.
Chaotic dynamics in dense fluids
Posch, H.A.; Hoover, W.G.
1987-09-01
We present calculations of the full spectra of Lyapunov exponents for 8- and 32-particle systems with periodic boundary conditions and interacting with the repulsive part of a Lennard-Jones potential both in equilibrium and nonequilibrium steady states. Lyapunov characteristic exponents lambda/sub n/ describe the mean exponential rates of divergence and convergence of neighbouring trajectories in phase-space. They are useful in characterizing the stochastic properties of a dynamical system. A new algorithm for their calculation is presented which incorporates ideas from control theory and constraint nonequilibrium molecular dynamics. 4 refs., 1 fig.
Topological analysis of chaotic dynamical systems
NASA Astrophysics Data System (ADS)
Gilmore, Robert
1998-10-01
Topological methods have recently been developed for the analysis of dissipative dynamical systems that operate in the chaotic regime. They were originally developed for three-dimensional dissipative dynamical systems, but they are applicable to all ``low-dimensional'' dynamical systems. These are systems for which the flow rapidly relaxes to a three-dimensional subspace of phase space. Equivalently, the associated attractor has Lyapunov dimension dL<3. Topological methods supplement methods previously developed to determine the values of metric and dynamical invariants. However, topological methods possess three additional features: they describe how to model the dynamics; they allow validation of the models so developed; and the topological invariants are robust under changes in control-parameter values. The topological-analysis procedure depends on identifying the stretching and squeezing mechanisms that act to create a strange attractor and organize all the unstable periodic orbits in this attractor in a unique way. The stretching and squeezing mechanisms are represented by a caricature, a branched manifold, which is also called a template or a knot holder. This turns out to be a version of the dynamical system in the limit of infinite dissipation. This topological structure is identified by a set of integer invariants. One of the truly remarkable results of the topological-analysis procedure is that these integer invariants can be extracted from a chaotic time series. Furthermore, self-consistency checks can be used to confirm the integer values. These integers can be used to determine whether or not two dynamical systems are equivalent; in particular, they can determine whether a model developed from time-series data is an accurate representation of a physical system. Conversely, these integers can be used to provide a model for the dynamical mechanisms that generate chaotic data. In fact, the author has constructed a doubly discrete classification of strange
Chaotic Pattern Dynamics in Spatially Ramped Turbulence
NASA Astrophysics Data System (ADS)
Wiener, R. J.; Ashbaker, E.; Olsen, T.; Bodenschatz, E.
2003-11-01
In previous experiments(Richard J. Wiener et al), Phys. Rev. E 55, 5489 (1997)., Taylor vortex flow in an hourglass geometry has demonstrated a period-doubling cascade to chaotic pattern dynamics. A spatial ramp exists in the Reynolds number. For low reduced Reynolds numbesr \\varepsilon, supercritical vortex flow occurs between regions of subcritical structureless flow with soft boundaries that allow for pattern dynamics. At \\varepsilon ≈ 0.5, the pattern exhibits phase slips that occur irregularly in time. At \\varepsilon ≈ 1.0 the entire system is supercritical, and the pattern is stabilized against phase slips. At \\varepsilon > 15, shear flow creates a spatial ramp in turbulence. Remarkably, the phase slip instability reoccurs. Vortex pairs are created chaotically, possibly due to the spatial variation of the turbulence. The variance and Fourier spectra of time series of light scattered off Kalliroscope tracer were measured. These indicate that a region of turbulence exists, within which phase slips occur, bounded by regions of laminar flow which may provide soft boundaries that allow for the phase dynamics. Despite the presence of turbulence, the dynamics might be describable by a phase equation.
Chaotic Dynamics of an Elastically Bouncing Dumbbell
NASA Astrophysics Data System (ADS)
Rees, Colin; Franklin, Scott
2009-03-01
The dynamics of an elastically bouncing dumbbell is analogous to those of an ball bouncing on a sinusoidally oscillating surface with one important exception: the dumbbell's angular velocity, analogous to the surface's oscillation frequency, changes with each bounce, making the subsequent motion significantly more complicated. We investigate this dynamical system over a range of aspect ratios and initial energy, finding periodic, quasi-periodic and chaotic motions. As the initial energy is increased, the dumbbell can flip over and tumble. We find for large aspect ratios, however, narrow bands of energies well above this minimum where tumbling suddenly ceases. Because energy is conserved, the dynamics of a bounce are uniquely determined by the angle and angular velocity. The Lyapunov exponents of paths in this two dimensional phase space can be calculated, with the hope of identifying periodic islands within the chaotic sea. Finally, for certain parameters, the angle at each collision moves from its initial value in a subdiffusive manner, and we determine the characteristic exponents.
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
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.
NASA Astrophysics Data System (ADS)
Altmann, Eduardo G.; Portela, Jefferson S. E.; Tél, Tamás
2015-02-01
We investigate chaotic dynamical systems for which the intensity of trajectories might grow unlimited in time. We show that i) the intensity grows exponentially in time and is distributed spatially according to a fractal measure with an information dimension smaller than that of the phase space, ii) such exploding cases can be described by an operator formalism similar to the one applied to chaotic systems with absorption (decaying intensities), but iii) the invariant quantities characterizing explosion and absorption are typically not directly related to each other, e.g., the decay rate and fractal dimensions of absorbing maps typically differ from the ones computed in the corresponding inverse (exploding) maps. We illustrate our general results through numerical simulation in the cardioid billiard mimicking a lasing optical cavity, and through analytical calculations in the baker map.
NASA Astrophysics Data System (ADS)
Zeyer, K.-P.; Münster, A. F.; Hauser, M. J. B.; Schneider, F. W.
1994-09-01
We extend previous work describing the passive electrical coupling of two periodic chemical states to include quasiperiodic and chaotic states. Our setup resembles an electrochemical concentration cell (a battery) whose half cells [continuous-flow stirred tank reactors (CSTRs)] each contain the Belousov-Zhabotinsky (BZ) reaction. For a closed electrical circuit the two half cells are weakly coupled by an external variable resistance and by a constant low mass flow. This battery may produce either periodic, quasiperiodic, or chaotic alternating current depending on the dynamic BZ states chosen in the half cells. A lower fractal dimensionality is calculated from the electrical potential of a single chaotic CSTR than from the difference potential (relative potential) of the two chaotic half cell potentials. A similar situation is observed in model calculations of a chaotic spatiotemporal system (the driven Brusselator in one space dimension) where the dimensionality derived from a local time series is lower than the dimensionality of the global trajectory calculated from the Karhunen-Loeve coefficients.
Chaotic dynamics of a candle oscillator
NASA Astrophysics Data System (ADS)
Lee, Mary Elizabeth; Byrne, Greg; Fenton, Flavio
The candle oscillator is a simple, fun experiment dating to the late nineteenth century. It consists of a candle with a rod that is transverse to its long axis, around which it is allowed to pivot. When both ends of the candle are lit, an oscillatory motion will initiate due to different mass loss as a function of the flame angle. Stable oscillations can develop due to damping when the system has friction between the rod and the base where the rod rests. However, when friction is minimized, it is possible for chaos to develop. In this talk we will show periodic orbits found in the system as well as calculated, maximal Lyapunov exponents. We show that the system can be described by three ordinary differential equations (one each for angle, angular velocity and mass loss) that can reproduce the experimental data and the transition from stable oscillations to chaotic dynamics as a function of damping.
Chaotic Dynamics in Partial Differential Equations.
NASA Astrophysics Data System (ADS)
Li, Yanguang
The existence of chaotic behavior, for a certain damped and driven perturbation of the nonlinear Schroedinger equation under even periodic boundary conditions, is established. More specifically, the existence of a symmetric pair of homoclinic orbits is established for the perturbed NLS equation through two main arguments: Argument 1 is a combination of Melnikov analysis and a geometric singular perturbation theory for the pde. The geometric singular perturbation theory involves the theory of persistence of invariant manifolds for the pde and the theory of Hadamard-Fenichel fiber coordinatization for those invariant manifolds. Argument 2 is a purely geometric argument. Finally, an argument is sketched which, we believe, provides a core of an existence proof for Smale "horseshoes" and a symbolic dynamics in a neighborhood of the persistent homoclinic orbits.
Forecasting catastrophe by exploiting chaotic dynamics
Stewart, H.B.; Lansbury, A.N.
1990-01-01
Our purpose here is to introduce a variation on the theme of short term forecasting from a chaotic time series. We show that for the lowest-dimensional chaotic attractors, it is possible to predict incipient catastrophes, or crises, by examining time series data taken near the catastrophic bifurcation threshold, but always remaining on the safe side of the threshold.
Traffic chaotic dynamics modeling and analysis of deterministic network
NASA Astrophysics Data System (ADS)
Wu, Weiqiang; Huang, Ning; Wu, Zhitao
2016-07-01
Network traffic is an important and direct acting factor of network reliability and performance. To understand the behaviors of network traffic, chaotic dynamics models were proposed and helped to analyze nondeterministic network a lot. The previous research thought that the chaotic dynamics behavior was caused by random factors, and the deterministic networks would not exhibit chaotic dynamics behavior because of lacking of random factors. In this paper, we first adopted chaos theory to analyze traffic data collected from a typical deterministic network testbed — avionics full duplex switched Ethernet (AFDX, a typical deterministic network) testbed, and found that the chaotic dynamics behavior also existed in deterministic network. Then in order to explore the chaos generating mechanism, we applied the mean field theory to construct the traffic dynamics equation (TDE) for deterministic network traffic modeling without any network random factors. Through studying the derived TDE, we proposed that chaotic dynamics was one of the nature properties of network traffic, and it also could be looked as the action effect of TDE control parameters. A network simulation was performed and the results verified that the network congestion resulted in the chaotic dynamics for a deterministic network, which was identical with expectation of TDE. Our research will be helpful to analyze the traffic complicated dynamics behavior for deterministic network and contribute to network reliability designing and analysis.
Fractal templates in the escape dynamics of trapped ultracold atoms
Mitchell, Kevin A.; Steck, Daniel A.
2007-09-15
We consider the dynamic escape of a small packet of ultracold atoms launched from within an optical dipole trap. Based on a theoretical analysis of the underlying nonlinear dynamics, we predict that fractal behavior can be seen in experimental escape data. These data can be collected by measuring the time-dependent escape rate for packets launched over a range of angles. This fractal pattern is particularly well resolved below the Bose-Einstein transition temperature - a direct result of the extreme phase-space localization of the condensate. We predict that several self-similar layers of this novel fractal should be measurable, and we explain how this fractal pattern can be predicted and analyzed with recently developed techniques in symbolic dynamics.
About Chaotic Dynamics in the Twisted Horseshoe Map
NASA Astrophysics Data System (ADS)
Sovrano, Elisa
2016-06-01
The twisted horseshoe map was developed in order to study a class of density dependent Leslie population models with two age classes. From the beginning, scientists have tried to prove that this map presents chaotic dynamics. Some demonstrations that have appeared in mathematical literature present some difficulties or delicate issues. In this paper, we give a simple and rigorous proof based on a different approach. We also highlight the possibility of getting chaotic dynamics for a broader class of maps.
Chaotic behavior in nonlinear polarization dynamics
David, D.; Holm, D.D.; Tratnik, M.V. )
1989-01-01
We analyze the problem of two counterpropagating optical laser beams in a slightly nonlinear medium from the point of view of Hamiltonian systems; the one-beam subproblem is also investigated as a special case. We are interested in these systems as integrable dynamical systems which undergo chaotic behavior under various types of perturbations. The phase space for the two-beam problem is C{sup 2} {times} C{sup 2} when we restricted the the regime of travelling-wave solutions. We use the method of reduction for Hamiltonian systems invariant under one-parameter symmetry groups to demonstrate that the phase space reduces to the two-sphere S{sup 2} and is therefore completely integrable. The phase portraits of the system are classified and we also determine the bifurcations that modify these portraits; some new degenerate bifurcations are presented in this context. Finally, we introduce various physically relevant perturbations and use the Melnikov method to prove that horseshoe chaos and Arnold diffusion occur as consequences of these perturbations. 10 refs., 7 figs., 1 tab.
Chaotic dynamics of weakly nonlinear systems
Vavriv, D.M.
1996-06-01
A review is given on the recent results in studying chaotic phenomena in weakly nonlinear systems. We are concerned with the class of chaotic states that can arise in physical systems with any degree of nonlinearity however small. The conditions for, and the mechanisms of, the transitions to chaos are discussed. These findings are illustrated by the results of the stability analysis of practical microwave and optical devices. {copyright} {ital 1996 American Institute of Physics.}
Chaotic dynamics in a simple dynamical green ocean plankton model
NASA Astrophysics Data System (ADS)
Cropp, Roger; Moroz, Irene M.; Norbury, John
2014-11-01
The exchange of important greenhouse gases between the ocean and atmosphere is influenced by the dynamics of near-surface plankton ecosystems. Marine plankton ecosystems are modified by climate change creating a feedback mechanism that could have significant implications for predicting future climates. The collapse or extinction of a plankton population may push the climate system across a tipping point. Dynamic green ocean models (DGOMs) are currently being developed for inclusion into climate models to predict the future state of the climate. The appropriate complexity of the DGOMs used to represent plankton processes is an ongoing issue, with models tending to become more complex, with more complicated dynamics, and an increasing propensity for chaos. We consider a relatively simple (four-population) DGOM of phytoplankton, zooplankton, bacteria and zooflagellates where the interacting plankton populations are connected by a single limiting nutrient. Chaotic solutions are possible in this 4-dimensional model for plankton population dynamics, as well as in a reduced 3-dimensional model, as we vary two of the key mortality parameters. Our results show that chaos is robust to the variation of parameters as well as to the presence of environmental noise, where the attractor of the more complex system is more robust than the attractor of its simplified equivalent. We find robust chaotic dynamics in low trophic order ecological models, suggesting that chaotic dynamics might be ubiquitous in the more complex models, but this is rarely observed in DGOM simulations. The physical equations of DGOMs are well understood and are constrained by conservation principles, but the ecological equations are not well understood, and generally have no explicitly conserved quantities. This work, in the context of the paucity of the empirical and theoretical bases upon which DGOMs are constructed, raises the interesting question of whether DGOMs better represent reality if they include
Design of Microlasers and Beam Splitters using chaotic ray dynamics
NASA Astrophysics Data System (ADS)
Luna-Acosta, German A.; Mendez-Bermudez, J. Antonio; Bewdix, Oliver
2005-03-01
. We consider chaotic waveguides formed by single or multiple 2D chaotic cavities connected to leads. The cavities are chaotic in the sense that the ray/particle dynamics within them is chaotic. Specifically the phase space is mixed, with chaotic regions surrounding stable islands where motion is regular. Stable islands are inaccessible to the incoming rays/particles. In contrast, incoming plane waves can dynamically tunnel into them at a certain set of discrete values of frequency/energy. The support of the corresponding quasi-bound state is along the trajectories of periodic orbits trapped within the cavity. We take advantage of this difference in the ray/wave behavior to demonstrate how chaotic waveguides, electromagnetic or electronic, can be used to design beam splitters and microlasers[1]. We also present some preliminary experimental results in a microwave realization of a chaotic waveguide. [1] J. A. M'edez-Berm'udez, G. A. Luna-Acosta, P. Seba, and K. N. Pichugin, Phys. Rev. B 67, 161104(R) (2003).
Dynamical features of reaction-diffusion fronts in fractals.
Méndez, Vicenç; Campos, Daniel; Fort, Joaquim
2004-01-01
The speed of front propagation in fractals is studied by using (i) the reduction of the reaction-transport equation into a Hamilton-Jacobi equation and (ii) the local-equilibrium approach. Different equations proposed for describing transport in fractal media, together with logistic reaction kinetics, are considered. Finally, we analyze the main features of wave fronts resulting from this dynamic process, i.e., why they are accelerated and what is the exact form of this acceleration. PMID:14995742
Chaotic dynamics of loosely supported tubes in crossflow
Cai, Y.; Chen, S.S.
1991-07-01
By means of the unsteady-flow theory and a bilinear mathematical model, a theoretical study was conducted of the chaotic dynamics associated with the fluidelastic instability of loosely supported tubes. Calculations were performed for the RMS of tube displacement, bifurcation diagram, phase portrait, power spectral density, and Poincare map. Analytical results show the existence of chaotic, quasiperiodic, and periodic regions when flow velocity exceeds a threshold value. 38 refs., 15 figs., 2 tabs.
Chaotic dynamics in optimal monetary policy
NASA Astrophysics Data System (ADS)
Gomes, O.; Mendes, V. M.; Mendes, D. A.; Sousa Ramos, J.
2007-05-01
There is by now a large consensus in modern monetary policy. This consensus has been built upon a dynamic general equilibrium model of optimal monetary policy as developed by, e.g., Goodfriend and King [ NBER Macroeconomics Annual 1997 edited by B. Bernanke and J. Rotemberg (Cambridge, Mass.: MIT Press, 1997), pp. 231 282], Clarida et al. [J. Econ. Lit. 37, 1661 (1999)], Svensson [J. Mon. Econ. 43, 607 (1999)] and Woodford [ Interest and Prices: Foundations of a Theory of Monetary Policy (Princeton, New Jersey, Princeton University Press, 2003)]. In this paper we extend the standard optimal monetary policy model by introducing nonlinearity into the Phillips curve. Under the specific form of nonlinearity proposed in our paper (which allows for convexity and concavity and secures closed form solutions), we show that the introduction of a nonlinear Phillips curve into the structure of the standard model in a discrete time and deterministic framework produces radical changes to the major conclusions regarding stability and the efficiency of monetary policy. We emphasize the following main results: (i) instead of a unique fixed point we end up with multiple equilibria; (ii) instead of saddle-path stability, for different sets of parameter values we may have saddle stability, totally unstable equilibria and chaotic attractors; (iii) for certain degrees of convexity and/or concavity of the Phillips curve, where endogenous fluctuations arise, one is able to encounter various results that seem intuitively correct. Firstly, when the Central Bank pays attention essentially to inflation targeting, the inflation rate has a lower mean and is less volatile; secondly, when the degree of price stickiness is high, the inflation rate displays a larger mean and higher volatility (but this is sensitive to the values given to the parameters of the model); and thirdly, the higher the target value of the output gap chosen by the Central Bank, the higher is the inflation rate and its
Chaotic dynamics in flow through unsaturated fractured media
NASA Astrophysics Data System (ADS)
Faybishenko, Boris
Predictions of flow and transport within fractured rock in the vadose zone cannot be made without first characterizing the physics of unstable flow phenomena in unsaturated fractures. This paper introduces a new approach for studying complex flow processes in heterogeneous fractured media, using the methods of nonlinear dynamics and chaos--in particular reconstructing the system dynamics and calculating chaotic diagnostic parameters from time-series data. To demonstrate the application of chaotic analysis, this author analyzed the time-series pressure fluctuations from two water-air flow experiments conducted by Persoff and Pruess [Water Resour. Res. 31 (1995) 1175] in replicas of rough-walled rock fractures under controlled boundary conditions. This analysis showed that chaotic flow in fractures creates relaxational oscillations of liquid, gas, and capillary pressures. These pressure oscillations were used to calculate the diagnostic parameters of deterministic chaos, including correlation time, global embedding dimension, local embedding dimension, Lyapunov dimension, Lyapunov exponents, and correlation dimension. The results of the Persoff-Pruess experiments were then compared with the chaotic analysis of laboratory dripping-water experiments in fracture models and field-infiltration experiments in fractured basalt. This comparison allowed us to conjecture that intrinsic fracture flow and dripping, as well as extrinsic water dripping (from a fracture) subjected to a capillary-barrier effect, are deterministic-chaotic processes with a certain random component. The unsaturated fractured rock is a dynamic system that exhibits chaotic behavior because the flow processes are nonlinear, dissipative, and sensitive to initial conditions, with chaotic fluctuations generated by intrinsic properties of the system, not random external factors. Identifying a system as deterministically chaotic is important for developing appropriate short- and long-term prediction models
Chaotic dynamics in a two-dimensional optical lattice.
Horsley, Eric; Koppell, Stewart; Reichl, L E
2014-01-01
The classical nonlinear dynamics of a dilute gas of rubidium atoms in an optical lattice is studied for a range of polarizations of the laser beams forming the lattice. The dynamics ranges from integrable to chaotic, and mechanisms leading to the onset of chaos in the lattice are described. PMID:24580307
Quantifying chaotic dynamics from integrate-and-fire processes
NASA Astrophysics Data System (ADS)
Pavlov, A. N.; Pavlova, O. N.; Mohammad, Y. K.; Kurths, J.
2015-01-01
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 periods of chaotic oscillations in the regime of phase-coherent chaos. Application to real data is discussed.
Quantifying chaotic dynamics from integrate-and-fire processes
Pavlov, A. N.; Pavlova, O. N.; Mohammad, Y. K.; Kurths, J.
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 periods of chaotic oscillations in the regime of phase-coherent chaos. Application to real data is discussed.
Nonlinear dynamics, fractals, cardiac physiology and sudden death
NASA Technical Reports Server (NTRS)
Goldberger, Ary L.
1987-01-01
The authors propose a diametrically opposite viewpoint to the generally accepted tendency of equating healthy function with order and disease with chaos. With regard to the question of sudden cardiac death and chaos, it is suggested that certain features of dynamical chaos related to fractal structure and fractal dynamics may be important organizing principles in normal physiology and that certain pathologies, including ventricular fibrillation, represent a class of 'pathological periodicities'. Some laboratory work bearing on the relation of nonlinear analysis to physiological and pathophysiological data is briefly reviewed, with tentative theories and models described in reference to the mechanism of ventricular fibrillation.
Wave dynamics of regular and chaotic rays
McDonald, S.W.
