Lyapunov exponents and kolmogorov-sinai entropy for a high-dimensional convex billiard

We compute the Lyapunov exponents and the Kolmogorov-Sinai (KS) entropy for a self-bound N-body system that is realized as a convex billiard. This system exhibits truly high-dimensional chaos, and 2N-4 Lyapunov exponents are found to be positive. The KS entropy increases linearly with the numbers of particles. We examine the chaos generating defocusing mechanism and investigate how high-dimensional chaos develops in this system with no dispersing elements.     

Hierarchy of Chaotic Maps with an Invariant Measure

We give hierarchy of one-parameter family (, x) of maps at the interval [0, 1] with an invariant measure. Using the measure, we calculate Kolmogorov-Sinai entropy, or equivalently Lyapunov characteristic exponent of these maps analytically, where the results thus obtained have been approved with the numerical simulation. In contrary to the usual one-parameter family of maps such as logistic and tent maps, these maps do not possess period doubling or period-n-tupling cascade bifurcation to chaos, but they have single fixed point attractor for certain values of the parameter, where they bifurcate directly to chaos without having period-n-tupling scenario exactly at those values of the parameter whose Lyapunov characteristic exponent begins to be positive.     

On the dynamics of the q-deformed logistic map

We analyze the q-deformed logistic map, where the q-deformation follows the scheme inspired in the Tsallis q-exponential function. We compute the topological entropy of the dynamical system, obtaining the parametric region in which the topological entropy is positive and hence the region in which chaos in the sense of Li and Yorke exists. In addition, it is shown the existence of the so-called Parrondo's paradox where two simple maps are combined to give a complicated dynamical behavior.     

Dynamical entropy production in spiking neuron networks in the balanced state

We demonstrate deterministic extensive chaos in the dynamics of large sparse networks of theta neurons in the balanced state. The analysis is based on numerically exact calculations of the full spectrum of Lyapunov exponents, the entropy production rate, and the attractor dimension. Extensive chaos is found in inhibitory networks and becomes more intense when an excitatory population is included. We find a strikingly high rate of entropy production that would limit information representation in cortical spike patterns to the immediate stimulus response.     

Rate of escape from nonattracting chaotic sets

Chaotic transient phenomena occur in the vicinity of nonattracting chaotic sets. The rate of escape measures the average length of the transients. There is a conjecture by Eckmann and Ruelle connecting the rate of escape to the Lyapunov exponents and entropy. We prove an inequality that partially supports the conjecture.     

Chaos and Hyperchaos in a Backward-Wave Oscillator

Based on a numerical solution of the equations of the nonstationary nonlinear theory, we study chaotic self-oscillation regimes in a backward-wave oscillator. For “weak” chaos, arising via a period-doubling cascade of self-modulation for moderate values of the normalized-length parameter, and for developed chaos, which corresponds to large values of this parameter, we present the temporal dependences of the output-signal amplitude, the phase portraits, and the statistical parameters of the dynamics. It is shown that developed chaos is characterized by the presence of more than one positive Lyapunov exponent (hyperchaos). We also present estimates of the Kolmogorov–Sinai entropy, the Lyapunov dimension, and the correlation dimension obtained from the Grassberger–Procaccia algorithm. The results confirm that a finite-dimensional strange attractor is responsible for the chaotic regimes in a backward-wave oscillator.     

Roundoff-induced attractors and reversibility in conservative two-dimensional maps

We numerically study two conservative two-dimensional maps, namely the baker map (whose Lyapunov exponent is known to be positive), and a typical one (exhibiting a vanishing Lyapunov exponent) chosen from the generalized shift family of maps introduced by C. Moore [Phys. Rev. Lett. 64 (1990) 2354] in the context of undecidability. We calculate the time evolution of the entropy (). We exhibit the dramatic effect introduced by numerical precision. Indeed, in spite of being area-preserving maps, they present, well after the initially concentrated ensemble has spread virtually all over the phase space, unexpected pseudo-attractors (fixed-point like for the baker map, and more complex structures for the Moore map). These pseudo-attractors, and the apparent time (partial) reversibility they provoke, gradually disappear for increasingly large precision. In the case of the Moore map, they are related to zero Lebesgue-measure effects associated with the frontiers existing in the definition of the map. In addition to the above, and consistent with the results by V. Latora and M. Baranger [Phys. Rev. Lett. 82 (1999) 520], we find that the rate of the far-from-equilibrium entropy production of baker map numerically coincides with the standard Kolmogorov-Sinai entropy of this strongly chaotic system.     