1983-09-01
In order to investigate general relationships between waves and rays in chaotic systems, I study the eigenfunctions and spectrum of a simple model, the two-dimensional Helmholtz equation in a stadium boundary, for which the rays are ergodic. Statistical measurements are performed so that the apparent randomness of the stadium modes can be quantitatively contrasted with the familiar regularities observed for the modes in a circular boundary (with integrable rays). The local spatial autocorrelation of the eigenfunctions is constructed in order to indirectly test theoretical predictions for the nature of the Wigner distribution corresponding to chaotic waves. A portion of the large-eigenvalue spectrum is computed and reported in an appendix; the probability distribution of successive level spacings is analyzed and compared with theoretical predictions. The two principal conclusions are: 1) waves associated with chaotic rays may exhibit randomly situated localized regions of high intensity; 2) the Wigner function for these waves may depart significantly from being uniformly distributed over the surface of constant frequency in the ray phase space.
Chaotic dynamics of magnetic domain walls in nanowires
NASA Astrophysics Data System (ADS)
Pivano, A.; Dolocan, V. O.
2016-04-01
The nonlinear dynamics of a transverse domain wall (TDW) in permalloy and nickel nanostrips with two artificially patterned pinning centers is studied numerically up to rf frequencies. The phase diagram frequency-driving amplitude shows a rich variety of dynamical behaviors depending on the material parameters and the type and shape of pinning centers. We find that T-shaped traps (antinotches) create a classical double well Duffing potential that leads to a small chaotic region in the case of nickel and a large one for Py. In contrast, the rectangular constrictions (notches) create an exponential potential that leads to larger chaotic regions interspersed with periodic windows for both Py and Ni. The influence of temperature manifests itself by enlarging the chaotic region and activating thermal jumps between the pinning sites while reducing the depinning field at low frequency in the notched strips.
Fermi resonance in dynamical tunneling in a chaotic billiard
NASA Astrophysics Data System (ADS)
Yi, Chang-Hwan; Kim, Ji-Hwan; Yu, Hyeon-Hye; Lee, Ji-Won; Kim, Chil-Min
2015-08-01
We elucidate that Fermi resonance ever plays a decisive role in dynamical tunneling in a chaotic billiard. Interacting with each other through an avoided crossing, a pair of eigenfunctions are coupled through tunneling channels for dynamical tunneling. In this case, the tunneling channels are an islands chain and its pair unstable periodic orbit, which equals the quantum number difference of the eigenfunctions. This phenomenon of dynamical tunneling is confirmed in a quadrupole billiard in relation with Fermi resonance.
Namazi, Hamidreza; Kulish, Vladimir V; Akrami, Amin
2016-01-01
One of the major challenges in vision research is to analyze the effect of visual stimuli on human vision. However, no relationship has been yet discovered between the structure of the visual stimulus, and the structure of fixational eye movements. This study reveals the plasticity of human fixational eye movements in relation to the 'complex' visual stimulus. We demonstrated that the fractal temporal structure of visual dynamics shifts towards the fractal dynamics of the visual stimulus (image). The results showed that images with higher complexity (higher fractality) cause fixational eye movements with lower fractality. Considering the brain, as the main part of nervous system that is engaged in eye movements, we analyzed the governed Electroencephalogram (EEG) signal during fixation. We have found out that there is a coupling between fractality of image, EEG and fixational eye movements. The capability observed in this research can be further investigated and applied for treatment of different vision disorders. PMID:27217194
Namazi, Hamidreza; Kulish, Vladimir V.; Akrami, Amin
2016-01-01
One of the major challenges in vision research is to analyze the effect of visual stimuli on human vision. However, no relationship has been yet discovered between the structure of the visual stimulus, and the structure of fixational eye movements. This study reveals the plasticity of human fixational eye movements in relation to the ‘complex’ visual stimulus. We demonstrated that the fractal temporal structure of visual dynamics shifts towards the fractal dynamics of the visual stimulus (image). The results showed that images with higher complexity (higher fractality) cause fixational eye movements with lower fractality. Considering the brain, as the main part of nervous system that is engaged in eye movements, we analyzed the governed Electroencephalogram (EEG) signal during fixation. We have found out that there is a coupling between fractality of image, EEG and fixational eye movements. The capability observed in this research can be further investigated and applied for treatment of different vision disorders. PMID:27217194
NASA Astrophysics Data System (ADS)
Namazi, Hamidreza; Kulish, Vladimir V.; Akrami, Amin
2016-05-01
One of the major challenges in vision research is to analyze the effect of visual stimuli on human vision. However, no relationship has been yet discovered between the structure of the visual stimulus, and the structure of fixational eye movements. This study reveals the plasticity of human fixational eye movements in relation to the ‘complex’ visual stimulus. We demonstrated that the fractal temporal structure of visual dynamics shifts towards the fractal dynamics of the visual stimulus (image). The results showed that images with higher complexity (higher fractality) cause fixational eye movements with lower fractality. Considering the brain, as the main part of nervous system that is engaged in eye movements, we analyzed the governed Electroencephalogram (EEG) signal during fixation. We have found out that there is a coupling between fractality of image, EEG and fixational eye movements. The capability observed in this research can be further investigated and applied for treatment of different vision disorders.
On the vortex dynamics in fractal Fourier turbulence.
Lanotte, Alessandra S; Malapaka, Shiva Kumar; Biferale, Luca
2016-04-01
Incompressible, homogeneous and isotropic turbulence is studied by solving the Navier-Stokes equations on a reduced set of Fourier modes, belonging to a fractal set of dimension D . By tuning the fractal dimension parameter, we study the dynamical effects of Fourier decimation on the vortex stretching mechanism and on the statistics of the velocity and the velocity gradient tensor. In particular, we show that as we move from D = 3 to D ∼ 2.8 , the statistics gradually turns into a purely Gaussian one. This result suggests that even a mild fractal mode reduction strongly depletes the stretching properties of the non-linear term of the Navier-Stokes equations and suppresses anomalous fluctuations. PMID:27125678
Active synchronization between two different chaotic dynamical system
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.
Active synchronization between two different chaotic dynamical system
NASA Astrophysics Data System (ADS)
Maheri, M.; Arifin, N. Md; Ismail, F.
2015-05-01
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.
Quantized chaotic dynamics and non-commutative KS entropy
Klimek, S.; Lesniewski, A.
1996-06-01
We study the quantization of two examples of classically chaotic dynamics, the Anosov dynamics of {open_quote}{open_quote}cat maps{close_quote}{close_quote} on a two dimensional torus, and the dynamics of baker{close_quote}s maps. Each of these dynamics is implemented as a discrete group of automorphisms of a von Neumann algebra of functions on a quantized torus. We compute the non-commutative generalization of the Kolmogorov-Sinai entropy, namely the Connes-Sto/rmer entropy, of the generator of this group, and find that its value is equal to the classical value. This can be interpreted as a sign of persistence of chaotic behavior in a dynamical system under quantization. Copyright {copyright} 1996 Academic Press, Inc.
Coevolutionary extremal dynamics on gasket fractal
NASA Astrophysics Data System (ADS)
Lee, Kyoung Eun; Sung, Joo Yup; Cha, Moon-Yong; Maeng, Seong Eun; Bang, Yu Sik; Lee, Jae Woo
2009-11-01
We considered a Bak-Sneppen model on a Sierpinski gasket fractal. We calculated the avalanche size distribution and the distribution of distances between subsequent minimal sites. To observe the temporal correlations of the avalanche, we estimated the return time distribution, the first-return time, and the all-return time distribution. The avalanche size distribution follows the power law, P(s)∼s, with the exponent τ=1.004(7). The distribution of jumping sites also follows the power law, P(r)∼r, with the critical exponent π=4.12(4). We observe the periodic oscillation of the distribution of the jumping distances which originated from the jumps of the level when the minimal site crosses the stage of the fractal. The first-return time distribution shows the power law, P(t)∼t, with the critical exponent τ=1.418(7). The all-return time distribution is also characterized by the power law, P(t)∼t, with the exponent τ=0.522(4). The exponents of the return time satisfy the scaling relation τ+τ=2 for τ⩽2.
Discriminating chaotic and stochastic dynamics through the permutation spectrum test
Kulp, C. W.; Zunino, L.
2014-09-01
In this paper, we propose a new heuristic symbolic tool for unveiling chaotic and stochastic dynamics: the permutation spectrum test. Several numerical examples allow us to confirm the usefulness of the introduced methodology. Indeed, we show that it is robust in situations in which other techniques fail (intermittent chaos, hyperchaotic dynamics, stochastic linear and nonlinear correlated dynamics, and deterministic non-chaotic noise-driven dynamics). We illustrate the applicability and reliability of this pragmatic method by examining real complex time series from diverse scientific fields. Taking into account that the proposed test has the advantages of being conceptually simple and computationally fast, we think that it can be of practical utility as an alternative test for determinism.
Discriminating chaotic and stochastic dynamics through the permutation spectrum test.
Kulp, C W; Zunino, L
2014-09-01
In this paper, we propose a new heuristic symbolic tool for unveiling chaotic and stochastic dynamics: the permutation spectrum test. Several numerical examples allow us to confirm the usefulness of the introduced methodology. Indeed, we show that it is robust in situations in which other techniques fail (intermittent chaos, hyperchaotic dynamics, stochastic linear and nonlinear correlated dynamics, and deterministic non-chaotic noise-driven dynamics). We illustrate the applicability and reliability of this pragmatic method by examining real complex time series from diverse scientific fields. Taking into account that the proposed test has the advantages of being conceptually simple and computationally fast, we think that it can be of practical utility as an alternative test for determinism. PMID:25273196
El Nino: a chaotic dynamical system
Vallis, G.K.
1986-04-11
Most of the principal qualitative features of the El Nino-Southern Oscillation phenomenon can be explained by a simple but physically motivated theory. These features are the occurrence of sea-surface warmings in the eastern equatorial Pacific and the associated trade wind reversal; the aperiodicity of these events; the preferred onset time with respect to the seasonal cycle; and the much weaker events in the Atlantic and Indian oceans. The theory, in its simplest form, is a conceptual model for the interaction of just three variables, namely near-surface temperatures in the east and west equatorial ocean and a wind-driven current advecting the temperature field. For a large range of parameters, the model is naturally chaotic and aperiodically produces El Nino-like events. For a smaller basin, representing a smaller ocean, the events are proportionally less intense.
Generalized correlation integral vectors: A distance concept for chaotic dynamical systems
Haario, Heikki; Kalachev, Leonid; 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. We 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.
Long-Range Correlations in Stride Intervals May Emerge from Non-Chaotic Walking Dynamics
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
Chaotic dynamics around astrophysical objects with nonisotropic stresses
Dubeibe, F. L.; Pachon, Leonardo A.; Sanabria-Gomez, Jose D.
2007-01-15
The existence of chaotic behavior for the geodesics of the test particles orbiting compact objects is a subject of much current research. Some years ago, Gueron and Letelier [Phys. Rev. E 66, 046611 (2002)] reported the existence of chaotic behavior for the geodesics of the test particles orbiting compact objects like black holes induced by specific values of the quadrupolar deformation of the source using as models the Erez--Rosen solution and the Kerr black hole deformed by an internal multipole term. In this work, we are interested in the study of the dynamic behavior of geodesics around astrophysical objects with intrinsic quadrupolar deformation or nonisotropic stresses, which induces nonvanishing quadrupolar deformation for the nonrotating limit. For our purpose, we use the Tomimatsu-Sato spacetime [Phys. Rev. Lett. 29 1344 (1972)] and its arbitrary deformed generalization obtained as the particular vacuum case of the five parametric solution of Manko et al. [Phys. Rev. D 62, 044048 (2000)] characterizing the geodesic dynamics throughout the Poincare sections method. We found only regular motion for the geodesics in the Tomimatsu-Sato {delta}=2 solution. Additionally, using the deformed generalization of Tomimatsu-Sato {delta}=2 solution given by Manko et al. we found chaotic motion for oblate deformation instead of prolate deformation, which is in contrast to the results by Gueron and Letelier. It opens the possibility that the particles forming the accretion disk around a large variety of different astrophysical bodies (nonprolate, e.g., neutron stars) could exhibit chaotic dynamics. We also conjecture that the existence of an arbitrary deformation parameter is necessary for the existence of chaotic dynamics.
Hypotheses on the functional roles of chaotic transitory dynamics
NASA Astrophysics Data System (ADS)
Tsuda, Ichiro
2009-03-01
In contrast to the conventional static view of the brain, recent experimental data show that an alternative view is necessary for an appropriate interpretation of its function. Some selected problems concerning the cortical transitory dynamics are discussed. For the first time, we propose five scenarios for the appearance of chaotic itinerancy, which provides typical transitory dynamics. Second, we describe the transitory behaviors that have been observed in human and animal brains. Finally, we propose nine hypotheses on the functional roles of such dynamics, focusing on the dynamics embedded in data and the dynamical interpretation of brain activity within the framework of cerebral hermeneutics.
Chaotic dynamics in cardiac aggregates induced by potassium channel block
NASA Astrophysics Data System (ADS)
Quail, Thomas; McVicar, Nevin; Aguilar, Martin; Kim, Min-Young; Hodge, Alex; Glass, Leon; Shrier, Alvin
2012-09-01
Chaotic rhythms in deterministic models can arise as a consequence of changes in model parameters. We carried out experimental studies in which we induced a variety of complex rhythms in aggregates of embryonic chick cardiac cells using E-4031 (1.0-2.5 μM), a drug that blocks the hERG potassium channel. Following the addition of the drug, the regular rhythm evolved to display a spectrum of complex dynamics: irregular rhythms, bursting oscillations, doublets, and accelerated rhythms. The interbeat intervals of the irregular rhythms can be described by one-dimensional return maps consistent with chaotic dynamics. A Hodgkin-Huxley-style cardiac ionic model captured the different types of complex dynamics following blockage of the hERG mediated potassium current.
Fractal and complex network analyses of protein molecular dynamics
NASA Astrophysics Data System (ADS)
Zhou, Yuan-Wu; Liu, Jin-Long; Yu, Zu-Guo; Zhao, Zhi-Qin; Anh, Vo
2014-12-01
Based on protein molecular dynamics, we investigate the fractal properties of energy, pressure and volume time series using the multifractal detrended fluctuation analysis (MF-DFA) and the topological and fractal properties of their converted horizontal visibility graphs (HVGs). The energy parameters of protein dynamics we considered are bonded potential, angle potential, dihedral potential, improper potential, kinetic energy, Van der Waals potential, electrostatic potential, total energy and potential energy. The shape of the h(q) curves from MF-DFA indicates that these time series are multifractal. The numerical values of the exponent h(2) of MF-DFA show that the series of total energy and potential energy are non-stationary and anti-persistent; the other time series are stationary and persistent apart from series of pressure (with H≈0.5 indicating the absence of long-range correlation). The degree distributions of their converted HVGs show that these networks are exponential. The results of fractal analysis show that fractality exists in these converted HVGs. For each energy, pressure or volume parameter, it is found that the values of h(2) of MF-DFA on the time series, exponent λ of the exponential degree distribution and fractal dimension dB of their converted HVGs do not change much for different proteins (indicating some universality). We also found that after taking average over all proteins, there is a linear relationship between
Synchronization in complex dynamical networks coupled with complex chaotic system
NASA Astrophysics Data System (ADS)
Wei, Qiang; Xie, Cheng-Jun; Wang, Bo
2015-11-01
This paper investigates synchronization in complex dynamical networks with time delay and perturbation. The node of complex dynamical networks is composed of complex chaotic system. A complex feedback controller is designed to realize different component of complex state variable synchronize up to different scaling complex function when complex dynamical networks realize synchronization. The synchronization scaling function is changed from real field to complex field. Synchronization in complex dynamical networks with constant delay and time-varying coupling delay are investigated, respectively. Numerical simulations show the effectiveness of the proposed method.
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 performed 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.
Chaotic dynamics of flexible Euler-Bernoulli beams
NASA Astrophysics Data System (ADS)
Awrejcewicz, J.; Krysko, A. V.; Kutepov, I. E.; Zagniboroda, N. A.; Dobriyan, V.; Krysko, V. A.
2013-12-01
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(c2) 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 is 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 q0 and frequency ωp 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.
Social opinion dynamics is not chaotic
NASA Astrophysics Data System (ADS)
Lim, Chjan; Zhang, Weituo
2016-08-01
Motivated by the research on social opinion dynamics over large and dense networks, a general framework for verifying the monotonicity property of multi-agent dynamics is introduced. This allows a derivation of sociologically meaningful sufficient conditions for monotonicity that are tailor-made for social opinion dynamics, which typically have high nonlinearity. A direct consequence of monotonicity is that social opinion dynamics is nonchaotic. A key part of this framework is the definition of a partial order relation that is suitable for a large class of social opinion dynamics such as the generalized naming games. Comparisons are made to previous techniques to verify monotonicity. Using the results obtained, we extend many of the consequences of monotonicity to this class of social dynamics, including several corollaries on their asymptotic behavior, such as global convergence to consensus and tipping points of a minority fraction of zealots or leaders.
Forward and adjoint sensitivity computation of chaotic dynamical systems
Wang, Qiqi
2013-02-15
This paper describes a forward algorithm and an adjoint algorithm for computing sensitivity derivatives in chaotic dynamical systems, such as the Lorenz attractor. The algorithms compute the derivative of long time averaged “statistical” quantities to infinitesimal perturbations of the system parameters. The algorithms are demonstrated on the Lorenz attractor. We show that sensitivity derivatives of statistical quantities can be accurately estimated using a single, short trajectory (over a time interval of 20) on the Lorenz attractor.
Chaotic dynamics in accelerator physics. Progress report
Cary, J.R.
1992-11-30
Substantial progress was in several areas of accelerator dynamics. For developing understanding of longitudinal adiabatic dynamics, and for creating efficiency enhancements of recirculating free-electron lasers, was substantially completed. A computer code for analyzing the critical KAM tori that bound the dynamic aperture in circular machines was developed. Studies of modes that arise due to the interaction of coating beams with a narrow-spectrum impedance have begun. During this research educational and research ties with the accelerator community at large have been strengthened.
Dynamics and Synchronization of Semiconductor Lasers for Chaotic Optical Communications
NASA Astrophysics Data System (ADS)
Liu, Jia-Ming; Chen, How-Foo; Tang, Shuo
The objective of this chapter is to provide a complete picture of the nonlinear dynamics and chaos synchronization of single-mode semiconductor lasers for chaotic optical communications. Basic concepts and theoretical framework are reviewed. Experimental results are presented to demonstrate the fundamental concepts. Numerical computations are employed for mapping the dynamical states and for illustrating certain detailed characteristics of the chaotic states. Three different semiconductor laser systems, namely, the optical injection system, the optical feedback system, and the optoelectronic feedback system, that are of most interest for high-bit-rate chaotic optical communications are considered. The optical injection system is a nonautonomous system that follows a period-doubling route to chaos. The optical feedback system is a phase-sensitive delayed-feedback autonomous system for which all three known routes, namely, period-doubling, quasiperiodicity, and intermittency, to chaos can be found. The optical feedback system is a phase-insensitive delayed-feedback autonomous system that follows a quasiperiodicity route to chaotic pulsing. Identical synchronization in unidirectionally coupled configurations is the focus of discussions for chaotic communications. For optical injection and optical feedback systems, the frequency, phase, and amplitude of the optical fields of both transmitter and receiver lasers are all locked in synchronism when complete synchronization is accomplished. For the optoelectronic feedback system, chaos synchronization involves neither the locking of the optical frequency nor the synchronization of the optical phase. For both optical feedback and optoelectronic feedback systems, where the transmitter is configured with a delayed feedback loop, anticipated and retarded synchronization can be observed as the difference between the feedback delay time and the propagation time from the transmitter laser to the receiver laser is varied. For a
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.
Chaotic electron dynamics in gyrotron resonators
Kominis, Y.; Dumbrajs, O.; Avramides, K.A.; Hizanidis, K.; Vomvoridis, J.L.
2005-04-15
Phase space analysis of electron dynamics is used in combination with the canonical perturbation method and the KAM (Kolmogorov-Arnold-Moser) theory in order to study the dependence of the efficient gyrotron operation on the rf field profile and frequency mismatch. Knowledge of the boundaries of the electron motion provided through robust (slightly distorted) KAM surfaces is useful for optimizing depressed collectors and thereby for enhancement of overall efficiency of gyrotron operation.
Chaotic dynamics in circulation with Tohoku University vibrating flow pump.
Nitta, S; Yambe, T; Kobayashi, S; Hashimoto, H; Yoshizawa, M; Mastuki, H; Tabayashi, K; Takeda, H
1999-01-01
For the development of a totally implantable ventricular assist system (VAS), we have been developing the vibrating flow pump (VFP), which can generate oscillated blood flow with a relative high frequency (10-50 Hz) for a totally implantable system. In this study, the effects of left ventricular assistance with this unique oscillated blood flow were analyzed by the use of nonlinear mathematics for evaluation as the whole circulatory regulatory system, not as the decomposed parts of the system. Left heart bypasses using the VFP from the left atrium to the descending aorta were performed in chronic animal experiments using healthy adult goats. The ECG, arterial blood pressure, VFP pump flow, and the flow of the descending aorta were recorded in the data recorder during awake conditions and analyzed in a personal computer system through an A-D convertor. By the use of nonlinear mathematics, time series data were embedded into the phase space, the Lyapunov numerical method, fractal dimension analysis, and power spectrum analysis were performed to evaluate nonlinear dynamics. During left ventricular assistance with the VFP, Mayer wave fluctuations were decreased in the power spectrum, the fractal dimension of the hemodynamics was significantly decreased, and peripheral vascular resistance was significantly decreased. These results suggest that nonlinear dynamics, which mediate the cardiovascular dynamics, may be affected during left ventricular (LV) bypass with oscillated flow. The decreased power of the Mayer wave in the spectrum caused the limit cycle attractor of the hemodynamics and decreased peripheral resistance. Decreased sympathetic discharges may be the origin of the decreased Mayer wave and fractal dimension. These nonlinear dynamic analyses may be useful to design optimal VAS control. PMID:9950190
A Brief Historical Introduction to Fractals and Fractal Geometry
ERIC Educational Resources Information Center
Debnath, Lokenath
2006-01-01
This paper deals with a brief historical introduction to fractals, fractal dimension and fractal geometry. Many fractals including the Cantor fractal, the Koch fractal, the Minkowski fractal, the Mandelbrot and Given fractal are described to illustrate self-similar geometrical figures. This is followed by the discovery of dynamical systems and…
Fractal analysis on human dynamics of library loans
NASA Astrophysics Data System (ADS)
Fan, Chao; Guo, Jin-Li; Zha, Yi-Long
2012-12-01
In this paper, the fractal characteristic of human behaviors is investigated from the perspective of time series constructed with the amount of library loans. The values of the Hurst exponent and length of non-periodic cycle calculated through rescaled range analysis indicate that the time series of human behaviors and their sub-series are fractal with self-similarity and long-range dependence. Then the time series are converted into complex networks by the visibility algorithm. The topological properties of the networks such as scale-free property and small-world effect imply that there is a close relationship among the numbers of repetitious behaviors performed by people during certain periods of time. Our work implies that there is intrinsic regularity in the human collective repetitious behaviors. The conclusions may be helpful to develop some new approaches to investigate the fractal feature and mechanism of human dynamics, and provide some references for the management and forecast of human collective behaviors.
A fractal approach to dynamic inference and distribution analysis
van Rooij, Marieke M. J. W.; Nash, Bertha A.; Rajaraman, Srinivasan; Holden, John G.
2013-01-01
Event-distributions inform scientists about the variability and dispersion of repeated measurements. This dispersion can be understood from a complex systems perspective, and quantified in terms of fractal geometry. The key premise is that a distribution's shape reveals information about the governing dynamics of the system that gave rise to the distribution. Two categories of characteristic dynamics are distinguished: additive systems governed by component-dominant dynamics and multiplicative or interdependent systems governed by interaction-dominant dynamics. A logic by which systems governed by interaction-dominant dynamics are expected to yield mixtures of lognormal and inverse power-law samples is discussed. These mixtures are described by a so-called cocktail model of response times derived from human cognitive performances. The overarching goals of this article are twofold: First, to offer readers an introduction to this theoretical perspective and second, to offer an overview of the related statistical methods. PMID:23372552
Chaotic dynamics of flexible Euler-Bernoulli beams.
Awrejcewicz, J; Krysko, A V; Kutepov, I E; Zagniboroda, N A; Dobriyan, V; Krysko, V A
2013-12-01
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(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 is 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(0) and frequency ω(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. PMID:24387569
Chaotic dynamics of flexible Euler-Bernoulli beams
Awrejcewicz, J.; Kutepov, I. E. Zagniboroda, N. A. Dobriyan, V. Krysko, V. A.
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 is 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.
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.
NASA Astrophysics Data System (ADS)
Kato, Tomohiro; Hasegawa, Mikio
Chaotic dynamics has been shown to be effective in improving the performance of combinatorial optimization algorithms. In this paper, the performance of chaotic dynamics in the asymmetric traveling salesman problem (ATSP) is investigated by introducing three types of heuristic solution update methods. Numerical simulation has been carried out to compare its performance with simulated annealing and tabu search; thus, the effectiveness of the approach using chaotic dynamics for driving heuristic methods has been shown. The chaotic method is also evaluated in the case of a combinatorial optimization problem in the real world, which can be solved by the same heuristic operation as that for the ATSP. We apply the chaotic method to the DNA fragment assembly problem, which involves building a DNA sequence from several hundred fragments obtained by the genome sequencer. Our simulation results show that the proposed algorithm using chaotic dynamics in a block shift operation exhibits the best performance for the DNA fragment assembly problem.
A review of sigma models for quantum chaotic dynamics
NASA Astrophysics Data System (ADS)
Altland, Alexander; Gnutzmann, Sven; Haake, Fritz; Micklitz, Tobias
2015-07-01
We review the construction of the supersymmetric sigma model for unitary maps, using the color-flavor transformation. We then illustrate applications by three case studies in quantum chaos. In two of these cases, general Floquet maps and quantum graphs, we show that universal spectral fluctuations arise provided the pertinent classical dynamics are fully chaotic (ergodic and with decay rates sufficiently gapped away from zero). In the third case, the kicked rotor, we show how the existence of arbitrarily long-lived modes of excitation (diffusion) precludes universal fluctuations and entails quantum localization.
A review of sigma models for quantum chaotic dynamics.
Altland, Alexander; Gnutzmann, Sven; Haake, Fritz; Micklitz, Tobias
2015-07-01
We review the construction of the supersymmetric sigma model for unitary maps, using the color-flavor transformation. We then illustrate applications by three case studies in quantum chaos. In two of these cases, general Floquet maps and quantum graphs, we show that universal spectral fluctuations arise provided the pertinent classical dynamics are fully chaotic (ergodic and with decay rates sufficiently gapped away from zero). In the third case, the kicked rotor, we show how the existence of arbitrarily long-lived modes of excitation (diffusion) precludes universal fluctuations and entails quantum localization. PMID:26181515
Lightning and the Heart: Fractal Behavior in Cardiac Function
BASSINGTHWAIGHTE, JAMES B.; van BEEK, J. H. G. M.
2010-01-01
Physical systems, from galactic clusters to diffusing molecules, often show fractal behavior. Likewise, living systems might often be well described by fractal algorithms. Such fractal descriptions in space and time imply that there is order in chaos, or put the other way around, chaotic dynamical systems in biology are more constrained and orderly than seen at first glance. The vascular network, the syncytium of cells, the processes of diffusion and transmembrane transport might be fractal features of the heart. These fractal features provide a basis which enables one to understand certain aspects of more global behavior such as atrial or ventricular fibrillation and perfusion heterogeneity. The heart might be regarded as a prototypical organ from these points of view. A particular example of the use of fractal geometry is in explaining myocardial flow heterogeneity via delivery of blood through an asymmetrical fractal branching network. PMID:21938081
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
Fractal dynamics of electric discharges in a thundercloud.
Iudin, D I; Trakhtengerts, V Y; Hayakawa, M
2003-07-01
We have investigated the fractal dynamics of intracloud microdischarges responsible for the formation of a so-called drainage system of electric charge transport inside a cloud volume. Microdischarges are related to the nonlinear stage of multiflow instability development, which leads to the generation of a small-scale intracloud electric structure. The latter is modeled by using a two-dimensional lattice of finite-state automata. The results of numerical simulations show that the developed drainage system belongs to the percolation-cluster family. We then point out the parameter region relevant to the proposed model, in which the thundercloud exhibits behavior corresponding to a regime of self-organized criticality. The initial development and statistical properties of dynamic conductive clusters are investigated, and a kinetic equation is introduced, which permits us to find state probabilities of electric cells and to estimate macroscopic parameters of the system. PMID:12935264
Coexisting chaotic and periodic dynamics in clock escapements.
Moon, Francis C; Stiefel, Preston D
2006-09-15
This paper addresses the nature of noise in machines. As a concrete example, we examine the dynamics of clock escapements from experimental, historical and analytical points of view. Experiments on two escapement mechanisms from the Reuleaux kinematic collection at Cornell University are used to illustrate chaotic-like noise in clocks. These vibrations coexist with the periodic dynamics of the balance wheel or pendulum. A mathematical model is presented that shows how self-generated chaos in clocks can break the dry friction in the gear train. This model is shown to exhibit a strange attractor in the structural vibration of the clock. The internal feedback between the oscillator and the escapement structure is similar to anti-control of chaos models. PMID:16893802
Detection of Ordered and Chaotic Motion Using the Dynamical Spectra
NASA Astrophysics Data System (ADS)
Voglis, N.; Contopoulos, G.; Efthymiopoulos, C.
1999-01-01
Two simple and efficient numerical methods to explore the phase space structure are presented, based on the properties of the "dynamical spectra". 1) We calculate a "spectral distance" D of the dynamical spectra for two different initial deviation vectors. D → 0 in the case of chaotic orbits, while D → const ≠ 0 in the case of ordered orbits. This method is by orders of magnitude faster than the method of the Lyapunov Characteristic Number (LCN). 2) We define a sensitive indicator called ROTOR (ROtational TOri Recongnizer) for 2D maps. The ROTOR remains zero in time on a rotational torus, while it tends to infinity at a rate ∝ N = number of iterations, in any case other than a rotational torus. We use this method to locate the last KAM torus of an island of stability, as well as the most important cantori causing stickiness near it.
Bifurcation Structures in a Bimodal Piecewise Linear Map: Chaotic Dynamics
NASA Astrophysics Data System (ADS)
Panchuk, Anastasiia; Sushko, Iryna; Avrutin, Viktor
In this work, we investigate the bifurcation structure of the parameter space of a generic 1D continuous piecewise linear bimodal map focusing on the regions associated with chaotic attractors (cyclic chaotic intervals). The boundaries of these regions corresponding to chaotic attractors with different number of intervals are identified. The results are obtained analytically using the skew tent map and the map replacement technique.
Escape dynamics and fractal basins boundaries in the three-dimensional Earth-Moon system
NASA Astrophysics Data System (ADS)
Zotos, Euaggelos E.
2016-03-01
The orbital dynamics of a spacecraft, or a comet, or an asteroid in the Earth-Moon system in a scattering region around the Moon using the three dimensional version of the circular restricted three-body problem is numerically investigated. The test particle can move in bounded orbits around the Moon or escape through the openings around the Lagrange points L1 and L2 or even collide with the surface of the Moon. We explore in detail the first four of the five possible Hill's regions configurations depending on the value of the Jacobi constant which is of course related with the total orbital energy. We conduct a thorough numerical analysis on the phase space mixing by classifying initial conditions of orbits in several two-dimensional types of planes and distinguishing between four types of motion: (i) ordered bounded, (ii) trapped chaotic, (iii) escaping and (iv) collisional. In particular, we locate the different basins and we relate them with the corresponding spatial distributions of the escape and collision times. Our outcomes reveal the high complexity of this planetary system. Furthermore, the numerical analysis suggests a strong dependence of the properties of the considered basins with both the total orbital energy and the initial value of the z coordinate, with a remarkable presence of fractal basin boundaries along all the regimes. Our results are compared with earlier ones regarding the planar version of the Earth-Moon system.
RAPID DYNAMICAL MASS SEGREGATION AND PROPERTIES OF FRACTAL STAR CLUSTERS
Yu Jincheng; Chen Li; De Grijs, Richard
2011-05-01
We investigate the evolution of young star clusters using N-body simulations. We confirm that subvirial and fractal-structured clusters will dynamically mass segregate on a short timescale (within 0.5 Myr). We adopt a modified minimum-spanning-tree method to measure the degree of mass segregation, demonstrating that the stars escaping from a cluster's potential are important for the temporal dependence of mass segregation in the cluster. The form of the initial velocity distribution will also affect the degree of mass segregation. If it depends on radius, the outer parts of the cluster would expand without undergoing collapse. In velocity space, we find 'inverse mass segregation', which indicates that massive stars have higher velocity dispersions than their lower-mass counterparts.
Chaotic Dynamics of Driven Flux Drops: A Superconducting ``Dripping Faucet''
NASA Astrophysics Data System (ADS)
Field, Stuart B.; Stan, Gheorghe
2008-02-01
When a current is applied to a type-I superconducting strip containing a narrow channel across its width, magnetic flux spots nucleate at the edge and are then driven along the channel by the current. These flux “drops” are reminiscent of water drops dripping from a faucet, a model system for studying low-dimensional chaos. We use a novel high-bandwidth Hall probe to detect in real time the motion of individual flux spots moving along the channel. Analyzing the time series consisting of the intervals between successive flux drops, we find distinct regions of chaotic behavior characterized by positive Lyapunov exponents, indicating that there is a close analogy between the dynamics of the superconducting and water drop systems.
Onset of Chaotic Dynamics in Vortex Sheet Roll-Up
NASA Astrophysics Data System (ADS)
Krasny, Robert; Nitsche, Monika
1997-11-01
Vortex sheet roll-up in planar and axisymmetric geometry is studied numerically. Starting from flat initial data, the sheet rolls up into either a vortex pair or a vortex ring, depending on the geometry. The spiral roll-up proceeds smoothly at early times, but at late times the sheet develops small-scale irregular features. The outer turn becomes folded and sheds a wake behind the vortex ring, and spiral turns in the vortex core become non-uniformly spaced for both cases. These features are attributed to the onset of chaotic dynamics, specifically a heteroclinic tangle in the case of the vortex ring, and a resonance band for both cases. A Poincaré section is presented to support this conjecture. Two factors account for the onset of chaos: the presence of hyperbolic and elliptic points in the instantaneous streamline pattern, and (2) self-sustained small amplitude oscillations in the core vorticity distribution.
Chaotic dynamics and diffusion in a piecewise linear equation
Shahrear, Pabel; Glass, Leon; Edwards, Rod
2015-03-15
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.
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.
Generalized Gaussian wave packet dynamics: Integrable and chaotic systems.
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. PMID:26871079
Regular and chaotic dynamics of a piecewise smooth bouncer
Langer, Cameron K. Miller, Bruce N.
2015-07-15
The dynamical properties of a particle in a gravitational field colliding with a rigid wall moving with piecewise constant velocity are studied. The linear nature of the wall's motion permits further analytical investigation than is possible for the system's sinusoidal counterpart. We consider three distinct approaches to modeling collisions: (i) elastic, (ii) inelastic with constant restitution coefficient, and (iii) inelastic with a velocity-dependent restitution function. We confirm the existence of distinct unbounded orbits (Fermi acceleration) in the elastic model, and investigate regular and chaotic behavior in the inelastic cases. We also examine in the constant restitution model trajectories wherein the particle experiences an infinite number of collisions in a finite time, i.e., the phenomenon of inelastic collapse. We address these so-called “sticking solutions” and their relation to both the overall dynamics and the phenomenon of self-reanimating chaos. Additionally, we investigate the long-term behavior of the system as a function of both initial conditions and parameter values. We find the non-smooth nature of the system produces novel bifurcation phenomena not seen in the sinusoidal model, including border-collision bifurcations. The analytical and numerical investigations reveal that although our piecewise linear bouncer is a simplified version of the sinusoidal model, the former not only captures essential features of the latter but also exhibits behavior unique to the discontinuous dynamics.
NASA Astrophysics Data System (ADS)
Zhang, Zhe; Xun, Zhi-Peng; Wu, Ling; Chen, Yi-Li; Xia, Hui; Hao, Da-Peng; Tang, Gang
2016-06-01
In order to study the effects of the microscopic details of fractal substrates on the scaling behavior of the growth model, a generalized linear fractal Langevin-type equation, ∂h / ∂t =(- 1) m + 1 ν∇ mzrw h (zrw is the dynamic exponent of random walk on substrates), driven by nonconserved and conserved noise is proposed and investigated theoretically employing scaling analysis. Corresponding dynamic scaling exponents are obtained.
Chaotic Soliton Dynamics in Photoexcited trans-Polyacetylene.
Bernasconi, Leonardo
2015-03-01
We study the photogeneration of topological solitons in trans-polyacetylene and their time evolution using ab initio excited-state dynamics. The system is excited to the optically allowed 1(1)Bu state, and the atoms are then propagated classically using quantum mechanical forces computed using hybrid time-dependent density functional theory (TD-DFT). A soliton/antisoliton pair nucleates spontaneously and creates two independent solitons moving at constant velocity, similar to simulations based on uncorrelated lattice models like the Su-Schrieffer-Heeger (SSH) Hamiltonian [Su, W. P.; Schrieffer, J. R.; Heeger, A. J. Phys. Rev. Lett. 1979, 42, 1698]. At T = 0, the solitons coalesce into bound pairs with a two-soliton functional form, whereas chaotic dynamics, in the form of 2-bounce resonances, is observed at soliton/antisoliton collisions at T ≠ 0. This behavior is related to the onset of a strong correlation regime at short intersoliton distance, which is not accounted for by SSH simulations. PMID:26262671
Blended particle filters for large-dimensional chaotic dynamical systems.
Majda, Andrew J; Qi, Di; Sapsis, Themistoklis P
2014-05-27
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
Chaotic magnetization dynamics in single-crystal thin-film structures
NASA Astrophysics Data System (ADS)
Shutyi, A. M.; Sementsov, D. I.
2009-01-01
The nonlinear dynamics of homogeneously precessing magnetization in perpendicularly magnetized single-crystal films has been investigated in a wide range of ac field frequencies on the basis of a numerical solution to the Landau-Lifshitz equation and construction of the spectrum of Lyapunov exponents. The conditions for implementing and specific features of chaotic dynamic modes are revealed for films of three basic crystallographic orientations: (100), (110), and (111). It is shown that chaotic precession modes can be controlled using external magnetic fields. Time analogs of the Poincaré section of chaotic mode trajectories are considered.
Chaotic magnetization dynamics in single-crystal thin-film structures
Shutyi, A. M. Sementsov, D. I.
2009-01-15
The nonlinear dynamics of homogeneously precessing magnetization in perpendicularly magnetized single-crystal films has been investigated in a wide range of ac field frequencies on the basis of a numerical solution to the Landau-Lifshitz equation and construction of the spectrum of Lyapunov exponents. The conditions for implementing and specific features of chaotic dynamic modes are revealed for films of three basic crystallographic orientations: (100), (110), and (111). It is shown that chaotic precession modes can be controlled using external magnetic fields. Time analogs of the Poincare section of chaotic mode trajectories are considered.
Universal behavior in the parametric evolution of chaotic saddles
Lai, Y.; Zyczkowski, K.; Grebogi, C.; Zyczkowski, K.; Grebogi, C.
1999-05-01
Chaotic saddles are nonattracting dynamical invariant sets that physically lead to transient chaos. As a system parameter changes, chaotic saddles can evolve via an infinite number of homoclinic or heteroclinic tangencies of their stable and unstable manifolds. Based on previous numerical evidence and a rigorous analysis of a class of representative models, we show that dynamical invariants such as the topological entropy and the fractal dimension of chaotic saddles obey a universal behavior: they exhibit a devil-staircase characteristic as a function of the system parameter. {copyright} {ital 1999} {ital The American Physical Society}
Efficient sensitivity analysis method for chaotic dynamical systems
NASA Astrophysics Data System (ADS)
Liao, Haitao
2016-05-01
The direct differentiation and improved least squares shadowing methods are both developed for accurately and efficiently calculating the sensitivity coefficients of time averaged quantities for chaotic dynamical systems. The key idea is to recast the time averaged integration term in the form of differential equation before applying the sensitivity analysis method. An additional constraint-based equation which forms the augmented equations of motion is proposed to calculate the time averaged integration variable and the sensitivity coefficients are obtained as a result of solving the augmented differential equations. The application of the least squares shadowing formulation to the augmented equations results in an explicit expression for the sensitivity coefficient which is dependent on the final state of the Lagrange multipliers. The LU factorization technique to calculate the Lagrange multipliers leads to a better performance for the convergence problem and the computational expense. Numerical experiments on a set of problems selected from the literature are presented to illustrate the developed methods. The numerical results demonstrate the correctness and effectiveness of the present approaches and some short impulsive sensitivity coefficients are observed by using the direct differentiation sensitivity analysis method.
OPEN PROBLEM: Orbits' statistics in chaotic dynamical systems
NASA Astrophysics Data System (ADS)
Arnold, V.
2008-07-01
This paper shows how the measurement of the stochasticity degree of a finite sequence of real numbers, published by Kolmogorov in Italian in a journal of insurances' statistics, can be usefully applied to measure the objective stochasticity degree of sequences, originating from dynamical systems theory and from number theory. Namely, whenever the value of Kolmogorov's stochasticity parameter of a given sequence of numbers is too small (or too big), one may conclude that the conjecture describing this sequence as a sample of independent values of a random variables is highly improbable. Kolmogorov used this strategy fighting (in a paper in 'Doklady', 1940) against Lysenko, who had tried to disprove the classical genetics' law of Mendel experimentally. Calculating his stochasticity parameter value for the numbers from Lysenko's experiment reports, Kolmogorov deduced, that, while these numbers were different from the exact fulfilment of Mendel's 3 : 1 law, any smaller deviation would be a manifestation of the report's number falsification. The calculation of the values of the stochasticity parameter would be useful for many other generators of pseudorandom numbers and for many other chaotically looking statistics, including even the prime numbers distribution (discussed in this paper as an example).
Fractal dynamics of heartbeat time series of young persons with metabolic syndrome
NASA Astrophysics Data System (ADS)
Muñoz-Diosdado, A.; Alonso-Martínez, A.; Ramírez-Hernández, L.; Martínez-Hernández, G.
2012-10-01
Many physiological systems have been in recent years quantitatively characterized using fractal analysis. We applied it to study heart variability of young subjects with metabolic syndrome (MS); we examined the RR time series (time between two R waves in ECG) with the detrended fluctuation analysis (DFA) method, the Higuchi's fractal dimension method and the multifractal analysis to detect the possible presence of heart problems. The results show that although the young persons have MS, the majority do not present alterations in the heart dynamics. However, there were cases where the fractal parameter values differed significantly from the healthy people values.
Coexisting chaotic and multi-periodic dynamics in a model of cardiac alternans
Skardal, Per Sebastian; Restrepo, Juan G.
2014-12-15
The spatiotemporal dynamics of cardiac tissue is an active area of research for biologists, physicists, and mathematicians. Of particular interest is the study of period-doubling bifurcations and chaos due to their link with cardiac arrhythmogenesis. In this paper, we study the spatiotemporal dynamics of a recently developed model for calcium-driven alternans in a one dimensional cable of tissue. In particular, we observe in the cable coexistence of regions with chaotic and multi-periodic dynamics over wide ranges of parameters. We study these dynamics using global and local Lyapunov exponents and spatial trajectory correlations. Interestingly, near nodes—or phase reversals—low-periodic dynamics prevail, while away from the nodes, the dynamics tend to be higher-periodic and eventually chaotic. Finally, we show that similar coexisting multi-periodic and chaotic dynamics can also be observed in a detailed ionic model.