de Broglie-Bohm formulation of quantum mechanics, quantum chaos and breaking of time-reversal invariance

A recently developed unified theory of classical and quantum chaos, based on the de Broglie-Bohm (Hamilton-Jacobi) formulation of quantum mechanics is presented and its consequences are discussed. The quantum dynamics is rigorously defined to be chaotic if the Lyapunov number, associated with the quantum trajectories in de Broglie-Bohm phase space, is positive definite. This definition of quantum chaos which under classical conditions goes over to the well-known definition of classical chaos in terms of positivity of Lyapunov numbers, provides a rigorous unified definition of chaos on the same footing for both the dynamics. A demonstration of the existence of positive Lyapunov numbers in a simple quantum system is given analytically, proving the existence of quantum chaos. Breaking of the time-reversal symmetry in the corresponding quantum dynamics under chaotic evolution is demonstrated. It is shown that the rigorous deterministic quantum chaos provides an intrinsic mechanism towards irreversibility of the Schrodinger evolution of the wave function, without invoking ‘wave function collapse’ or ‘measurements’     

Emergent horizon,Hawking radiation and chaos in the collapsed polymer model of a black hole

We have proposed that the interior of a macroscopic Schwarzschild black hole (BH) consists of highly excited, long, closed, interacting strings and, as such, can be modeled as a collapsed polymer. It was previously shown that the scaling relations of the collapsed‐polymer model agree with those of the BH. The current paper further substantiates this proposal with an investigation into some of its dynamical consequences. In particular, we show that the model predicts, without relying on gravitational effects, an emergent horizon. We further show that the horizon fluctuates quantum mechanically as it should and that the strength of the fluctuations is inversely proportional to the BH entropy. It is then demonstrated that the emission of Hawking radiation is realized microscopically by the quantum‐induced escape of small pieces of string, with the rate of escape and the energy per emitted piece both parametrically matching the Hawking temperature. We also show, using standard methods from statistical mechanics and chaos theory, how our model accounts for some other known properties of BHs. These include the accepted results for the scrambling time and the viscosity‐to‐entropy ratio, which in our model apply not only at the horizon but throughout the BH interior.     

二维logistic映射的动力学行为和奇怪吸引子的分形特征

研究了二维logistic映射的动力学行为和奇怪吸引子的分形特征.利用分岔图、相图和Lyapunov指数谱分析系统的分岔过程,研究系统通向混沌的道路并确定系统处于混沌运动的参数区间;采用G-P算法计算奇怪吸引子的关联维数和Kolmogorov熵,对奇怪吸引子的分形特征定量刻画;采用逃逸时间算法构造奇怪吸引子的彩色广义M-J集,对奇怪吸引子的分形特征定性表征.结果表明,这些分析方法的配合使用可以更全面、形象地描述奇怪吸引子的分形特征.     

Characterization of the spatial complex behavior and transition to chaos in flow systems

We introduce a “spatial” Lyapunov exponent to characterize the complex behavior of non-chaotic but convectively unstable flow sytems. This complexity is of spatial type and is due to sensitivity to the boundary conditions. We show that there exists a relation between the spatial-complexity index we define and the comoving Lyapunov exponents. In such systems the transition to chaos, i.e., the occurrence of a positive Lyapunov exponent, can manifest itself in two different ways. In the first case (from neither chaotic nor spatially complex behavior to chaos) one observes the typical scenario; i.e., as the system size grows up the spectrum of the Lyapunov exponents gives rise to a density. In the second case (when the chaos develops from a convectively unstable situation) one observes only a finite number of positive Lyapunov exponents.     