Celso Grebogi
2000-02-29
This is the final report on a research project that explored (a) controlling complex dynamical systems; (b) using controlled chaotic signals for communication (c) methods of controlling chaos via targeting; (d) deterministic modeling; and miscellaneous work on the interface between chaotic and stable periodic behavior as system parameters vary, bifurcations of non-smooth systems that describe impact oscillators; phenomena that occur in quasiperiodically forced systems, and the fractal and topological properties of chaotic inveriant sets, in particular those arising in fluid flow.
Kinematic variability, fractal dynamics and local dynamic stability of treadmill walking
2011-01-01
Background Motorized treadmills are widely used in research or in clinical therapy. Small kinematics, kinetics and energetics changes induced by Treadmill Walking (TW) as compared to Overground Walking (OW) have been reported in literature. The purpose of the present study was to characterize the differences between OW and TW in terms of stride-to-stride variability. Classical (Standard Deviation, SD) and non-linear (fractal dynamics, local dynamic stability) methods were used. In addition, the correlations between the different variability indexes were analyzed. Methods Twenty healthy subjects performed 10 min TW and OW in a random sequence. A triaxial accelerometer recorded trunk accelerations. Kinematic variability was computed as the average SD (MeanSD) of acceleration patterns among standardized strides. Fractal dynamics (scaling exponent α) was assessed by Detrended Fluctuation Analysis (DFA) of stride intervals. Short-term and long-term dynamic stability were estimated by computing the maximal Lyapunov exponents of acceleration signals. Results TW did not modify kinematic gait variability as compared to OW (multivariate T2, p = 0.87). Conversely, TW significantly modified fractal dynamics (t-test, p = 0.01), and both short and long term local dynamic stability (T2 p = 0.0002). No relationship was observed between variability indexes with the exception of significant negative correlation between MeanSD and dynamic stability in TW (3 × 6 canonical correlation, r = 0.94). Conclusions Treadmill induced a less correlated pattern in the stride intervals and increased gait stability, but did not modify kinematic variability in healthy subjects. This could be due to changes in perceptual information induced by treadmill walking that would affect locomotor control of the gait and hence specifically alter non-linear dependencies among consecutive strides. Consequently, the type of walking (i.e. treadmill or overground) is important to consider in each protocol
A fast chaotic cryptographic scheme with dynamic look-up table
NASA Astrophysics Data System (ADS)
Wong, K. W.
2002-06-01
We propose a fast chaotic cryptographic scheme based on iterating a logistic map. In particular, no random numbers need to be generated and the look-up table used in the cryptographic process is updated dynamically. Simulation results show that the proposed method leads to a substantial reduction in the encryption and decryption time. As a result, chaotic cryptography becomes more practical in the secure transmission of large multi-media files over public data communication network.
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.
NASA Astrophysics Data System (ADS)
Igeta, Hideki; Hasegawa, Mikio
Chaotic dynamics have been effectively applied to improve various heuristic algorithms for combinatorial optimization problems in many studies. Currently, the most used chaotic optimization scheme is to drive heuristic solution search algorithms applicable to large-scale problems by chaotic neurodynamics including the tabu effect of the tabu search. Alternatively, meta-heuristic algorithms are used for combinatorial optimization by combining a neighboring solution search algorithm, such as tabu, gradient, or other search method, with a global search algorithm, such as genetic algorithms (GA), ant colony optimization (ACO), or others. In these hybrid approaches, the ACO has effectively optimized the solution of many benchmark problems in the quadratic assignment problem library. In this paper, we propose a novel hybrid method that combines the effective chaotic search algorithm that has better performance than the tabu search and global search algorithms such as ACO and GA. Our results show that the proposed chaotic hybrid algorithm has better performance than the conventional chaotic search and conventional hybrid algorithms. In addition, we show that chaotic search algorithm combined with ACO has better performance than when combined with GA.
Self-affine fractal variability of human heartbeat interval dynamics in health and disease.
Meyer, M; Stiedl, O
2003-10-01
The complexity of the cardiac rhythm is demonstrated to exhibit self-affine multifractal variability. The dynamics of heartbeat interval time series was analyzed by application of the multifractal formalism based on the Cramèr theory of large deviations. The continuous multifractal large deviation spectrum uncovers the nonlinear fractal properties in the dynamics of heart rate and presents a useful diagnostic framework for discrimination and classification of patients with cardiac disease, e.g., congestive heart failure. The characteristic multifractal pattern in heart transplant recipients or chronic heart disease highlights the importance of neuroautonomic control mechanisms regulating the fractal dynamics of the cardiac rhythm. PMID:12942331
Nonlinear dynamics of drops and bubbles and chaotic phenomena
NASA Technical Reports Server (NTRS)
Trinh, Eugene H.; Leal, L. G.; Feng, Z. C.; Holt, R. G.
1994-01-01
Nonlinear phenomena associated with the dynamics of free drops and bubbles are investigated analytically, numerically and experimentally. Although newly developed levitation and measurement techniques have been implemented, the full experimental validation of theoretical predictions has been hindered by interfering artifacts associated with levitation in the Earth gravitational field. The low gravity environment of orbital space flight has been shown to provide a more quiescent environment which can be utilized to better match the idealized theoretical conditions. The research effort described in this paper is a closely coupled collaboration between predictive and guiding theoretical activities and a unique experimental program involving the ultrasonic and electrostatic levitation of single droplets and bubbles. The goal is to develop and to validate methods based on nonlinear dynamics for the understanding of the large amplitude oscillatory response of single drops and bubbles to both isotropic and asymmetric pressure stimuli. The first specific area on interest has been the resonant coupling between volume and shape oscillatory modes isolated gas or vapor bubbles in a liquid host. The result of multiple time-scale asymptotic treatment, combined with domain perturbation and bifurcation methods, has been the prediction of resonant and near-resonant coupling between volume and shape modes leading to stable as well as chaotic oscillations. Experimental investigations of the large amplitude shape oscillation modes of centimeter-size single bubbles trapped in water at 1 G and under reduced hydrostatic pressure, have suggested the possibility of a low gravity experiment to study the direct coupling between these low frequency shape modes and the volume pulsation, sound-radiating mode. The second subject of interest has involved numerical modeling, using the boundary integral method, of the large amplitude shape oscillations of charged and uncharged drops in the presence
Experimental measurement of chaotic attractors in solid mechanics(a)).
Moon, Francis C.
1991-07-01
In this paper a review is given of experimental techniques in chaotic dynamics of solid mechanical systems based on modern ideas of nonlinear dynamics. These methods include Poincare maps, double Poincare sections, symbol dynamics, and fractal dimension. The physical problems discussed include nonlinear elastic beams, forced motion of a string, flow-induced vibration of a rod, forced motions of a magnetic pendulum, and rigid body dynamics of a magnet and high-temperature superconductor. PMID:12779894
Recurrence Quantification of Fractal Structures
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
Periodic and Chaotic Dynamics of the Ehrhard-Müller System
NASA Astrophysics Data System (ADS)
Park, Junho; Lee, Hyunho; Baik, Jong-Jin
2016-06-01
This paper investigates nonlinear ordinary differential equations of the Ehrhard-Müller system which describes natural convection in a single-phase loop in the presence of nonsymmetric heating. Stability and dynamics of periodic and chaotic behaviors of the equations are investigated and the periodicity diagram is obtained in wide ranges of parameters. Regimes of both periodic and chaotic solutions are observed with complex behaviors such that the periodic regimes enclose the chaotic regime while they are also immersed inside the chaotic regime with various shapes. An asymptotic analysis is performed for sufficiently large parameters to understand the enclosure by the periodic regimes and asymptotic limit cycles are obtained to compare with limit cycles obtained from numerical results.
Regular and chaotic dynamics of magnetization precession in ferrite-garnet films
NASA Astrophysics Data System (ADS)
Shutyĭ, Anatoliy M.; Sementsov, Dmitriy I.
2009-03-01
By numerically solving equations of motion and constructing the spectrum of Lyapunov exponents, nonlinear dynamics of uniformly precessing magnetization in (110) thin film structures with perpendicular magnetic bias is investigated over a wide frequency range of the alternating field. Bifurcational changes in magnetization precession and the states of dynamical bistability are discovered. Conditions for the realization of high-amplitude regular and chaotic dynamic regimes are revealed. The possibility of controlling those precession regimes by using external magnetic fields is shown. The features of time analogs of the Poincaré section of trajectories in the chaotic regimes are studied.
Richness of chaotic dynamics in nonholonomic models of a celtic stone
NASA Astrophysics Data System (ADS)
Gonchenko, Alexander S.; Gonchenko, Sergey V.; Kazakov, Alexey O.
2013-09-01
We study the regular and chaotic dynamics of two nonholonomic models of a Celtic stone. We show that in the first model (the so-called BM-model of a Celtic stone) the chaotic dynamics arises sharply, during a subcritical period doubling bifurcation of a stable limit cycle, and undergoes certain stages of development under the change of a parameter including the appearance of spiral (Shilnikov-like) strange attractors and mixed dynamics. For the second model, we prove (numerically) the existence of Lorenz-like attractors (we call them discrete Lorenz attractors) and trace both scenarios of development and break-down of these attractors.
Delayed feedback control method for dynamical systems with chaotic saddles
NASA Astrophysics Data System (ADS)
Kobayashi, Miki U.; Aihara, Kazuyuki
2012-08-01
We consider systems whose orbits diverge after chaotic transient for a finite time, and propose a controlmethod for preventing the divergence. These systems generally possess not chaotic attractors but some chaotic saddles. Our aim of control, i.e., the prevention of divergence, is achieved through the stabilization of unstable periodic orbits embedded in the chaotic saddle by making use of the delayed feedback controlmethod. The key concept of our control strategy is the application of the Proper Interior Maximum (PIM) triple method and the method to detect unstable periodic orbits from time series, originally developed by Lathrop and Kostelich, as initial steps before adding the delayed feedback control input. We show that our control method can be applied to the Hénon map and an intermittent androgen suppression (IAS) therapy model, which is a model for therapy of advanced prostate cancer. The fact that our method can be applied to the IAS therapy model indicates that our control strategy may be useful in the therapy of advanced prostate cancer.
Resistive magnetohydrodynamic reconnection: Resolving long-term, chaotic dynamics
Keppens, R.; Restante, A. L.; Lapenta, G.; Porth, O.; Galsgaard, K.; Frederiksen, J. T.; Parnell, C.
2013-09-15
In this paper, we address the long-term evolution of an idealised double current system entering reconnection regimes where chaotic behavior plays a prominent role. Our aim is to quantify the energetics in high magnetic Reynolds number evolutions, enriched by secondary tearing events, multiple magnetic island coalescence, and compressive versus resistive heating scenarios. Our study will pay particular attention to the required numerical resolutions achievable by modern (grid-adaptive) computations, and comment on the challenge associated with resolving chaotic island formation and interaction. We will use shock-capturing, conservative, grid-adaptive simulations for investigating trends dominated by both physical (resistivity) and numerical (resolution) parameters, and confront them with (visco-)resistive magnetohydrodynamic simulations performed with very different, but equally widely used discretization schemes. This will allow us to comment on the obtained evolutions in a manner irrespective of the adopted discretization strategy. Our findings demonstrate that all schemes used (finite volume based shock-capturing, high order finite differences, and particle in cell-like methods) qualitatively agree on the various evolutionary stages, and that resistivity values of order 0.001 already can lead to chaotic island appearance. However, none of the methods exploited demonstrates convergence in the strong sense in these chaotic regimes. At the same time, nonperturbed tests for showing convergence over long time scales in ideal to resistive regimes are provided as well, where all methods are shown to agree. Both the advantages and disadvantages of specific discretizations as applied to this challenging problem are discussed.
Jamming and chaotic dynamics in different granular systems
NASA Astrophysics Data System (ADS)
Maghsoodi, Homayoon; Luijten, Erik
Although common in nature and industry, the jamming transition has long eluded a concrete, mechanistic explanation. Recently, Banigan et al. (Nat. Phys. 9, 288-292, 2013) proposed a method for characterizing this transition in a granular system in terms of the system's chaotic properties, as quantified by the largest Lyapunov exponent. They demonstrated that in a two-dimensional shear cell the jamming transition coincides with the bulk density at which the system's largest Lyapunov exponent changes sign, indicating a transition between chaotic and non-chaotic regimes. To examine the applicability of this observation to realistic granular systems, we study a model that includes frictional forces within an expanded phase space. Furthermore, we test the generality of the relation between chaos and jamming by investigating the relationship between jamming and the chaotic properties of several other granular systems, notably sheared systems (Howell, D., Behringer R. P., Veje C., Phys. Rev. Lett. 82, 5241-5244, 1999) and systems with a free boundary. Finally, we quantify correlations between the largest Lyapunov vector and collective rearrangements of the system to demonstrate the predictive capabilities enabled by adopting this perspective of jamming.
Resistive magnetohydrodynamic reconnection: Resolving long-term, chaotic dynamics
NASA Astrophysics Data System (ADS)
Keppens, R.; Porth, O.; Galsgaard, K.; Frederiksen, J. T.; Restante, A. L.; Lapenta, G.; Parnell, C.
2013-09-01
In this paper, we address the long-term evolution of an idealised double current system entering reconnection regimes where chaotic behavior plays a prominent role. Our aim is to quantify the energetics in high magnetic Reynolds number evolutions, enriched by secondary tearing events, multiple magnetic island coalescence, and compressive versus resistive heating scenarios. Our study will pay particular attention to the required numerical resolutions achievable by modern (grid-adaptive) computations, and comment on the challenge associated with resolving chaotic island formation and interaction. We will use shock-capturing, conservative, grid-adaptive simulations for investigating trends dominated by both physical (resistivity) and numerical (resolution) parameters, and confront them with (visco-)resistive magnetohydrodynamic simulations performed with very different, but equally widely used discretization schemes. This will allow us to comment on the obtained evolutions in a manner irrespective of the adopted discretization strategy. Our findings demonstrate that all schemes used (finite volume based shock-capturing, high order finite differences, and particle in cell-like methods) qualitatively agree on the various evolutionary stages, and that resistivity values of order 0.001 already can lead to chaotic island appearance. However, none of the methods exploited demonstrates convergence in the strong sense in these chaotic regimes. At the same time, nonperturbed tests for showing convergence over long time scales in ideal to resistive regimes are provided as well, where all methods are shown to agree. Both the advantages and disadvantages of specific discretizations as applied to this challenging problem are discussed.
Chaotic pendulum: The complete attractor
NASA Astrophysics Data System (ADS)
DeSerio, Robert
2003-03-01
A commercial chaotic pendulum is modified to study nonlinear dynamics, including the determination of Poincaré sections, fractal dimensions, and Lyapunov exponents. The apparatus is driven by a simple oscillating mechanism powered by a 200 pulse per revolution stepper motor running at constant angular velocity. A computer interface generates the uniform pulse train needed to run the stepper motor and, with each pulse, reads a rotary encoder attached to the pendulum axle. Ten million readings from overnight runs of 50 000 drive cycles were smoothed and differentiated to obtain the pendulum angle θ and the angular velocity ω at each pulse of the drive. A plot of the 50 000 (θ,ω) phase points corresponding to one phase of the drive system produces a single Poincaré section. Thus, 200 Poincaré sections are experimentally available, one at each step of the drive. Viewed separately, any one of them strikingly illustrates the fractal geometry of the underlying chaotic attractor. Viewed sequentially in a repeating loop, they demonstrate the stretching and folding of phase point density typical of chaotic dynamics. Results for four pendulum damping conditions are presented and compared.
Takada, Ryu; Munetaka, Daigo; Kobayashi, Shoji; Suemitsu, Yoshikazu; Nara, Shigetoshi
2007-09-01
Chaotic dynamics in a recurrent neural network model and in two-dimensional cellular automata, where both have finite but large degrees of freedom, are investigated from the viewpoint of harnessing chaos and are applied to motion control to indicate that both have potential capabilities for complex function control by simple rule(s). An important point is that chaotic dynamics generated in these two systems give us autonomous complex pattern dynamics itinerating through intermediate state points between embedded patterns (attractors) in high-dimensional state space. An application of these chaotic dynamics to complex controlling is proposed based on an idea that with the use of simple adaptive switching between a weakly chaotic regime and a strongly chaotic regime, complex problems can be solved. As an actual example, a two-dimensional maze, where it should be noted that the spatial structure of the maze is one of typical ill-posed problems, is solved with the use of chaos in both systems. Our computer simulations show that the success rate over 300 trials is much better, at least, than that of a random number generator. Our functional simulations indicate that both systems are almost equivalent from the viewpoint of functional aspects based on our idea, harnessing of chaos. PMID:19003512
NASA Astrophysics Data System (ADS)
Martienssen, W.; Hübinger, B.; Doerner, R.
A method to transfer secret information using chaotic dynamical systems is proposed. It is based on modulating a chaotic system with the message such that its time evolution contains the hidden information. Decryption of the cipher is achieved by chaos control. Operation of the scheme is demonstrated by en- and decoding a short german text.
Estimating the level of dynamical noise in time series by using fractal dimensions
NASA Astrophysics Data System (ADS)
Sase, Takumi; Ramírez, Jonatán Peña; Kitajo, Keiichi; Aihara, Kazuyuki; Hirata, Yoshito
2016-03-01
We present a method for estimating the dynamical noise level of a 'short' time series even if the dynamical system is unknown. The proposed method estimates the level of dynamical noise by calculating the fractal dimensions of the time series. Additionally, the method is applied to EEG data to demonstrate its possible effectiveness as an indicator of temporal changes in the level of dynamical noise.
Periodic, Quasi-periodic and Chaotic Dynamics in Simple Gene Elements with Time Delays
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
Robust PRNG based on homogeneously distributed chaotic dynamics
NASA Astrophysics Data System (ADS)
Garasym, Oleg; Lozi, René; Taralova, Ina
2016-02-01
This paper is devoted to the design of new chaotic Pseudo Random Number Generator (CPRNG). Exploring several topologies of network of 1-D coupled chaotic mapping, we focus first on two dimensional networks. Two topologically coupled maps are studied: TTL rc non-alternate, and TTL SC alternate. The primary idea of the novel maps has been based on an original coupling of the tent and logistic maps to achieve excellent random properties and homogeneous /uniform/ density in the phase plane, thus guaranteeing maximum security when used for chaos base cryptography. In this aim two new nonlinear CPRNG: MTTL 2 sc and NTTL 2 are proposed. The maps successfully passed numerous statistical, graphical and numerical tests, due to proposed ring coupling and injection mechanisms.
Fractal and Small-World Networks Formed by Self-Organized Critical Dynamics
NASA Astrophysics Data System (ADS)
Watanabe, Akitomo; Mizutaka, Shogo; Yakubo, Kousuke
2015-11-01
We propose a dynamical model in which a network structure evolves in a self-organized critical (SOC) manner and explain a possible origin of the emergence of fractal and small-world networks. Our model combines a network growth and its decay by failures of nodes. The decay mechanism reflects the instability of large functional networks against cascading overload failures. It is demonstrated that the dynamical system surely exhibits SOC characteristics, such as power-law forms of the avalanche size distribution, the cluster size distribution, and the distribution of the time interval between intermittent avalanches. During the network evolution, fractal networks are spontaneously generated when networks experience critical cascades of failures that lead to a percolation transition. In contrast, networks far from criticality have small-world structures. We also observe the crossover behavior from fractal to small-world structure in the network evolution.
The Geometry and Dynamics of a Propagating Front in a Chaotic Flow Field
NASA Astrophysics Data System (ADS)
Paul, Mark
There are many important problems regarding transport in complex fluid flows with implications in science, nature, and technology. Examples include the combustion of pre-mixed gases in a turbulent flow, the complex patterns of reagents in a chemical system, the spread of a forest fire, and the outbreak of an epidemic. This talk explores the transport and dynamics of a reacting species in a chaotic fluid flow field. Large-scale parallel numerical simulations are used to explore the dynamics of propagating fronts in complex three-dimensional time-dependent fluid flows for the precise conditions of the laboratory. It is shown that a chaotic flow field enhances the front propagation when compared with a purely cellular flow field. This enhancement is quantified by computing measures of the spreading rate of the products and by quantifying the complexity of the three-dimensional front geometry for a range of chaotic flow conditions.
Allegrini, Paolo; Paradisi, Paolo; Menicucci, Danilo; Gemignani, Angelo
2010-01-01
Resting-state EEG signals undergo rapid transition processes (RTPs) that glue otherwise stationary epochs. We study the fractal properties of RTPs in space and time, supporting the hypothesis that the brain works at a critical state. We discuss how the global intermittent dynamics of collective excitations is linked to mentation, namely non-constrained non-task-oriented mental activity. PMID:21423370
Fractal dimensions of soy protein nanoparticle aggregates determined by dynamic mechanical method
Technology Transfer Automated Retrieval System (TEKTRAN)
The fractal dimension of the protein aggregates can be estimated by dynamic mechanical methods when the particle aggregates are imbedded in a polymer matrix. Nanocomposites were formed by mixing hydrolyzed soy protein isolate (HSPI) nanoparticle aggregates with styrene-butadiene (SB) latex, followe...
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.
[Regular and chaotic dynamics with applications in nonlinear optics]. Final report
Kovacic, G.
1998-10-12
The following major pieces of work were completed under the sponsorship of this grant: (1) singular perturbation theory for dynamical systems; (2) homoclinic orbits and chaotic dynamics in second-harmonic generating, optically pumped, passive optical cavities; (3) chaotic dynamics in short ring-laser cavities; (4) homoclinic orbits in moderately-long ring-laser cavities; (5) finite-dimensional attractor in ring-laser cavities; (6) turbulent dynamics in long ring-laser cavities; (7) bifurcations in a model for a free-boundary problem for the heat equation; (8) weakly nonlinear dynamics of interface propagation; (9) slowly periodically forced planar Hamiltonian systems; and (10) soliton spectrum of the solutions of the nonlinear Schroedinger equation. A brief summary of the research is given for each project.