Statistical mechanics of two-dimensional point vortices: relaxation equations and strong mixing limit

We complement the literature on the statistical mechanics of point vortices in two-dimensional hydrodynamics. Using a maximum entropy principle, we determine the multi-species Boltzmann-Poisson equation and establish a form of Virial theorem. Using a maximum entropy production principle (MEPP), we derive a set of relaxation equations towards statistical equilibrium. These relaxation equations can be used as a numerical algorithm to compute the maximum entropy state. We mention the analogies with the Fokker-Planck equations derived by Debye and Hückel for electrolytes. We then consider the limit of strong mixing (or low energy). To leading order, the relationship between the vorticity and the stream function at equilibrium is linear and the maximization of the entropy becomes equivalent to the minimization of the enstrophy. This expansion is similar to the Debye-Hückel approximation for electrolytes, except that the temperature is negative instead of positive so that the effective interaction between like-sign vortices is attractive instead of repulsive. This leads to an organization at large scales presenting geometry-induced phase transitions, instead of Debye shielding. We compare the results obtained with point vortices to those obtained in the context of the statistical mechanics of continuous vorticity fields described by the Miller-Robert-Sommeria (MRS) theory. At linear order, we get the same results but differences appear at the next order. In particular, the MRS theory predicts a transition between sinh and tanh-like ω ? ψ relationships depending on the sign of Ku ? 3 (where Ku is the Kurtosis) while there is no such transition for point vortices which always show a sinh-like ω ? ψ relationship. We derive the form of the relaxation equations in the strong mixing limit and show that the enstrophy plays the role of a Lyapunov functional.     

Noisy one-dimensional maps near a crisis. I. Weak Gaussian white and colored noise

We study one-dimensional single-humped maps near the boundary crisis at fully developed chaos in the presence of additive weak Gaussian white noise. By means of a new perturbation-like method the quasi-invariant density is calculated from the invariant density at the crisis in the absence of noise. In the precritical regime, where the deterministic map may show periodic windows, a necessary and sufficient condition for the validity of this method is derived. From the quasi-invariant density we determine the escape rate, which has the form of a scaling law and compares excellently with results from numerical simulations. We find that deterministic transient chaos is stabilized by weak noise whenever the maximum of the map is of orderz>1. Finally, we extend our method to more general maps near a boundary crisis and to multiplicative as well as colored weak Gaussian noise. Within this extended class of noises and for single-humped maps with any fixed orderz>0 of the maximum, in the scaling law for the escape rate both the critical exponents and the scaling function are universal.     

Quantifying chaos for ecological stoichiometry

The theory of ecological stoichiometry considers ecological interactions among species with different chemical compositions. Both experimental and theoretical investigations have shown the importance of species composition in the outcome of the population dynamics. A recent study of a theoretical three-species food chain model considering stoichiometry [B. Deng and I. Loladze, Chaos 17, 033108 (2007)] shows that coexistence between two consumers predating on the same prey is possible via chaos. In this work we study the topological and dynamical measures of the chaotic attractors found in such a model under ecological relevant parameters. By using the theory of symbolic dynamics, we first compute the topological entropy associated with unimodal Poincare? return maps obtained by Deng and Loladze from a dimension reduction. With this measure we numerically prove chaotic competitive coexistence, which is characterized by positive topological entropy and positive Lyapunov exponents, achieved when the first predator reduces its maximum growth rate, as happens at increasing δ1. However, for higher values of δ1 the dynamics become again stable due to an asymmetric bubble-like bifurcation scenario. We also show that a decrease in the efficiency of the predator sensitive to prey's quality (increasing parameter ζ) stabilizes the dynamics. Finally, we estimate the fractal dimension of the chaotic attractors for the stoichiometric ecological model.     