Minati, Ludovico E-mail: ludovico.minati@unitn.it
2014-09-01
In this paper, an experimental characterization of the dynamical properties of five autonomous chaotic oscillators, based on bipolar-junction transistors and obtained de-novo through a genetic algorithm in a previous study, is presented. In these circuits, a variable resistor connected in series to the DC voltage source acts as control parameter, for a range of which the largest Lyapunov exponent, correlation dimension, approximate entropy, and amplitude variance asymmetry are calculated, alongside bifurcation diagrams and spectrograms. Numerical simulations are compared to experimental measurements. The oscillators can generate a considerable variety of regular and chaotic sine-like and spike-like signals.
Minati, Ludovico
2014-09-01
In this paper, an experimental characterization of the dynamical properties of five autonomous chaotic oscillators, based on bipolar-junction transistors and obtained de-novo through a genetic algorithm in a previous study, is presented. In these circuits, a variable resistor connected in series to the DC voltage source acts as control parameter, for a range of which the largest Lyapunov exponent, correlation dimension, approximate entropy, and amplitude variance asymmetry are calculated, alongside bifurcation diagrams and spectrograms. Numerical simulations are compared to experimental measurements. The oscillators can generate a considerable variety of regular and chaotic sine-like and spike-like signals. PMID:25273190
Nonlinear enhancement of the fractal structure in the escape dynamics of Bose-Einstein condensates
Mitchell, Kevin A.; Ilan, Boaz
2009-10-15
We consider the escape dynamics of an ensemble of Bose-Einstein-condensed atoms from an optical-dipole trap consisting of two overlapping Gaussian wells. Earlier theoretical studies (based on a model of quantum evolution using ensembles of classical trajectories) predicted that self-similar fractal features could be visible in this system by measuring the escaping flux as a function of time for varying initial conditions. Here, direct numerical quantum simulations show the clear influence of quantum interference on the escape data. Fractal features are still evident in the data, albeit with interference fringes superposed. Furthermore, the nonlinear influence of atom-atom interactions is also considered, in the context of the (2+1)-dimensional Gross-Pitaevskii equation. Of particular note is that an attractive nonlinear interaction enhances the resolution of fractal structures in the escape data. Thus, the interplay between nonlinear focusing and dispersion results in dynamics that resolve the underlying classical fractal more faithfully than the noninteracting quantum (or classical) dynamics.
NASA Astrophysics Data System (ADS)
Ge, Zheng-Ming; Leu, Jia-Haur; Lin, Tsung-Nan
The paper is to present the detailed dynamic analysis of a vertically vibrating and rotating elliptic tube containing a particle. By subjecting to an external periodic excitation, it has shown that the system exhibits both regular and chaotic motions. By using the Lyapunov direct method and Chetaev’s theorem, the stability and instability of the relative equilibrium position of the particle in the tube can be determined. The center manifold theorem is applied to verify the conditions of stability when system is under the critical case. The effects of the changes of parameters in the system can be found in the bifurcation and parametric diagrams. By applying various numerical results such as phase plane, Poincaré map and power spectrum analysis, a variety of the periodic solutions and the phenomena of the chaotic motion can be presented. Further, chaotic behavior can be verified by using Lyapunov exponents and Lyapunov dimensions.
Emission from dielectric cavities in terms of invariant sets of the chaotic ray dynamics
NASA Astrophysics Data System (ADS)
Altmann, Eduardo G.
2009-01-01
The chaotic ray dynamics inside dielectric cavities is described by the properties of an invariant chaotic saddle. The localization of the far-field emission in specific directions, recently observed in different experiments and wave simulations, is found to be a consequence of the filamentary pattern of the saddle’s unstable manifold. For cavities with mixed phase space, the chaotic saddle is divided in hyperbolic and nonhyperbolic components, related, respectively, to the intermediate exponential (t
Analysis and circuitry realization of a novel three-dimensional chaotic system
NASA Astrophysics Data System (ADS)
Abooee, A.; Yaghini-Bonabi, H. A.; Jahed-Motlagh, M. R.
2013-05-01
In this paper a new three-dimensional chaotic system is introduced. Some basic dynamical properties are analyzed to show chaotic behavior of the presented system. These properties are covered by dissipation of system, instability of equilibria, strange attractor, Lyapunov exponents, fractal dimension and sensitivity to initial conditions. Through altering one of the system parameters, various dynamical behaviors are observed which included chaos, periodic and convergence to an equilibrium point. Eventually, an analog circuit is designed and implemented experimentally to realize the chaotic system.
Observation of chaotic dynamics of coupled nonlinear oscillators
NASA Astrophysics Data System (ADS)
van Buskirk, R.; Jeffries, C.
1985-05-01
Experimental data are employed as bases for theoretically modelling the behavior of a finite number of driven nonlinear coupled oscillators. Attention is focused on Si p-n junction resonators exposed to an external inductance. A junction oscillator displays period doubling, Hopf figuracions to quasi-periodicity, entrainment horns and breakup of the invariant torus. Calculated and measured data are compared, with favorable results, by means of Poincare' sections, bifurcation diagrams and parameter phase space diagrams for the drive voltage and frequency. Fractal dimensions 2.03 and 2.33 are expressed in Poincare' sections to illustrate the behavior of single and dual coupled resonators which experience a breakup of the strange attractor.
Chaotic dynamics of cardioventilatory coupling in humans: effects of ventilatory modes
Mangin, Laurence; Clerici, Christine; Similowski, Thomas; Poon, Chi-Sang
2009-01-01
Cardioventilatory coupling (CVC), a transient temporal alignment between the heartbeat and inspiratory activity, has been studied in animals and humans mainly during anesthesia. The origin of the coupling remains uncertain, whether or not ventilation is a main determinant in the CVC process and whether the coupling exhibits chaotic behavior. In this frame, we studied sedative-free, mechanically ventilated patients experiencing rapid sequential changes in breathing control during ventilator weaning during a switch from a machine-controlled assistance mode [assist-controlled ventilation (ACV)] to a patient-driven mode [inspiratory pressure support (IPS) and unsupported spontaneous breathing (USB)]. Time series were computed as R to start inspiration (RI) and R to the start of expiration (RE). Chaos was characterized with the noise titration method (noise limit), largest Lyapunov exponent (LLE) and correlation dimension (CD). All the RI and RE time series exhibit chaotic behavior. Specific coupling patterns were displayed in each ventilatory mode, and these patterns exhibited different linear and chaotic dynamics. When switching from ACV to IPS, partial inspiratory loading decreases the noise limit value, the LLE, and the correlation dimension of the RI and RE time series in parallel, whereas decreasing intrathoracic pressure from IPS to USB has the opposite effect. Coupling with expiration exhibits higher complexity than coupling with inspiration during mechanical ventilation either during ACV or IPS, probably due to active expiration. Only 33% of the cardiac time series (RR interval) exhibit complexity either during ACV, IPS, or USB making the contribution of the cardiac signal to the chaotic feature of the coupling minimal. We conclude that 1) CVC in unsedated humans exhibits a complex dynamic that can be chaotic, and 2) ventilatory mode has major effects on the linear and chaotic features of the coupling. Taken together these findings reinforce the role of
Effects of correlation among stored patterns on associative dynamics of chaotic neural network
NASA Astrophysics Data System (ADS)
Iwai, Toshiya; Matsuzaki, Fuminari; Kuroiwa, Jousuke; Miyake, Shogo
2005-12-01
We numerically investigate the effects of correlation among stored patterns on the associative dynamics in a chaotic neural network model. In the model, there are two kinds of parameters: one is a measure of the Hopfield like behavior of the retrieval process and another controls the chaotic behavior. The parameter dependence of the associative dynamics is also examined. The following results are found. (i) Two dimensional parameter space is divided into two kinds of associative states by a distinct boundary. One is the retrieval state of the association such as the Hopfield like retrieval state, and another is the wandering state of the associative dynamics where the network retrieves stored patterns and their reverse patterns. (ii) The area of the wandering state becomes larger as the degree of correlation becomes larger. (iii) As the degree of correlation becomes larger, both the recall ratio of correlated patterns and the transition frequency between correlated patterns becomes larger in the wandering state. (iv) The whole region of the wandering state in the parameter space is not necessarily chaotic from the view point of the Lyapunov dimension, but most of the region of the wandering state is chaotic.
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.
Structure of the Asteroid Belt from the Gas Giants' Growth and Chaotic Dynamics
NASA Astrophysics Data System (ADS)
Izidoro, André; Raymond, Sean N.; Pierens, Arnaud; Morbidelli, Alessandro; Winter, Othon; Nesvorny, David
2016-05-01
The structure of the asteroid belt holds a record of the Solar System's dynamical history. The current belt only contains 10-3 Earth masses yet the asteroids' orbits are dynamically excited, with a large spread in eccentricity and inclination. The belt is also chemically segregated: the inner belt is dominated by dry S-types and the outer belt by hydrated C-types. Here we propose a new model in which the asteroid belt was always low-mass and was partially populated and sculpted by the giant planets on chaotic, resonant orbits. We first show that the compositional dichotomy of the asteroid belt is a simple consequence of Jupiter's growth in the gaseous protoplanetary disk. As Jupiter's core rapidly grew by accreting gas, orbits of nearby planetesimals were perturbed onto Jupiter-crossing trajectories. A significant fraction (~10%) of objects in the neighborhood exterior of Jupiter's orbit were implanted by gas drag into the outer parts of the asteroid belt as C-types. While the gas giants were likely in mean motion resonance at the end of the gaseous disk phase, we show that small perturbations may have driven them into a chaotic but stable state. After the dissipation of the gaseous disk, stochastic variations in the gas giants orbits caused resonances to chaotically jump across the main belt and excite the asteroids' orbits. Our results suggest that the early Solar System was chaotic and introduce a simple framework to understand the origins of the asteroid belt.
Multifractality and the effect of turbulence on the chaotic dynamics of a HeNe laser
NASA Astrophysics Data System (ADS)
Gulich, Damián.; Zunino, Luciano; Pérez, Darío.; Garavaglia, Mario
2013-09-01
We propose the use of multifractal detrended fluctuation analysis (MF-DFA) to measure the influence of atmospheric turbulence on the chaotic dynamics of a HeNe laser. Fit ranges for MF-DFA are obtained with goodness of linear fit (GoLF) criterion. The chaotic behavior is generated by means of a simple interferometric setup with a feedback to the cavity of the gas laser. Such dynamics have been studied in the past and modeled as a function of the feedback level. Different intensities of isotropic turbulence have been generated with a turbulator device, allowing a structure constant for the index of refraction of air adjustable by means of a temperature difference parameter in the unit. Considering the recent interest in message encryption with this kind of setups, the study of atmospheric turbulence effects plays a key role in the field of secure laser communication through the atmosphere. In principle, different intensities of turbulence may be interpreted as different levels of white noise on the original chaotic series. These results can be of utility for performance optimization in chaotic free-space laser communication systems.
Quantum chaotic scattering in graphene systems in the absence of invariant classical dynamics
NASA Astrophysics Data System (ADS)
Wang, Guang-Lei; Ying, Lei; Lai, Ying-Cheng; Grebogi, Celso
2013-05-01
Quantum chaotic scattering is referred to as the study of quantum behaviors of open Hamiltonian systems that exhibit transient chaos in the classical limit. Traditionally a central issue in this field is how the elements of the scattering matrix or their functions fluctuate as a system parameter, e.g., the electron Fermi energy, is changed. A tacit hypothesis underlying previous works was that the underlying classical phase-space structure remains invariant as the parameter varies, so semiclassical theory can be used to explain various phenomena in quantum chaotic scattering. There are, however, experimental situations where the corresponding classical chaotic dynamics can change characteristically with some physical parameter. Multiple-terminal quantum dots are one such example where, when a magnetic field is present, the classical chaotic-scattering dynamics can change between being nonhyperbolic and being hyperbolic as the Fermi energy is changed continuously. For such systems semiclassical theory is inadequate to account for the characteristics of conductance fluctuations with the Fermi energy. To develop a general framework for quantum chaotic scattering associated with variable classical dynamics, we use multi-terminal graphene quantum-dot systems as a prototypical model. We find that significant conductance fluctuations occur with the Fermi energy even for fixed magnetic field strength, and the characteristics of the fluctuation patterns depend on the energy. We propose and validate that the statistical behaviors of the conductance-fluctuation patterns can be understood by the complex eigenvalue spectrum of the generalized, complex Hamiltonian of the system which includes self-energies resulted from the interactions between the device and the semi-infinite leads. As the Fermi energy is increased, complex eigenvalues with extremely smaller imaginary parts emerge, leading to sharp resonances in the conductance.
New developments in classical chaotic scattering.
Seoane, Jesús M; Sanjuán, Miguel A F
2013-01-01
Classical chaotic scattering is a topic of fundamental interest in nonlinear physics due to the numerous existing applications in fields such as celestial mechanics, atomic and nuclear physics and fluid mechanics, among others. Many new advances in chaotic scattering have been achieved in the last few decades. This work provides a current overview of the field, where our attention has been mainly focused on the most important contributions related to the theoretical framework of chaotic scattering, the fractal dimension, the basins boundaries and new applications, among others. Numerical techniques and algorithms, as well as analytical tools used for its analysis, are also included. We also show some of the experimental setups that have been implemented to study diverse manifestations of chaotic scattering. Furthermore, new theoretical aspects such as the study of this phenomenon in time-dependent systems, different transitions and bifurcations to chaotic scattering and a classification of boundaries in different types according to symbolic dynamics are also shown. Finally, some recent progress on chaotic scattering in higher dimensions is also described. PMID:23242261
Chaotic dynamics and basin erosion in nanomagnets subject to time-harmonic magnetic fields
NASA Astrophysics Data System (ADS)
d'Aquino, M.; Quercia, A.; Serpico, C.; Bertotti, G.; Mayergoyz, I. D.; Perna, S.; Ansalone, P.
2016-04-01
Magnetization dynamics in uniformly magnetized particles subject to time-harmonic (AC) external fields is considered. The study is focused on the behavior of the AC-driven dynamics close to saddle equilibria. It happens that such dynamics has chaotic nature at moderately low power level, due to the heteroclinic tangle phenomenon which is produced by the combined effect of AC-excitations and saddle type dynamics. By using analytical theory for the threshold AC excitation amplitudes necessary to create the heteroclinic tangle together with numerical simulations, we quantify and show how the tangle produces the erosion of the safe basin around the stable equilibria.
Electric field driven fractal growth dynamics in polymeric medium
NASA Astrophysics Data System (ADS)
Dawar, Anit; Chandra, Amita
2014-08-01
This paper reports the extension of earlier work (Dawar and Chandra, 2012) [27] by including the influence of low values of electric field on diffusion limited aggregation (DLA) patterns in polymer electrolyte composites. Subsequently, specified cut-off value of voltage has been determined. Below the cut-off voltage, the growth becomes direction independent (i.e., random) and gives rise to ramified DLA patterns while above the cut-off, growth is governed by diffusion, convection and migration. These three terms (i.e., diffusion, convection and migration) lead to structural transition that varies from dense branched morphology (DBM) to chain-like growth to dendritic growth, i.e., from high field region (A) to constant field region (B) to low field region (C), respectively. The paper further explores the growth under different kinds of electrode geometries (circular and square electrode geometry). A qualitative explanation for fractal growth phenomena at applied voltage based on Nernst-Planck equation has been proposed.
The Retrospective Iterated Analysis Scheme for Nonlinear Chaotic Dynamics
NASA Technical Reports Server (NTRS)
Todling, Ricardo
2002-01-01
Atmospheric data assimilation is the name scientists give to the techniques of blending atmospheric observations with atmospheric model results to obtain an accurate idea of what the atmosphere looks like at any given time. Because two pieces of information are used, observations and model results, the outcomes of data assimilation procedure should be better than what one would get by using one of these two pieces of information alone. There is a number of different mathematical techniques that fall under the data assimilation jargon. In theory most these techniques accomplish about the same thing. In practice, however, slight differences in the approaches amount to faster algorithms in some cases, more economical algorithms in other cases, and even give better overall results in yet some other cases because of practical uncertainties not accounted for by theory. Therefore, the key is to find the most adequate data assimilation procedure for the problem in hand. In our Data Assimilation group we have been doing extensive research to try and find just such data assimilation procedure. One promising possibility is what we call retrospective iterated analysis (RIA) scheme. This procedure has recently been implemented and studied in the context of a very large data assimilation system built to help predict and study weather and climate. Although the results from that study suggest that the RIA scheme produces quite reasonable results, a complete evaluation of the scheme is very difficult due to the complexity of that problem. The present work steps back a little bit and studies the behavior of the RIA scheme in the context of a small problem. The problem is small enough to allow full assessment of the quality of the RIA scheme, but it still has some of the complexity found in nature, namely, its chaotic-type behavior. We find that the RIA performs very well for this small but still complex problem which is a result that seconds the results of our early studies.
Mixmaster universe: A chaotic Farey tale
NASA Astrophysics Data System (ADS)
Cornish, Neil J.; Levin, Janna J.
1997-06-01
When gravitational fields are at their strongest, the evolution of spacetime is thought to be highly erratic. Over the past decade debate has raged over whether this evolution can be classified as chaotic. The debate has centered on the homogeneous but anisotropic mixmaster universe. A definite resolution has been lacking as the techniques used to study the mixmaster dynamics yield observer-dependent answers. Here we resolve the conflict by using observer-independent fractal methods. We prove the mixmaster universe is chaotic by exposing the fractal strange repellor that characterizes the dynamics. The repellor is laid bare in both the six-dimensional minisuperspace of the full Einstein equations and in a two-dimensional discretization of the dynamics. The chaos is encoded in a special set of numbers that form the irrational Farey tree. We quantify the chaos by calculating the strange repellor's Lyapunov dimension, topological entropy, and multifractal dimensions. As all of these quantities are coordinate or gauge independent, there is no longer any ambiguity-the mixmaster universe is indeed chaotic.
Numerical test for hyperbolicity of chaotic dynamics in time-delay systems
NASA Astrophysics Data System (ADS)
Kuptsov, Pavel V.; Kuznetsov, Sergey P.
2016-07-01
We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting, and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them, previously predicted hyperbolicity is confirmed. The third one provides an example of a time-delay system with nonhyperbolic chaos.
Numerical test for hyperbolicity of chaotic dynamics in time-delay systems.
Kuptsov, Pavel V; Kuznetsov, Sergey P
2016-07-01
We develop a numerical test of hyperbolicity of chaotic dynamics in time-delay systems. The test is based on the angle criterion and includes computation of angle distributions between expanding, contracting, and neutral manifolds of trajectories on the attractor. Three examples are tested. For two of them, previously predicted hyperbolicity is confirmed. The third one provides an example of a time-delay system with nonhyperbolic chaos. PMID:27575062
Magma Mixing by Chaotic Dynamics: Results from a New Experimental Device
NASA Astrophysics Data System (ADS)
de Campos, C. P.; Perugini, D.; Ingrisch, W. E.; Dingwell, D. B.; Poli, G.
2009-12-01
In this work we present a new experimental device, based on the Journal Bearing System (JBS), to perform chaotic mixing of high viscosity melts under controlled fluid-dynamics and temperature conditions. The system consists of an outer cylinder, hosting the melts of interest and, an inner cylinder, eccentrically located, whose motions are independent. This way the development of chaotic streamlines in the mixing system is induced. An experiment was performed, using as end-members a peralkaline haplogranite (HPG8) 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 under laminar fluid dynamic condition (Re of the order of 10-7). Viscosity ratio between the two melts, at the beginning of the experiment, was of the order of 1,000. Analyses of experimental samples revealed, at short length scale (of the order of μm), a complex pattern of mixing structures. These consisted of an intimate distribution of filaments of the two melts, a typical feature in rocks produced by magma mixing processes. Stretching and folding dynamics between both melts induced chaotic flow fields and generated wide compositional interfaces. This way, chemical diffusion processes acted efficiently, producing melts with highly heterogeneous compositions. Despite a short running time, very low Re and a high viscosity ratio, a clear modulation of compositional fields has been obtained (fig.1). This indicates that chaotic mixing can be a very efficient process in enhancing the compositional variability in igneous systems, even under extreme rheological conditions and laminar fluid-dynamics. The excellence of our experimental device to replicate natural magma mixing features may open new frontiers in the study of this important petrological and volcanological process. Figure 1
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.
Cycles, randomness, and transport from chaotic dynamics to stochastic processes
NASA Astrophysics Data System (ADS)
Gaspard, Pierre
2015-09-01
An overview of advances at the frontier between dynamical systems theory and nonequilibrium statistical mechanics is given. Sensitivity to initial conditions is a mechanism at the origin of dynamical randomness—alias temporal disorder—in deterministic dynamical systems. In spatially extended systems, sustaining transport processes, such as diffusion, relationships can be established between the characteristic quantities of dynamical chaos and the transport coefficients, bringing new insight into the second law of thermodynamics. With methods from dynamical systems theory, the microscopic time-reversal symmetry can be shown to be broken at the statistical level of description in nonequilibrium systems. In this way, the thermodynamic entropy production turns out to be related to temporal disorder and its time asymmetry away from equilibrium.
Cycles, randomness, and transport from chaotic dynamics to stochastic processes.
Gaspard, Pierre
2015-09-01
An overview of advances at the frontier between dynamical systems theory and nonequilibrium statistical mechanics is given. Sensitivity to initial conditions is a mechanism at the origin of dynamical randomness-alias temporal disorder-in deterministic dynamical systems. In spatially extended systems, sustaining transport processes, such as diffusion, relationships can be established between the characteristic quantities of dynamical chaos and the transport coefficients, bringing new insight into the second law of thermodynamics. With methods from dynamical systems theory, the microscopic time-reversal symmetry can be shown to be broken at the statistical level of description in nonequilibrium systems. In this way, the thermodynamic entropy production turns out to be related to temporal disorder and its time asymmetry away from equilibrium. PMID:26428559
NASA Astrophysics Data System (ADS)
Mohammad, Yasir K.; Pavlova, Olga N.; Pavlov, Alexey N.
2016-04-01
We discuss the problem of quantifying chaotic dynamics at the input of the "integrate-and-fire" (IF) model from the output sequences of interspike intervals (ISIs) for the case when the fluctuating threshold level leads to the appearance of noise in ISI series. We propose a way to detect an ability of computing dynamical characteristics of the input dynamics and the level of noise in the output point processes. The proposed approach is based on the dependence of the largest Lyapunov exponent from the maximal orientation error used at the estimation of the averaged rate of divergence of nearby phase trajectories.
Dynamics of the stochastic Lorenz chaotic system with long memory effects
Zeng, Caibin Yang, Qigui
2015-12-15
Little seems to be known about the ergodic dynamics of stochastic systems with fractional noise. This paper is devoted to discern such long time dynamics through the stochastic Lorenz chaotic system (SLCS) with long memory effects. By a truncation technique, the SLCS is proved to generate a continuous stochastic dynamical system Λ. Based on the Krylov-Bogoliubov criterion, the required Lyapunov function is further established to ensure the existence of the invariant measure of Λ. Meanwhile, the uniqueness of the invariant measure of Λ is proved by examining the strong Feller property, together with an irreducibility argument. Therefore, the SLCS has exactly one adapted stationary solution.
Regular and chaotic dynamics of a chain of magnetic dipoles with moments of inertia
Shutyi, A. M.
2009-05-15
The nonlinear dynamic modes of a chain of coupled spherical bodies having dipole magnetic moments that are excited by a homogeneous ac magnetic field are studied using numerical analysis. Bifurcation diagrams are constructed and used to find conditions for the presence of several types of regular, chaotic, and quasi-periodic oscillations. The effect of the coupling of dipoles on the excited dynamics of the system is revealed. The specific features of the Poincare time sections are considered for the cases of synchronous chaos with antiphase synchronization and asynchronous chaos. The spectrum of Lyapunov exponents is calculated for the dynamic modes of an individual dipole.
A New Image Encryption Scheme Based on Dynamic S-Boxes and Chaotic Maps
NASA Astrophysics Data System (ADS)
Rehman, Atique Ur; Khan, Jan Sher; Ahmad, Jawad; Hwang, Soeng Oun
2016-03-01
Substitution box is a unique and nonlinear core component of block ciphers. A better designing technique of substitution box can boost up the quality of ciphertexts. In this paper, a new encryption method based on dynamic substitution boxes is proposed via using two chaotic maps. To break the correlation in an original image, pixels values of the original plaintext image are permuted row- and column-wise through random sequences. The aforementioned random sequences are generated by 2-D Burgers chaotic map. For the generation of dynamic substitution boxes, Logistic chaotic map is employed. In the process of diffusion, the permuted image is divided into blocks and each block is substituted via different dynamic substitution boxes. In contrast to conventional encryption schemes, the proposed scheme does not undergo the fixed block cipher and hence the security level can be enhanced. Extensive security analysis including histogram test is applied on the proposed image encryption technique. All experimental results reveal that the proposed scheme has a high level of security and robustness for transmission of digital images on insecure communication channels.
NASA Technical Reports Server (NTRS)
Makikallio, T. H.; Ristimae, T.; Airaksinen, K. E.; Peng, C. K.; Goldberger, A. L.; Huikuri, H. V.
1998-01-01
Dynamic analysis techniques may uncover abnormalities in heart rate (HR) behavior that are not easily detectable with conventional statistical measures. However, the applicability of these new methods for detecting possible abnormalities in HR behavior in various cardiovascular disorders is not well established. Conventional measures of HR variability were compared with short-term (< or = 11 beats, alpha1) and long-term (> 11 beats, alpha2) fractal correlation properties and with approximate entropy of RR interval data in 38 patients with stable angina pectoris without previous myocardial infarction or cardiac medication at the time of the study and 38 age-matched healthy controls. The short- and long-term fractal scaling exponents (alpha1, alpha2) were significantly higher in the coronary patients than in the healthy controls (1.34 +/- 0.15 vs 1.11 +/- 0.12 [p <0.001] and 1.10 +/- 0.08 vs 1.04 +/- 0.06 [p <0.01], respectively), and they also had lower approximate entropy (p <0.05), standard deviation of all RR intervals (p <0.01), and high-frequency spectral component of HR variability (p <0.05). The short-term fractal scaling exponent performed better than other heart rate variability parameters in differentiating patients with coronary artery disease from healthy subjects, but it was not related to the clinical or angiographic severity of coronary artery disease or any single nonspectral or spectral measure of HR variability in this retrospective study. Patients with stable angina pectoris have altered fractal properties and reduced complexity in their RR interval dynamics relative to age-matched healthy subjects. Dynamic analysis may complement traditional analyses in detecting altered HR behavior in patients with stable angina pectoris.
Fractal structures and processes
Bassingthwaighte, J.B.; Beard, D.A.; Percival, D.B.; Raymond, G.M.
1996-06-01
Fractals and chaos are closely related. Many chaotic systems have fractal features. Fractals are self-similar or self-affine structures, which means that they look much of the same when magnified or reduced in scale over a reasonably large range of scales, at least two orders of magnitude and preferably more (Mandelbrot, 1983). The methods for estimating their fractal dimensions or their Hurst coefficients, which summarize the scaling relationships and their correlation structures, are going through a rapid evolutionary phase. Fractal measures can be regarded as providing a useful statistical measure of correlated random processes. They also provide a basis for analyzing recursive processes in biology such as the growth of arborizing networks in the circulatory system, airways, or glandular ducts. {copyright} {ital 1996 American Institute of Physics.}
Chaotic and ballistic dynamics in time-driven quasiperiodic lattices
NASA Astrophysics Data System (ADS)
Wulf, Thomas; Schmelcher, Peter
2016-04-01
We investigate the nonequilibrium dynamics of classical particles in a driven quasiperiodic lattice based on the Fibonacci sequence. An intricate transient dynamics of extraordinarily long ballistic flights at distinct velocities is found. We argue how these transients are caused and can be understood by a hierarchy of block decompositions of the quasiperiodic lattice. A comparison to the cases of periodic and fully randomized lattices is performed.
Chaotic and ballistic dynamics in time-driven quasiperiodic lattices.
Wulf, Thomas; Schmelcher, Peter
2016-04-01
We investigate the nonequilibrium dynamics of classical particles in a driven quasiperiodic lattice based on the Fibonacci sequence. An intricate transient dynamics of extraordinarily long ballistic flights at distinct velocities is found. We argue how these transients are caused and can be understood by a hierarchy of block decompositions of the quasiperiodic lattice. A comparison to the cases of periodic and fully randomized lattices is performed. PMID:27176301
Fractal dynamics of light scattering intensity fluctuation in disordered dusty plasmas
Safaai, S. S.; Muniandy, S. V.; Chew, W. X.; Asgari, H.; Yap, S. L.; Wong, C. S.
2013-10-15
Dynamic light scattering (DLS) technique is a simple and yet powerful technique for characterizing particle properties and dynamics in complex liquids and gases, including dusty plasmas. Intensity fluctuation in DLS experiments often studied using correlation analysis with assumption that the fluctuation is statistically stationary. In this study, the temporal variation of the nonstationary intensity fluctuation is analyzed directly to show the existence of fractal characteristics by employing wavelet scalogram approach. Wavelet based scale decomposition approach is used to separate non-scaling background noise (without dust) from scaling intensity fluctuation from dusty plasma. The Hurst exponents for light intensity fluctuation in dusty plasma at different neutral gas pressures are determined. At low pressures, weaker damping of dust motions via collisions with neutral gases results in stronger persistent behavior in the fluctuation of DLS time series. The fractal scaling Hurst exponent is demonstrated to be useful for characterizing structural phases in complex disordered dusty plasma, especially when particle configuration or sizes are highly inhomogeneous which makes the standard pair-correlation function difficult to interpret. The results from fractal analysis are compared with alternative interpretation of disorder based on approximate entropy and particle transport using mean square displacement.
The Analysis of the Influence of Odorant's Complexity on Fractal Dynamics of Human Respiration.
Namazi, Hamidreza; Akrami, Amin; Kulish, Vladimir V
2016-01-01
One of the major challenges in olfaction research is to relate the structural features of the odorants to different features of olfactory system. However, no relationship has been yet discovered between the structure of the olfactory stimulus, and the structure of respiratory signal. This study reveals the plasticity of human respiratory signal in relation to 'complex' olfactory stimulus (odorant). We demonstrated that fractal temporal structure of respiration dynamics shifts towards the properties of the odorants used. The results show for the first time that more structurally complex a monomolecular odorant will result in less fractal respiratory signal. On the other hand, odorant with higher entropy will result the respiratory signal with lower entropy. The capability observed in this research can be further investigated and applied for treatment of patients with different respiratory diseases. PMID:27244590
Bells Galore: Oscillations and circle-map dynamics from space-filling fractal functions
Puente, C.E.; Cortis, A.; Sivakumar, B.
2008-10-15
The construction of a host of interesting patterns over one and two dimensions, as transformations of multifractal measures via fractal interpolating functions related to simple affine mappings, is reviewed. It is illustrated that, while space-filling fractal functions most commonly yield limiting Gaussian distribution measures (bells), there are also situations (depending on the affine mappings parameters) in which there is no limit. Specifically, the one-dimensional case may result in oscillations between two bells, whereas the two-dimensional case may give rise to unexpected circle map dynamics of an arbitrary number of two-dimensional circular bells. It is also shown that, despite the multitude of bells over two dimensions, whose means dance making regular polygons or stars inscribed on a circle, the iteration of affine maps yields exotic kaleidoscopes that decompose such an oscillatory pattern in a way that is similar to the many cases that converge to a single bell.
The Analysis of the Influence of Odorant’s Complexity on Fractal Dynamics of Human Respiration
Namazi, Hamidreza; Akrami, Amin; Kulish, Vladimir V.
2016-01-01
One of the major challenges in olfaction research is to relate the structural features of the odorants to different features of olfactory system. However, no relationship has been yet discovered between the structure of the olfactory stimulus, and the structure of respiratory signal. This study reveals the plasticity of human respiratory signal in relation to ‘complex’ olfactory stimulus (odorant). We demonstrated that fractal temporal structure of respiration dynamics shifts towards the properties of the odorants used. The results show for the first time that more structurally complex a monomolecular odorant will result in less fractal respiratory signal. On the other hand, odorant with higher entropy will result the respiratory signal with lower entropy. The capability observed in this research can be further investigated and applied for treatment of patients with different respiratory diseases. PMID:27244590
The Analysis of the Influence of Odorant’s Complexity on Fractal Dynamics of Human Respiration
NASA Astrophysics Data System (ADS)
Namazi, Hamidreza; Akrami, Amin; Kulish, Vladimir V.
2016-05-01
One of the major challenges in olfaction research is to relate the structural features of the odorants to different features of olfactory system. However, no relationship has been yet discovered between the structure of the olfactory stimulus, and the structure of respiratory signal. This study reveals the plasticity of human respiratory signal in relation to ‘complex’ olfactory stimulus (odorant). We demonstrated that fractal temporal structure of respiration dynamics shifts towards the properties of the odorants used. The results show for the first time that more structurally complex a monomolecular odorant will result in less fractal respiratory signal. On the other hand, odorant with higher entropy will result the respiratory signal with lower entropy. The capability observed in this research can be further investigated and applied for treatment of patients with different respiratory diseases.
Dynamics of chaotic systems with attractive and repulsive couplings.
Chen, Yuehua; Xiao, Jinghua; Liu, Weiqing; Li, Lixiang; Yang, Yixian
2009-10-01
Together with attractive couplings, repulsive couplings play crucial roles in determining important evolutions in natural systems, such as in learning and oscillatory processes of neural networks. The complex interactions between them have great influence on the systems. A detailed understanding of the dynamical properties under this type of couplings is of practical significance. In this paper, we propose a model to investigate the dynamics of attractive and repulsive couplings, which give rise to rich phenomena, especially for amplitude death (AD). The relationship among various dynamics and possible transitions to AD are illustrated. When the system is in the maximally stable AD, we observe the transient behavior of in-phase (high frequency) and out-of-phase (low frequency) motions. The mechanism behind the phenomenon is given. PMID:19905414
Exact coherent structures and chaotic dynamics in a model of cardiac tissue
Byrne, Greg; Marcotte, Christopher D.; Grigoriev, Roman O.
2015-03-15
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 is 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.
Exact coherent structures and chaotic dynamics in a model of cardiac tissue.
Byrne, Greg; Marcotte, Christopher D; Grigoriev, Roman O
2015-03-01
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 is 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. PMID:25833430
NASA Astrophysics Data System (ADS)
Xia, H. M.; Shu, C.; Wan, S. Y. M.; Chew, Y. T.
2006-01-01
A micromixer is a key component of various microfluidic systems, such as microreactors and μ-total analysis systems. One important strategy for passive mixer design is to generate chaotic advection using channel geometry, which usually has spatially periodic structures. In this paper, the influence of the Reynolds number on chaotic mixing in such mixers is studied with three mixer models. Characterization of the mixer with dynamical system techniques is also studied. The influence of fluid inertial effects on the occurrence of chaotic advection is first discussed. It is found that at low Re(Re < 1), the flow could become reversible in the mixer, which raises the difficulty to generate chaotic advection. In this case, specific fluid manipulations, such as stretching and folding processes, are necessary. This study also proposes a characterization method using Lyapunov exponent (λ) and Poincaré mapping information, which allows us to analyze the mixing performance of the mixer with one single mixer unit. Results show that it objectively reflects the dynamical properties of the mixers, such as being globally chaotic, partially chaotic or stable. So it can be used as an analytical tool to differentiate, evaluate and optimize various chaotic micromixers.
Wang, Zhiheng; Huo, Zhanqiang; Shi, Wenbo
2015-01-01
With rapid development of computer technology and wide use of mobile devices, the telecare medicine information system has become universal in the field of medical care. To protect patients' privacy and medial data's security, many authentication schemes for the telecare medicine information system have been proposed. Due to its better performance, chaotic maps have been used in the design of authentication schemes for the telecare medicine information system. However, most of them cannot provide user's anonymity. Recently, Lin proposed a dynamic identity based authentication scheme using chaotic maps for the telecare medicine information system and claimed that their scheme was secure against existential active attacks. In this paper, we will demonstrate that their scheme cannot provide user anonymity and is vulnerable to the impersonation attack. Further, we propose an improved scheme to fix security flaws in Lin's scheme and demonstrate the proposed scheme could withstand various attacks. PMID:25486894
Spatial-temporal dynamics of chaotic behavior in cultured hippocampal networks.
Chen, Wenjuan; Li, Xiangning; Pu, Jiangbo; Luo, Qingming
2010-06-01
Using multiple nonlinear techniques, we revealed the existence of chaos in the spontaneous activity of neuronal networks in vitro. The spatial-temporal dynamics of these networks indicated that emergent transition between chaotic behavior and superburst occurred periodically in low-frequency oscillations. An analysis of network-wide activity indicated that chaos was synchronized among different sites. Moreover, we found that the degree of chaos increased as the number of active sites in the network increased during long-term development (over three months in vitro). The chaotic behavior of the dissociated networks had similar spatial-temporal characteristics (rapid transition, periodicity, and synchronization) as the intact brain; however, the degree of chaos depended on the number of active sites at the mesoscopic level. This work could provide insight into neural coding and neurocybernetics. PMID:20866436
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).
NASA Astrophysics Data System (ADS)
Che, Yanqiu; Yang, Tingting; Li, Ruixue; Li, Huiyan; Han, Chunxiao; Wang, Jiang; Wei, Xile
2015-09-01
In this paper, we propose a dynamic delayed feedback control approach or desynchronization of chaotic-bursting synchronous activities in an ensemble of globally coupled neuronal oscillators. We demonstrate that the difference signal between an ensemble's mean field and its time delayed state, filtered and fed back to the ensemble, can suppress the self-synchronization in the ensemble. These individual units are decoupled and stabilized at the desired desynchronized states while the stimulation signal reduces to the noise level. The effectiveness of the method is illustrated by examples of two different populations of globally coupled chaotic-bursting neurons. The proposed method has potential for mild, effective and demand-controlled therapy of neurological diseases characterized by pathological synchronization.
Exploring the Spatiotemporal Dynamics of Covariant Lyapunov Vectors for Chaotic Convection
NASA Astrophysics Data System (ADS)
Xu, Mu; Paul, Mark
Covariant Lyapunov vectors provide access to fundamental features of chaos in high-dimensional systems that are driven far-from-equilibrium. We explore the spatiotemporal dynamics of covariant Lyapunov vectors for chaotic Rayleigh-Bénard convection to provide new physical insights. We use the covariant Lyapunov vectors to quantify the transition from hyperbolic to non-hyperbolic dynamics, to determine the degree of Oseledec splitting exhibited by the dynamics, and to shed light upon upon the tangled nature of the Lyapunov vectors. In this talk, we will explore the spatiotemporal dynamics of the Lyapunov vectors and their relation with the chaotic pattern dynamics of the flow field. Our results suggest that the Lyapunov vectors contain two distinct spatiotemporal features consisting of highly localized regions near defect structures and a spatially distributed checkerboard pattern. We will explore the connection between these features and the ideas of physical and spurious modes that may compose the dynamics. This research was funded by NSF Grant No. DMS-1125234.
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
On the Large Scale Dynamics in the Wake of a Fractal Obstacle
NASA Astrophysics Data System (ADS)
Higham, Jonathan; Brevis, Wernher
2015-11-01
In a water flume three-dimensional Particle Tracking Velocimetry is used to capture the turbulent wake of two full-width and wall-mounted obstacles: The first obstacle is a uniformly spaced array of square cylinders of same length-scale; the second is a three-iteration pre-fractal based on a the deterministic Sierpinski Carpet. Both obstacles emerge from the water surface and had the same porosity. For the description of the instantaneous vortical structures the velocity gradient tensor is analysed. It is found that whilst the largest length scales of the fractal dominated the vorticity field in the wake, the smaller length-scale within the obstacle caused intense vortical structures within the near field of the wake. To further investigate the spatio-temporal behaviour of the wake a simple and integrated use of the Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) is introduced. POD is used to rank the spatial structures relatable to the total variance (i.e. vorticity) while DMD is used to identify their dominant oscillation frequencies and spatial characteristics. From the POD it is clear that the largest length-scale creates spatially dominant structures, whilst the DMD extracts a set of oscillatory frequencies relatable to each fractal length-scale.
Dynamics of fractal cluster colloidal gels with embedded active Janus particles
NASA Astrophysics Data System (ADS)
Solomon, Michael; Szakasits, Megan; Zhang, Wenxuan
We find that fractal cluster gels of colloids in which platinum-coated Janus particles have been embedded exhibit enhanced mobility when the Janus particles are made active by the addition of hydrogen peroxide. Gelation is induced through addition of a divalent salt, magnesium chloride, to an initially stable suspension of Janus and polystyrene colloids, each of size about 1 micron. After the gels have been created, the embedded Janus colloids are activated by hydrogen peroxide, which is delivered to the system through a porous hydrogel membrane. We vary the ratio of active to passive colloids in the gels from about 1:20 to 1:8. Changes in structure and dynamics are visualized by two channel confocal laser scanning microscopy. By image analysis, we determine the particle positions and compute the mean squared displacement (MSD) of all particles in the gel. We measure the mobility enhancement in the fractal gels as a function of hydrogen peroxide concentration and Janus particle concentration and discuss the results in terms of the force provided by each active particle to the fractal gel network.
Nonequilibrium dynamics in lattice ecosystems: Chaotic stability and dissipative structures
NASA Astrophysics Data System (ADS)
Solé, Ricard V.; Bascompte, Jordi; Valls, Joaquim
1992-07-01
A generalized coupled map lattice (CML) model of ecosystem dynamics is presented. We consider the spatiotemporal behavior of a prey-predator map, a model of host-parasitoid interactions, and two-species competition. The latter model can show phase separation of domains (Turing-like structures) even when chaos is present. We also use this CML model to explore the time evolution and structural properties of ecological networks built with a set of N competing species. The May-Wigner criterion is applied as a measure of stability, and some regularities in the stable networks observed are discussed.
Stabilization of chaotic and non-permanent food-web dynamics
NASA Astrophysics Data System (ADS)
Williams, R. J.; Martinez, N. D.
2004-03-01
Several decades of dynamical analyses of food-web networks[CITE] have led to important insights into the effects of complexity, omnivory and interaction strength on food-web stability[CITE]. Several recent insights[CITE] are based on nonlinear bioenergetic consumer-resource models[CITE] that display chaotic behavior in three species food chains[CITE] which can be stabilized by omnivory[CITE] and weak interaction of a fourth species[CITE]. We slightly relax feeding on low-density prey in these models by modifying standard food-web interactions known as “typeII” functional responses[CITE]. This change drastically alters the dynamics of realistic systems containing up to ten species. Our modification stabilizes chaotic dynamics in three species systems and reduces or eliminates extinctions and non-persistent chaos[CITE] in ten species systems. This increased stability allows analysis of systems with greater biodiversity than in earlier work and suggests that dynamic stability is not as severe a constraint on the structure of large food webs as previously thought. The sensitivity of dynamical models to small changes in the predator-prey functional response well within the range of what is empirically observed suggests that functional response is a crucial aspect of species interactions that must be more precisely addressed in empirical studies.
The role of model dynamics in ensemble Kalman filter performance for chaotic systems
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.
Regular and chaotic motions in applied dynamics of a rigid body.
Beletskii, V. V.; Pivovarov, M. L.; Starostin, E. L.