Strange Attractors with One Direction of Instability

We give simple conditions that guarantee, for strongly dissipative maps, the existence of strange attractors with a single direction of instability and certain controlled behaviors. Only the d= 2 case is treated in this paper, although our approach is by no means limited to two phase-dimensions. We develop a dynamical picture for the attractors in this class, proving they have many of the statistical properties associated with chaos: positive Lyapunov exponents, existence of SRB measures, and exponential decay of correlations. Other results include the geometry of fractal critical sets, nonuniform hyperbolic behavior, symbolic coding of orbits, and formulas for topological entropy. Received: 25 April 2000 / Accepted: 17 October 2000     

Positive cosmological constant,non-local gravity and horizon entropy

We discuss a class of (local and non-local) theories of gravity that share same properties: (i) they admit the Einstein spacetime with arbitrary cosmological constant as a solution; (ii) the on-shell action of such a theory vanishes and (iii) any (cosmological or black hole) horizon in the Einstein spacetime with a positive cosmological constant does not have a non-trivial entropy. The main focus is made on a recently proposed non-local model. This model has two phases: with a positive cosmological constant Λ>0$\Lambda >0$ and with zero Λ. The effective gravitational coupling differs essentially in these two phases. Generalizing the previous result of Barvinsky we show that the non-local theory in question is free of ghosts on the background of any Einstein spacetime and that it propagates a standard spin-2 particle. Contrary to the phase with a positive Λ, where the entropy vanishes for any type of horizon, in an Einstein spacetime with zero cosmological constant the horizons have the ordinary entropy proportional to the area. We conclude that, somewhat surprisingly, the presence of any, even extremely tiny, positive cosmological constant should be important for the proper resolution of the entropy problem and, possibly, the information puzzle.     

Anti-control of chaos in Bose-Einstein condensates

The spatial structure of a Bose-Einstein Condensate (BEC) loaded into an optical lattice potential is investigated. We suggest a method for generating chaos in BEC by modulating periodic signals to convert the regular states into chaotic states. The maximal Lyapunov exponent is calculated as a function of modulation intensity and modulation frequency respectively, and the chaotic orbits associated with the positive Lyapunov exponents.      

Characterizing the dynamics of coupled pendulums via symbolic time series analysis

We propose a novel method of symbolic time-series analysis aimed at characterizing the regular or chaotic dynamics of coupled oscillators. The method is applied to two identical pendulums mounted on a frictionless platform, resembling Huygens’ clocks. Employing a transformation rule inspired in ordinal analysis [C. Bandt and B. Pompe, Phys. Rev. Lett. 88, 174102 (2002)], the dynamics of the coupled system is represented by a sequence of symbols that are determined by the order in which the trajectory of each pendulum intersects an appropriately chosen hyperplane in the phase space. For two coupled pendulums we use four symbols corresponding to the crossings of the vertical axis (at the bottom equilibrium point), either clock-wise or anti-clock wise. The complexity of the motion, quantified in terms of the entropy of the symbolic sequence, is compared with the degree of chaos, quantified in terms of the largest Lyapunov exponent. We demonstrate that the symbolic entropy sheds light into the large variety of different periodic and chaotic motions, with different types synchronization, that cannot be inferred from the Lyapunov analysis.     

Lyapunov exponent diagrams of a 4-dimensional Chua system

We report numerical results on the existence of periodic structures embedded in chaotic and hyperchaotic regions on the Lyapunov exponent diagrams of a 4-dimensional Chua system. The model was obtained from the 3-dimensional Chua system by the introduction of a feedback controller. Both the largest and the second largest Lyapunov exponents were considered in our colorful Lyapunov exponent diagrams, and allowed us to characterize periodic structures and regions of chaos and hyperchaos. The shrimp-shaped periodic structures appear to be malformed on some of Lyapunov exponent diagrams, and they present two different bifurcation scenarios to chaos when passing the boundaries of itself, namely via period-doubling and crisis. Hyperchaos-chaos transition can also be observed on the Lyapunov exponent diagrams for the second largest exponent.     

Decoherence and the rate of entropy production in chaotic quantum systems

We show that for an open quantum system which is classically chaotic (a quartic double well with harmonic driving coupled to a sea of harmonic oscillators) the rate of entropy production has, as a function of time, two relevant regimes: For short times it is proportional to the diffusion coefficient (fixed by the system-environment coupling strength). For longer times (but before equilibration) there is a regime where the entropy production rate is fixed by the Lyapunov exponent. The nature of the transition time between both regimes is investigated.