1996-06-01
Periodic and regular motions, having a predictable functioning mode, play an important role in many problems of dynamics. The achievements of mathematics and mechanics (beginning with Poincare) have made it possible to establish that such motion modes, generally speaking, are local and form "islands" of regularity in a "chaotic sea" of essentially unpredictable trajectories. The development of computer techniques together with theoretical investigations makes it possible to study the global structure of the phase space of many problems having applied significance. A review of a number of such problems, considered by the authors in the past four or five years, is given in this paper. These include orientation and rotation problems of artificial and natural celestial bodies and the problem of controlling the motion of a locomotion robot. The structure of phase space is investigated for these problems. The phase trajectories of the motion are constructed by a numerical implementation of the Poincare point map method. Distinctions are made between regular (or resonance), quasiregular (or conditionally periodic), and chaotic trajectories. The evolution of the phase picture as the parameters are varied is investigated. A large number of "phase portraits" gives a notion of the arrangement and size of the stability islands in the "sea" of chaotic motions, about the appearance and disappearance of these islands as the parameters are varied, etc. (c) 1996 American Institute of Physics. PMID:12780243
Ławniczak, Michał; Białous, Małgorzata; Yunko, Vitalii; Bauch, Szymon; Sirko, Leszek
2015-03-01
We present the results of an experimental study of the elastic enhancement factor W for a microwave rectangular cavity simulating a two-dimensional quantum billiard in a transient region between regular and chaotic dynamics. The cavity was coupled to a vector network analyzer via two microwave antennas. The departure of the system from an integrable one due to the presence of antennas acting as scatterers is characterized by the parameter of chaoticity κ=2.8. The experimental results for the rectangular cavity are compared with those obtained for a microwave rough cavity simulating a chaotic quantum billiard. The experimental results were obtained for the frequency range ν=16-18.5 GHz and moderate absorption strength γ=5.2-7.4. We show that the elastic enhancement factor for the rectangular cavity lies below the theoretical value W=3 predicted for integrable systems, and it is significantly higher than that obtained for the rough cavity. The results obtained for the microwave rough cavity are smaller than those obtained within the framework of random matrix theory, and they lie between them and those predicted within a recently introduced model of the two-channel coupling [V. V. Sokolov and O. V. Zhirov, arXiv:1411.6211 [nucl-th
Wind tunnel experiments on chaotic dynamics of a flexible tube row in a cross flow
Muntean, G.; Moon, F.C.
1994-12-31
Flow visualization and dynamics measurements of flexible cylindrical tubes in a cross-flow are described. Five tubes mounted on flexible supports were subjected to cross flow in a low turbulence wind tunnel. Dynamic measurements of the tube motion are presented. The data suggests that a low dimensional attractor exists for tube flutter under impact constraints using fractal dimension calculations. There is also qualitative evidence for single tube flutter in-line with the flow. In another set of experiments, a flow visualization technique is used to examine the flow behind the vibrating cylinders. Four different configurations of the jet flow behind the cylinders are observed. Coupling of the jet dynamics and tube motion seems apparent from the video data. These experiments are being used to try and construct a low order nonlinear model for the tube-flow dynamics.
Fractal dynamics in self-evaluation reveal self-concept clarity.
Wong, Alexander E; Vallacher, Robin R; Nowak, Andrzej
2014-10-01
The structural account of self-esteem and self-evaluation maintains that they are distinct constructs. Trait self-esteem is stable and is expressed over macro timescales, whereas state self-evaluation is unstable and experienced on micro timescales. We compared predictions based on the structural account with those derived from a dynamical systems perspective on the self, which maintains that self-esteem and self-evaluation are hierarchically related and share basic dynamic properties. Participants recorded a 3-minute narrative about themselves, then used the mouse paradigm (Vallacher, Nowak, Froehlich, & Rockloff, 2002) to track the momentary self-evaluation in their narrative. Multiple methods converged to reveal fractal patterns in the resultant temporal patterns, indicative of nested timescales that link micro and macro selfevaluation and thus supportive of the dynamical account. The fractal dynamics were associated with participants' self-concept clarity, suggesting that the hierarchical relation between macro self-evaluation (self-esteem) and momentary self-evaluation is predicted by the coherence of self-concept organization. PMID:25196705
Fuzzy fractals, chaos, and noise
Zardecki, A.
1997-05-01
To distinguish between chaotic and noisy processes, the authors analyze one- and two-dimensional chaotic mappings, supplemented by the additive noise terms. The predictive power of a fuzzy rule-based system allows one to distinguish ergodic and chaotic time series: in an ergodic series the likelihood of finding large numbers is small compared to the likelihood of finding them in a chaotic series. In the case of two dimensions, they consider the fractal fuzzy sets whose {alpha}-cuts are fractals, arising in the context of a quadratic mapping in the extended complex plane. In an example provided by the Julia set, the concept of Hausdorff dimension enables one to decide in favor of chaotic or noisy evolution.
Periodic and chaotic dynamics in a map-based model of tumor-immune interaction.
Moghtadaei, Motahareh; Hashemi Golpayegani, Mohammad Reza; Malekzadeh, Reza
2013-10-01
Clinicians and oncologists believe that tumor growth has unpredictable dynamics. For this reason they encounter many difficulties in the treatment of cancer. Mathematical modeling is a great tool to improve our better understanding of the complicated biological system of tumor growth. Also, it can help to identify states of the disease and as a result help to predict later behaviors of the tumor. Having an insight into the future behaviors of the tumor can be very useful for the oncologists and clinicians to decide on the treatment method and dosage of the administered drug. This paper suggests that a suitable model for the tumor growth system should be a discrete model capable of exhibiting periodic and complex chaotic dynamics. This is the key feature of the proposed model. The model is validated here through experimental data and its potential dynamics are analyzed. The model can explain many biologically observed tumor states and dynamics, such as exponential growth, and periodic and chaotic behaviors in the steady states. The model shows that even an avascular tumor could become invasive under certain conditions. PMID:23770106
Modulated point-vortex pairs on a rotating sphere: Dynamics and chaotic advection
NASA Astrophysics Data System (ADS)
Drótos, Gábor; Tél, Tamás; Kovács, Gergely
2013-06-01
The dynamics of modulated point-vortex pairs is investigated on a rotating sphere, where modulation is chosen to reflect the conservation of angular momentum (potential vorticity). For sufficiently close vortices (dipoles) the trajectories of their center-of-mass are shown to correspond to those of a point particle moving freely on a rotating sphere. For finite size vortex pairs, a qualitative similarity to the geodesic dynamics is found. The advection dynamics generated by vortex pairs on a rotating sphere is found to be chaotic. In the short time dynamics we point out a transition from closed to open chaotic advection, which implies that the transport properties of the flow might drastically be altered by changing the initial conditions of the pair on the sphere. Due to spherical topology, for long times, even the open advection patterns are found to gradually cross over to that corresponding to a homogeneous closed mixing. This pattern extends along a zonal band, whereas short term closed mixing remains always bounded to the moving pair.
Modulated point-vortex pairs on a rotating sphere: dynamics and chaotic advection.
Drótos, Gábor; Tél, Tamás; Kovács, Gergely
2013-06-01
The dynamics of modulated point-vortex pairs is investigated on a rotating sphere, where modulation is chosen to reflect the conservation of angular momentum (potential vorticity). For sufficiently close vortices (dipoles) the trajectories of their center-of-mass are shown to correspond to those of a point particle moving freely on a rotating sphere. For finite size vortex pairs, a qualitative similarity to the geodesic dynamics is found. The advection dynamics generated by vortex pairs on a rotating sphere is found to be chaotic. In the short time dynamics we point out a transition from closed to open chaotic advection, which implies that the transport properties of the flow might drastically be altered by changing the initial conditions of the pair on the sphere. Due to spherical topology, for long times, even the open advection patterns are found to gradually cross over to that corresponding to a homogeneous closed mixing. This pattern extends along a zonal band, whereas short term closed mixing remains always bounded to the moving pair. PMID:23848782
Chaotic dynamics of dilute thermal atom clouds on stationary optical Bessel beams
NASA Astrophysics Data System (ADS)
Castañeda, J. A.; Pérez-Pascual, R.; Jáuregui, R.
2013-07-01
We characterize the semiclassical dynamics of dilute thermal atom clouds located in three-dimensional optical lattices generated by stationary optical Bessel beams. The dynamics of the cold atoms is explored in the quasi-Hamiltonian regime that arises using laser beams with far-off resonance detuning. Although the transverse structure of Bessel beams exhibits a complex topological structure, it is found that the longitudinal motion along the main propagation axis of the beam is the detonator of a high sensitivity of the atoms' motion to the initial conditions. This effect would not be properly described by bidimensional models. We show that an experimental implementation can be highly simplified by an analysis of the behaviour of the dynamical system under scale transformations. Experimentally feasible signatures of the chaotic dynamics of the atom clouds are also identified.
Lifetime statistics in chaotic dielectric microresonators
Schomerus, Henning; Wiersig, Jan; Main, Joerg
2009-05-15
We discuss the statistical properties of lifetimes of electromagnetic quasibound states in dielectric microresonators with fully chaotic ray dynamics. Using the example of a resonator of stadium geometry, we find that a recently proposed random-matrix model very well describes the lifetime statistics of long-lived resonances, provided that two effective parameters are appropriately renormalized. This renormalization is linked to the formation of short-lived resonances, a mechanism also known from the fractal Weyl law and the resonance-trapping phenomen0008.
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.
Random matrix theory for mixed regular-chaotic dynamics in the super-extensive regime
El-Hady, A. Abd; Abul-Magd, A. Y.
2011-10-27
We apply Tsallis's q-indexed nonextensive entropy to formulate a random matrix theory (RMT), which may be suitable for systems with mixed regular-chaotic dynamics. We consider the super-extensive regime of q<1. We obtain analytical expressions for the level-spacing distributions, which are strictly valid for 2 X2 random-matrix ensembles, as usually done in the standard RMT. We compare the results with spacing distributions, numerically calculated for random matrix ensembles describing a harmonic oscillator perturbed by Gaussian orthogonal and unitary ensembles.
Haotic, Fractal, and Nonlinear Signal Processing. Proceedings
Katz, R.A.
1996-10-01
These proceedings include papers presented at the Third Technical Conference on Nonlinear Dynamics and Full{minus}Spectrum Processing held in Mystic, Connecticut. The Conference focus was on the latest advances in chaotic, fractal and nonlinear signal processing methods. Topics of discussion covered in the Conference include: mathematical frontiers; predictability and control of chaos, detection and classification with applications in acoustics; advanced applied signal processing methods(linear and nonlinear); stochastic resonance; machinery diagnostics; turbulence; geophysics; medicine; and recent novel approaches to modeling nonlinear systems. There were 58 papers in the conference and all have been abstracted for the Energy Science and Technology database. (AIP)
Chaotic dynamics of one-dimensional systems with periodic boundary conditions
NASA Astrophysics Data System (ADS)
Kumar, Pankaj; Miller, Bruce N.
2014-12-01
We provide appropriate tools for the analysis of dynamics and chaos for one-dimensional systems with periodic boundary conditions. Our approach allows for the investigation of the dependence of the largest Lyapunov exponent on various initial conditions of the system. The method employs an effective approach for defining the phase-space distance appropriate for systems with periodic boundaries and allows for an unambiguous test-orbit rescaling in the phase space required to calculate the Lyapunov exponents. We elucidate our technique by applying it to investigate the chaotic dynamics of a one-dimensional plasma with periodic boundaries. Exact analytic expressions are derived for the electric field and potential using Ewald sums, thereby making it possible to follow the time evolution of the plasma in simulations without any special treatment of the boundary. By employing a set of event-driven algorithms, we calculate the largest Lyapunov exponent, the radial distribution function, and the pressure by following the evolution of the system in phase space without resorting to numerical manipulation of the equations of motion. Simulation results are presented and analyzed for the one-dimensional plasma with a view to examining the dynamical and chaotic behavior exhibited by small and large versions of the system.
Age-related alterations in the fractal scaling of cardiac interbeat interval dynamics
NASA Technical Reports Server (NTRS)
Iyengar, N.; Peng, C. K.; Morin, R.; Goldberger, A. L.; Lipsitz, L. A.
1996-01-01
We postulated that aging is associated with disruption in the fractallike long-range correlations that characterize healthy sinus rhythm cardiac interval dynamics. Ten young (21-34 yr) and 10 elderly (68-81 yr) rigorously screened healthy subjects underwent 120 min of continuous supine resting electrocardiographic recording. We analyzed the interbeat interval time series using standard time and frequency domain statistics and using a fractal measure, detrended fluctuation analysis, to quantify long-range correlation properties. In healthy young subjects, interbeat intervals demonstrated fractal scaling, with scaling exponents (alpha) from the fluctuation analysis close to a value of 1.0. In the group of healthy elderly subjects, the interbeat interval time series had two scaling regions. Over the short range, interbeat interval fluctuations resembled a random walk process (Brownian noise, alpha = 1.5), whereas over the longer range they resembled white noise (alpha = 0.5). Short (alpha s)- and long-range (alpha 1) scaling exponents were significantly different in the elderly subjects compared with young (alpha s = 1.12 +/- 0.19 vs. 0.90 +/- 0.14, respectively, P = 0.009; alpha 1 = 0.75 +/- 0.17 vs. 0.99 +/- 0.10, respectively, P = 0.002). The crossover behavior from one scaling region to another could be modeled as a first-order autoregressive process, which closely fit the data from four elderly subjects. This implies that a single characteristic time scale may be dominating heartbeat control in these subjects. The age-related loss of fractal organization in heartbeat dynamics may reflect the degradation of integrated physiological regulatory systems and may impair an individual's ability to adapt to stress.
NASA Technical Reports Server (NTRS)
Pikkujamsa, S. M.; Makikallio, T. H.; Sourander, L. B.; Raiha, I. J.; Puukka, P.; Skytta, J.; Peng, C. K.; Goldberger, A. L.; Huikuri, H. V.
1999-01-01
BACKGROUND: New methods of R-R interval variability based on fractal scaling and nonlinear dynamics ("chaos theory") may give new insights into heart rate dynamics. The aims of this study were to (1) systematically characterize and quantify the effects of aging from early childhood to advanced age on 24-hour heart rate dynamics in healthy subjects; (2) compare age-related changes in conventional time- and frequency-domain measures with changes in newly derived measures based on fractal scaling and complexity (chaos) theory; and (3) further test the hypothesis that there is loss of complexity and altered fractal scaling of heart rate dynamics with advanced age. METHODS AND RESULTS: The relationship between age and cardiac interbeat (R-R) interval dynamics from childhood to senescence was studied in 114 healthy subjects (age range, 1 to 82 years) by measurement of the slope, beta, of the power-law regression line (log power-log frequency) of R-R interval variability (10(-4) to 10(-2) Hz), approximate entropy (ApEn), short-term (alpha(1)) and intermediate-term (alpha(2)) fractal scaling exponents obtained by detrended fluctuation analysis, and traditional time- and frequency-domain measures from 24-hour ECG recordings. Compared with young adults (<40 years old, n=29), children (<15 years old, n=27) showed similar complexity (ApEn) and fractal correlation properties (alpha(1), alpha(2), beta) of R-R interval dynamics despite lower spectral and time-domain measures. Progressive loss of complexity (decreased ApEn, r=-0.69, P<0.001) and alterations of long-term fractal-like heart rate behavior (increased alpha(2), r=0.63, decreased beta, r=-0.60, P<0.001 for both) were observed thereafter from middle age (40 to 60 years, n=29) to old age (>60 years, n=29). CONCLUSIONS: Cardiac interbeat interval dynamics change markedly from childhood to old age in healthy subjects. Children show complexity and fractal correlation properties of R-R interval time series comparable to those
Regular and Chaotic Flow Behavior and Orientational Dynamics of Tumbling Nematics
NASA Astrophysics Data System (ADS)
Hess, S.; Heidenreich, S.; Ilg, P.; Kröger, M.
2006-05-01
We consider liquid crystalline polymers under plane Couette flow and investigate the influence of fluctuating shear rates on the orientational dynamics. With help of phase portraits and time evolution diagrams of the alignment tensor components, we discuss the effect of fluctuations on the flow-aligned, isotropic and periodic solutions. To explore the effect of fluctuations on the chaotic behavior we calculated the greatest Lyapunov exponent for different fluctuation strengths. We found that fluctuations of the shear rate in general have little effect on the dynamics of tumbling nematics. Further we present a new amended potential modeling the isotropic-to-nematic transition. In contrast to the Landau-de Gennes potential our potential has the advantage to restrict the order parameter to physically admissible values. In the end we present some results of the orientational dynamics for a spatially inhomogeneous system.
Dynamic interpretation of atomic and molecular spectra in the chaotic regime
Taylor, H.S.; Zakrzewski, J.
1988-10-01
A quantum partitioning theory is given for extracting dynamic information from the high-resolution spectra of highly excited atoms and molecules that is relatively simple to apply. The presented approach is applicable whenever the classical counterpart of the system studied is chaotic. The theory allows a picture of the underlying non-statistically-describable part of the dynamics to be obtained from the spectra. The theory presented effectively uses and unifies many aspects of classical trajectory approaches, Feshbach resonant-scattering partitioning theory, semiclassical periodic-orbit theory, ''scars'' theory, bright- and dark-state concepts, and Fourier transforms of the spectra. The power of the theory is demonstrated quantitatively by interpreting the dynamics underlying the absorption spectra of the hydrogen atom in a strong uniform magnetic field.
NASA Technical Reports Server (NTRS)
Hausdorff, J. M.; Mitchell, S. L.; Firtion, R.; Peng, C. K.; Cudkowicz, M. E.; Wei, J. Y.; Goldberger, A. L.
1997-01-01
Fluctuations in the duration of the gait cycle (the stride interval) display fractal dynamics and long-range correlations in healthy young adults. We hypothesized that these stride-interval correlations would be altered by changes in neurological function associated with aging and certain disease states. To test this hypothesis, we compared the stride-interval time series of 1) healthy elderly subjects and young controls and of 2) subjects with Huntington's disease and healthy controls. Using detrended fluctuation analysis we computed alpha, a measure of the degree to which one stride interval is correlated with previous and subsequent intervals over different time scales. The scaling exponent alpha was significantly lower in elderly subjects compared with young subjects (elderly: 0.68 +/- 0.14; young: 0.87 +/- 0.15; P < 0.003). The scaling exponent alpha was also smaller in the subjects with Huntington's disease compared with disease-free controls (Huntington's disease: 0.60 +/- 0.24; controls: 0.88 +/-0.17; P < 0.005). Moreover, alpha was linearly related to degree of functional impairment in subjects with Huntington's disease (r = 0.78, P < 0.0005). These findings demonstrate that strike-interval fluctuations are more random (i.e., less correlated) in elderly subjects and in subjects with Huntington's disease. Abnormal alterations in the fractal properties of gait dynamics are apparently associated with changes in central nervous system control.
Fractal structure and the dynamics of aggregation of synthetic melanin in low pH aqueous solutions
Huang, J.S.; Sung, J.; Eisner, M.; Moss, S.C.; Gallas, J.
1989-01-01
We have used static and dynamic light scattering to study the dynamics of aggregation of synthetic melanin, an amorphous biopolymeric substance, in low pH aqueous solution. We have found that, depending on the final pH value of the solutions, there existed two regimes of the aggregation kinetics, one corresponding to diffusion limited aggregation (DLA), and the other corresponding to reaction limited aggregation (RLA). The precipitates formed in these two regimes can be characterized by fractal structures. We have found fractal dimensions of d/sub f/ = 1.8 for the DLA clusters and d/sub f/ = 2.2 for the RLA clusters. These results agree well with the proposed limits of the fractal dimensions of the gold aggregates formed in aqueous solutions by Weitz et al.
Experimental and theoretical evidence for the chaotic dynamics of complex structures
NASA Astrophysics Data System (ADS)
Agop, M.; Dimitriu, D. G.; Niculescu, O.; Poll, E.; Radu, V.
2013-04-01
This paper presents the experimental results on the formation, dynamics and evolution towards chaos of complex space charge structures that emerge in front of a positively biased electrode immersed in a quiescent plasma. In certain experimental conditions, we managed to obtain the so-called multiple double layers (MDLs) with non-concentric configuration. Our experiments show that the interactions between each MDL's constituent entities are held responsible for the complex dynamics and eventually for its transition to chaos through cascades of spatio-temporal sub-harmonic bifurcations. Further, we build a theoretical model based on the fractal approximation (scale relativity theory) in order to reproduce the experimental results (plasma self-structuring and scenario of evolution to chaos). Comparing the experimental results with the theoretical ones, we observe a good correlation between them.
Fractal 1/f Dynamics Suggest Entanglement of Measurement and Human Performance
ERIC Educational Resources Information Center
Holden, John G.; Choi, Inhyun; Amazeen, Polemnia G.; Van Orden, Guy
2011-01-01
Variability of repeated measurements in human performances exhibits fractal 1/f noise. Yet the relative strength of this fractal pattern varies widely across conditions, tasks, and individuals. Four experiments illustrate how subtle details of the conditions of measurement change the fractal patterns observed across task conditions. The results…
High-frequency chaotic dynamics enabled by optical phase-conjugation
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
Zunino, L; Soriano, M C; Rosso, O A
2012-10-01
In this paper we introduce a multiscale symbolic information-theory approach for discriminating nonlinear deterministic and stochastic dynamics from time series associated with complex systems. More precisely, we show that the multiscale complexity-entropy causality plane is a useful representation space to identify the range of scales at which deterministic or noisy behaviors dominate the system's dynamics. Numerical simulations obtained from the well-known and widely used Mackey-Glass oscillator operating in a high-dimensional chaotic regime were used as test beds. The effect of an increased amount of observational white noise was carefully examined. The results obtained were contrasted with those derived from correlated stochastic processes and continuous stochastic limit cycles. Finally, several experimental and natural time series were analyzed in order to show the applicability of this scale-dependent symbolic approach in practical situations. PMID:23214666
High-frequency chaotic dynamics enabled by optical phase-conjugation.
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
High-frequency chaotic dynamics enabled by optical phase-conjugation
NASA Astrophysics Data System (ADS)
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.
Cheng, Mengfan; Deng, Lei; Li, Hao; Liu, Deming
2014-03-10
We propose a scheme whereby a time domain fractional Fourier transform (FRFT) is used to post process the optical chaotic carrier generated by an electro-optic oscillator. The time delay signature of the delay dynamics is successfully masked by the FRFT when some conditions are satisfied. Meanwhile the dimension space of the physical parameters is increased. Pseudo random binary sequence (PRBS) with low bit rate (hundreds of Mbps) is introduced to control the parameters of the FRFT. The chaotic optical carrier, FRFT parameters and the PRBS are covered by each other so that the eavesdropper has to search the whole key space to crack the system. The scheme allows enhancing the security of communication systems based on delay dynamics without modifying the chaotic source. In this way, the design of chaos based communication systems can be implemented in a modular manner. PMID:24663864
Escape dynamics and fractal basin boundaries in the planar Earth-Moon system
NASA Astrophysics Data System (ADS)
de Assis, Sheila C.; Terra, Maisa O.
2014-10-01
The escape of trajectories of a spacecraft, or comet or asteroid in the presence of the Earth-Moon system is investigated in detail in the context of the planar circular restricted three-body problem, in a scattering region around the Moon. The escape through the necks around the collinear points and as well as the leaking produced by considering collisions with the Moon surface, taking the lunar mean radius into account, were considered. Given that different transport channels are available as a function of the Jacobi constant, four distinct escape regimes are analyzed. Besides the calculation of exit basins and of the spatial distribution of escape time, the qualitative dynamical investigation through Poincaré sections is performed in order to elucidate the escape process. Our analyses reveal the dependence of the properties of the considered escape basins with the energy, with a remarkable presence of fractal basin boundaries along all the escape regimes. Finally, we observe the plentiful presence of stickiness motion near stability islands which plays a remarkable role in the longest escape time behavior. The application of this analysis is important both in space mission design and study of natural systems, given that fractal boundaries are related with high sensitivity to initial conditions, implying in uncertainty between safe and unsafe solutions, as well as between escaping solutions that evolve to different phase space regions.
Multiparticle sintering dynamics: from fractal-like aggregates to compact structures.
Eggersdorfer, Max L; Kadau, Dirk; Herrmann, Hans J; Pratsinis, Sotiris E
2011-05-17
Multiparticle sintering is encountered in almost all high temperature processes for material synthesis (titania, silica, and nickel) and energy generation (e.g., fly ash formation) resulting in aggregates of primary particles (hard- or sinter-bonded agglomerates). This mechanism of particle growth is investigated quantitatively by mass and energy balances during viscous sintering of amorphous aerosol materials (e.g., SiO(2) and polymers) that typically have a distribution of sizes and complex morphology. This model is validated at limited cases of sintering between two (equally or unequally sized) particles, and chains of particles. The evolution of morphology, surface area and radii of gyration of multiparticle aggregates are elucidated for various sizes and initial fractal dimension. For each of these structures that had been generated by diffusion limited (DLA), cluster-cluster (DLCA), and ballistic particle-cluster agglomeration (BPCA) the surface area evolution is monitored and found to scale differently than that of the radius of gyration (moment of inertia). Expressions are proposed for the evolution of fractal dimension and the surface area of aggregates undergoing viscous sintering. These expressions are important in design of aerosol processes with population balance equations (PBE) and/or fluid dynamic simulations for material synthesis or minimization and even suppression of particle formation. PMID:21488641
Chaotic dynamics outside Saturn’s main rings: The case of Atlas
NASA Astrophysics Data System (ADS)
Renner, Stéfan; Cooper, Nicholas J.; El Moutamid, Maryame; Evans, Mike W.; Murray, Carl D.; Sicardy, Bruno
2014-11-01
We revisit in detail the dynamics of Atlas. From a fit to new Cassini ISS astrometric observations spanning February 2004 to August 2013, we estimate GM_Atlas=0.384+/-0.001 x 10^(-3)km^3s^(-2), a value 13% smaller than the previously published estimate but with an order of magnitude reduction in the uncertainty. Our numerically-derived orbit shows that Atlas is currently librating in both a 54:53 corotation eccentricity resonance (CER) and a 54:53 Lindblad eccentricity resonance (LER) with Prometheus. We demonstrate 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. The interactions between the two resonances is investigated using the CoraLin analytical model (El Moutamid et al., 2014), showing that the chaotic zone fills almost all the corotation site 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 on Atlas.We estimate the capture probabilities of Atlas into resonances with Prometheus as the orbits expand through tidal effects, and discuss the implications for the orbital evolution.
Chaotic dynamics in charged-particle beams: Possible analogs of galactic evolution
Bohn, Courtlandt L.; /Northern Illinois U. /Fermilab
2004-12-01
During the last couple of years of his life, Henry Kandrup became intensely interested in using charged-particle beams as a tool for exploring the dynamics of evolving galaxies. He and I recognized that both galaxies and charged-particle beams can exhibit collisionless relaxation on surprisingly short time scales, and that this circumstance can be attributed to phase mixing of chaotic orbits. The chaos is often triggered by resonances caused by time dependence in the bulk potential, which acts almost identically for attractive gravitational forces as for repulsive electrostatic forces superposed on external focusing forces. Together we published several papers concerning evolving beams and galaxies, papers that relate to diverse topics such as the physics of chaotic mixing, the applicability of the Vlasov-Poisson formalism, and the production of diffuse halos. We also teamed with people from the University of Maryland to begin designing controlled experiments to be done at the University of Maryland Electron Ring. This paper highlights our collaborative findings as well as plans for future investigations that the findings have motivated.
A simple non-chaotic map generating subdiffusive, diffusive, and superdiffusive dynamics.
Salari, Lucia; Rondoni, Lamberto; Giberti, Claudio; Klages, Rainer
2015-07-01
Analytically tractable dynamical systems exhibiting a whole range of normal and anomalous deterministic diffusion are rare. Here, we introduce a simple non-chaotic model in terms of an interval exchange transformation suitably lifted onto the whole real line which preserves distances except at a countable set of points. This property, which leads to vanishing Lyapunov exponents, is designed to mimic diffusion in non-chaotic polygonal billiards that give rise to normal and anomalous diffusion in a fully deterministic setting. As these billiards are typically too complicated to be analyzed from first principles, simplified models are needed to identify the minimal ingredients generating the different transport regimes. For our model, which we call the slicer map, we calculate all its moments in position analytically under variation of a single control parameter. We show that the slicer map exhibits a transition from subdiffusion over normal diffusion to superdiffusion under parameter variation. Our results may help to understand the delicate parameter dependence of the type of diffusion generated by polygonal billiards. We argue that in different parameter regions the transport properties of our simple model match to different classes of known stochastic processes. This may shed light on difficulties to match diffusion in polygonal billiards to a single anomalous stochastic process. PMID:26232964
NASA Astrophysics Data System (ADS)
Small, Michael; Walker, David M.; Tordesillas, Antoinette; Tse, Chi K.
2013-03-01
For a given observed time series, it is still a rather difficult problem to provide a useful and compelling description of the underlying dynamics. The approach we take here, and the general philosophy adopted elsewhere, is to reconstruct the (assumed) attractor from the observed time series. From this attractor, we then use a black-box modelling algorithm to estimate the underlying evolution operator. We assume that what cannot be modeled by this algorithm is best treated as a combination of dynamic and observational noise. As a final step, we apply an ensemble of techniques to quantify the dynamics described in each model and show that certain types of dynamics provide a better match to the original data. Using this approach, we not only build a model but also verify the performance of that model. The methodology is applied to simulations of a granular assembly under compression. In particular, we choose a single time series recording of bulk measurements of the stress ratio in a biaxial compression test of a densely packed granular assembly—observed during the large strain or so-called critical state regime in the presence of a fully developed shear band. We show that the observed behavior may best be modeled by structures capable of exhibiting (hyper-) chaotic dynamics.
ERIC Educational Resources Information Center
Esbenshade, Donald H., Jr.
1991-01-01
Develops the idea of fractals through a laboratory activity that calculates the fractal dimension of ordinary white bread. Extends use of the fractal dimension to compare other complex structures as other breads and sponges. (MDH)
The influence of auditory-motor coupling on fractal dynamics in human gait.
Hunt, Nathaniel; McGrath, Denise; Stergiou, Nicholas
2014-01-01
Humans exhibit an innate ability to synchronize their movements to music. The field of gait rehabilitation has sought to capitalize on this phenomenon by invoking patients to walk in time to rhythmic auditory cues with a view to improving pathological gait. However, the temporal structure of the auditory cue, and hence the temporal structure of the target behavior has not been sufficiently explored. This study reveals the plasticity of auditory-motor coupling in human walking in relation to 'complex' auditory cues. The authors demonstrate that auditory-motor coupling can be driven by different coloured auditory noise signals (e.g. white, brown), shifting the fractal temporal structure of gait dynamics towards the statistical properties of the signals used. This adaptive capability observed in whole-body movement, could potentially be harnessed for targeted neuromuscular rehabilitation in patient groups, depending on the specific treatment goal. PMID:25080936
The influence of auditory-motor coupling on fractal dynamics in human gait
Hunt, Nathaniel; McGrath, Denise; Stergiou, Nicholas
2014-01-01
Humans exhibit an innate ability to synchronize their movements to music. The field of gait rehabilitation has sought to capitalize on this phenomenon by invoking patients to walk in time to rhythmic auditory cues with a view to improving pathological gait. However, the temporal structure of the auditory cue, and hence the temporal structure of the target behavior has not been sufficiently explored. This study reveals the plasticity of auditory-motor coupling in human walking in relation to ‘complex' auditory cues. The authors demonstrate that auditory-motor coupling can be driven by different coloured auditory noise signals (e.g. white, brown), shifting the fractal temporal structure of gait dynamics towards the statistical properties of the signals used. This adaptive capability observed in whole-body movement, could potentially be harnessed for targeted neuromuscular rehabilitation in patient groups, depending on the specific treatment goal. PMID:25080936
Micro and MACRO Fractals Generated by Multi-Valued Dynamical Systems
NASA Astrophysics Data System (ADS)
Banakh, T.; Novosad, N.
2014-08-01
Given a multi-valued function Φ : X \\mumap X on a topological space X we study the properties of its fixed fractal \\malteseΦ, which is defined as the closure of the orbit Φω(*Φ) = ⋃n∈ωΦn(*Φ) of the set *Φ = {x ∈ X : x ∈ Φ(x)} of fixed points of Φ. A special attention is paid to the duality between micro-fractals and macro-fractals, which are fixed fractals \\maltese Φ and \\maltese {Φ -1} for a contracting compact-valued function Φ : X \\mumap X on a complete metric space X. With help of algorithms (described in this paper) we generate various images of macro-fractals which are dual to some well-known micro-fractals like the fractal cross, the Sierpiński triangle, Sierpiński carpet, the Koch curve, or the fractal snowflakes. The obtained images show that macro-fractals have a large-scale fractal structure, which becomes clearly visible after a suitable zooming.
NASA Astrophysics Data System (ADS)
Yuan, Fang; Wang, Guang-Yi; Wang, Xiao-Yuan
2015-06-01
To develop real world memristor application circuits, an equivalent circuit model which imitates memductance (memory conductance) of the HP memristor is presented. The equivalent circuit can be used for breadboard experiments for various application circuit designs of memristor. Based on memductance of the realistic HP memristor and Chua’s circuit a new chaotic oscillator is designed. Some basic dynamical behaviors of the oscillator, including equilibrium set, Lyapunov exponent spectrum, and bifurcations with various circuit parameters are investigated theoretically and numerically. To confirm the correction of the proposed oscillator an analog circuit is designed using the proposed equivalent circuit model of an HP memristor, and the circuit simulations and the experimental results are given. Project supported by the National Natural Science Foundation of China (Grant Nos. 61271064 and 60971046), the Natural Science Foundation of Zhejiang Province, China (Grant No. LZ12F01001), and the Program for Zhejiang Leading Team of Science and Technology Innovation, China (Grant No. 2010R50010-07).
Chaotic dynamics of stellar spin in binaries and the production of misaligned hot Jupiters.
Storch, Natalia I; Anderson, Kassandra R; Lai, Dong
2014-09-12
Many exoplanetary systems containing hot Jupiters are observed to have highly misaligned orbital axes relative to the stellar spin axes. Kozai-Lidov oscillations of orbital eccentricity and inclination induced by a binary companion, in conjunction with tidal dissipation, constitute a major channel for the production of hot Jupiters. We demonstrate that gravitational interaction between the planet and its oblate host star can lead to chaotic evolution of the stellar spin axis during Kozai cycles. As parameters such as the planet mass and stellar rotation period are varied, periodic islands can appear in an ocean of chaos, in a manner reminiscent of other dynamical systems. In the presence of tidal dissipation, the complex spin evolution can leave an imprint on the final spin-orbit misalignment angles. PMID:25214623
NASA Astrophysics Data System (ADS)
Druzgalski, Clara; Mani, Ali
2014-11-01
We have investigated the transport dynamics of an electrokinetic instability that occurs when ions are driven from bulk fluids to ion-selective membranes due to externally applied electric fields. This phenomenon is relevant to a wide range of electrochemical applications including electrodialysis for fresh water production. Using data from our 3D DNS, we show how electroconvective instability, arising from concentration polarization, results in a chaotic flow that significantly alters the net ion transport rate across the membrane surface. The 3D DNS results, which fully resolve the spatiotemporal scales including the electric double layers, enable visualization of instantaneous snapshots of current density directly on the membrane surface, as well as analysis of transport statistics such as concentration variance and fluctuating advective fluxes. Furthermore, we present a full spectral analysis revealing broadband spectra in both concentration and flow fields and deduce the key parameter controlling the range of contributing scales.
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.
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 using 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.
Kengne, Jacques; Kenmogne, Fabien
2014-12-01
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 using 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. PMID:25554054
NASA Astrophysics Data System (ADS)
Kengne, Jacques; Kenmogne, Fabien
2014-12-01
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 using 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.
Chaotic behavior of collective ion dynamics in the presence of an external static magnetic field
NASA Astrophysics Data System (ADS)
Poria, Swarup; Ghosh, Samiran
2016-06-01
The two-dimensional nonlinear collective ion dynamics in the presence of external magnetic field in an electron-ion plasma is investigated. The analysis is performed for traveling plane waves to elucidate the various aspects of the phase-space dynamics. The presence of magnetic field makes the dynamics of the nonlinear wave complex with a complicated phase-space behavior. Thus, the nonlinear wave supports a wide class of nonlinear structures viz., single soliton, multi-soliton, periodic, and quasi-periodic oscillations depending on the values of M (Mach number) and Ω (the ratio of ion gyro-frequency to the ion plasma frequency). The computational results predict the chaotic behavior of the nonlinear wave and the transition to chaos takes place when Ω ≳ 0.35 depending on the direction of propagation and the value of M. The amplitude of the wave depends on the obliqueness of the propagation and Mach number, whereas the magnetic field changes the dispersion properties of the wave.
Chaos vs linear instability in the Vlasov equation: A fractal analysis characterization
Atalmi, A.; Baldo, M.; Burgio, G.F.; Rapisarda, A.
1996-05-01
In this paper we discuss the most recent results concerning the Vlasov dynamics inside the spinodal region. The chaotic behavior which follows an initial regular evolution is characterized through the calculation of the fractal dimension of the distribution of the final modes excited. The ambiguous role of the largest Lyapunov exponent for unstable systems is also critically reviewed. This investigation seems to confirm the crucial role played by deterministic chaos in nuclear multifragmentation. {copyright} {ital 1996 The American Physical Society.}
My chaotic trajectory: A brief (personalized) history of solar-system dynamics.
NASA Astrophysics Data System (ADS)
Burns, Joseph A.
2014-05-01
I will use this opportunity to recall my professional career. Like many, I was drawn into the space program during the mid-60s and early 70s when the solar system’s true nature was being revealed. Previously, dynamical astronomy discussed the short-term, predictable motions of point masses; simultaneously, small objects (e.g., satellites, asteroids, dust) were thought boring rather than dynamically rich. Many of today’s most active research subjects were unknown: TNOs, planetary rings, exoplanets and debris disks. The continuing stream of startling findings by spacecraft, ground-based surveys and numerical simulations forced a renaissance in celestial mechanics, incorporating new dynamical paradigms and additional physics (e.g., energy loss, catastrophic events, radiation forces). My interests evolved as the space program expanded outward: dust, asteroids, natural satellites, rings; rotations, orbital evolution, origins. Fortunately for me, in the early days, elementary models with simple solutions were often adequate to gain a first-order explanation of many puzzles. One could be a generalist, always learning new things.My choice of research subjects was influenced greatly by: i) Cornell colleagues involved in space missions who shared results: the surprising diversity of planetary satellites, the unanticipated orbital and rotational dynamics of asteroids, the chaotic histories of solar system bodies, the non-intuitive behavior of dust and planetary rings, irregular satellites. ii) Teaching introductory courses in applied math, dynamics and planetary science encouraged understandable models. iii) The stimulation of new ideas owing to service at Icarus and on space policy forums. iv) Most importantly, excellent students and colleagues who pushed me into new research directions, and who then stimulated and educated me about those topics.If time allows, I will describe some of today’s puzzles for me and point out similarities between the past development in our
Conduction block and chaotic dynamics in an asymmetrical model of coupled cardiac cells.
Landau, M; Lorente, P
1997-05-01
The initiation and propagation of the cardiac impulse depends on intrinsic properties of cells, geometrical arrangements, and intercellular coupling resistances. To address the issue of the interplay between these factors in a simple way, we have used a system, based on the van Capelle and Dürrer model, including a pacemaker and a non-pacemaker cell linked by an ohmic coupling resistance. The influence of asymmetrical cell sizes and electronic load was investigated by using numerical simulations and continuation-bifurcation techniques. The loading of a small pacemaker cell by a large non-pacemaker one (pacemaker: non-pacemaker size ratio = 0.3) was expressed as a pronounced early repolarization in the pacemaker cell and a quite long latency for the impulse propagation. Using coupling resistance as the continuation parameter, three behavioral zones were detected from low to high coupling resistance values: a zone of total quiescence (0:0), a zone of effective entertainment (1:1), and a zone of total block (1:0 pattern). At the boundary between 1:1 and 1:0 patterns, for relatively high coupling resistance values, a cascade of period doubling bifurcations emerged, corresponding to discrete changes of propagation patterns leading into irregular dynamics. Another route to irregular dynamics was also observed in the parameter space. The high sensitivity of the detected irregular dynamics to initial conditions and the positive value of the maximum Lyapunov exponent allowed us to identify these dynamics as being chaotic. Since neither intermediate block patterns nor irregular dynamics were observed with larger size ratios, we suggest that the interplay between resting membrane conductance of the non-pacemaker cell and intercellular coupling may bring about these rhythmic disturbances. PMID:9176640
NASA Astrophysics Data System (ADS)
Avellar, J.; Duarte, L. G. S.; da Mota, L. A. C. P.; de Melo, N.; Skea, J. E. F.
2012-09-01
A set of Maple routines is presented, fully compatible with the new releases of Maple (14 and higher). The package deals with the numerical evolution of dynamical systems and provide flexible plotting of the results. The package also brings an initial conditions generator, a numerical solver manager, and a focusing set of routines that allow for better analysis of the graphical display of the results. The novelty that the package presents an optional C interface is maintained. This allows for fast numerical integration, even for the totally inexperienced Maple user, without any C expertise being required. Finally, the package provides the routines to calculate the fractal dimension of boundaries (via box counting). New version program summary Program Title: Ndynamics Catalogue identifier: %Leave blank, supplied by Elsevier. Licensing provisions: no. Programming language: Maple, C. Computer: Intel(R) Core(TM) i3 CPU M330 @ 2.13 GHz. Operating system: Windows 7. RAM: 3.0 GB Keywords: Dynamical systems, Box counting, Fractal dimension, Symbolic computation, Differential equations, Maple. Classification: 4.3. Catalogue identifier of previous version: ADKH_v1_0. Journal reference of previous version: Comput. Phys. Commun. 119 (1999) 256. Does the new version supersede the previous version?: Yes. Nature of problem Computation and plotting of numerical solutions of dynamical systems and the determination of the fractal dimension of the boundaries. Solution method The default method of integration is a fifth-order Runge-Kutta scheme, but any method of integration present on the Maple system is available via an argument when calling the routine. A box counting [1] method is used to calculate the fractal dimension [2] of the boundaries. Reasons for the new version The Ndynamics package met a demand of our research community for a flexible and friendly environment for analyzing dynamical systems. All the user has to do is create his/her own Maple session, with the system to
NASA Astrophysics Data System (ADS)
Martelloni, Gianluca; Bagnoli, Franco
2016-04-01
Richardson's treatise on turbulent diffusion in 1926 [24] and today, the list of system displaying anomalous dynamical behavior is quite extensive. We only report some examples: charge carrier transport in amorphous semiconductors [25], porous systems [26], reptation dynamics in polymeric systems [27, 28], transport on fractal geometries [29], the long-time dynamics of DNA sequences [30]. In this scenario, the fractional calculus is used to generalized the Fokker-Planck linear equation -∂P (x,t)=D ∇2P (x,t), ∂t (3) where P (x,t) is the density of probability in the space x=[x1, x2, x3] and time t, while D >0 is the diffusion coefficient. Such processes are characterized by Eq. (1). An example of Eq. (3) generalization is ∂∂tP (x,t)=D∇ αP β(x,t) ‑ ∞ < α ≤ 2 β > ‑ 1 , (4) where the fractional based-derivatives Laplacian Σ(∂α/∂xα)i, (i = 1, 2, 3), of non-linear term Pβ(x,t) is taken into account [31]. Another generalized form is represented by equation ∂∂tδδP(x,t)=D ∇ αP(x,t) δ > 0 α ≤ 2 , (5) that considers also the fractional time-derivative [32]. These fractional-described processes exhibit a power law patters as expressed by Eq. (2). This general introduction introduces the presented work, whose aim is to develop a theoretical model in order to forecast the triggering and propagation of landslides, using the techniques of fractional calculus. The latter is suitable for modeling the water infiltration (i.e., the pore water pressure diffusion in the soil) and the dynamical processes in the fractal media [33]. Alternatively the fractal representation of temporal and spatial derivative (the fractal order only appears in the denominator of the derivative) is considered and the results are compared to the fractional one. The prediction of landslides and the discovering of the triggering mechanism, is one of the challenging problems in earth science. Landslides can be triggered by different factors but in most cases the trigger is an