Chaos control in a discrete time system through asymmetric coupling

We study a pair of asymmetrically coupled identical chaotic quadratic maps. We investigate, via numerical simulations, chaos suppression associated with the variation of both parameters, the coupling parameter and the parameter which measures the asymmetry. This is a new technique recently introduced for chaos suppression in continuous systems and, as far we know, not yet tested for discrete systems. Parameter-space regions where the chaotic dynamics is driven towards regular dynamics are shown. Lyapunov exponents and phase-space plots are also used to characterize the phenomenon observed as the parameters are changed.     

Long time tail correlations in discrete chaotic dynamics

A scaling relation is derived connecting the exponent of the algebraically decaying correlation and response functions with the degree of intermittency and the order of the maximum. It is universal, i.e. within a large class independent of the correlated variables. This implies universal 1/f-like spectra. The corrections to scaling are investigated, too.     

Chaos in discrete fractional difference equations

Recently, the discrete fractional calculus (DFC) is receiving attention due to its potential applications in the mathematical modelling of real-world phenomena with memory effects. In the present paper, the chaotic behaviour of fractional difference equations for the tent map, Gauss map and 2x(mod 1) map are studied numerically. We analyse the chaotic behaviour of these fractional difference equations and compare them with their integer counterparts. It is observed that fractional difference equations for the Gauss and tent maps are more stable compared to their integer-order version.     

Chaos and neural dynamics

Small neural networks (central pattern generators), which control rythmic behavior of animals, are considered. The mechanisms responsible for regularity, reliability, and plasticity of such behavior (swimming, running, walking, etc.) are analyzed. New data on the origin of regular cooperative behavior of neurons are obtained by using methods of nonlinear dynamics.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 39, No. 6, pp. 757–770, June, 1996.     

Chaos without nonlinear dynamics

A linear, second-order filter driven by randomly polarized pulses is shown to generate a waveform that is chaotic under time reversal. That is, the filter output exhibits determinism and a positive Lyapunov exponent when viewed backward in time. The filter is demonstrated experimentally using a passive electronic circuit, and the resulting waveform exhibits a Lorenz-like butterfly structure. This phenomenon suggests that chaos may be connected to physical theories whose underlying framework is not that of a traditional deterministic nonlinear dynamical system.     

Mean-field approximations of fixation time distributions of evolutionary game dynamics on graphs

The mean fixation time is often not accurate for describing the timescales of fixation probabilities of evolutionary games taking place on complex networks. We simulate the game dynamics on top of complex network topologies and approximate the fixation time distributions using a mean-field approach. We assume that there are two absorbing states. Numerically, we show that the mean fixation time is sufficient in characterizing the evolutionary timescales when network structures are close to the well-mixing condition. In contrast, the mean fixation time shows large inaccuracies when networks become sparse. The approximation accuracy is determined by the network structure, and hence by the suitability of the mean-field approach. The numerical results show good agreement with the theoretical predictions.     

More relaxed condition for dynamics of discrete time delayed Hopfield neural networks

The dynamics of discrete time delayed Hopfield neural networks is investigated. By using a difference inequality combining with the linear matrix inequality, a sufficient condition ensuring global exponential stability of the unique equilibrium point of the networks is found. The result obtained holds not only for constant delay but also for time-varying delays.     

More relaxed condition for dynamics of discrete time delayed Hopfield neural networks

The dynamics of discrete time delayed Hopfield neural networks is investigated. By using a difference inequality combining with the linear matrix inequality, a sufficient condition ensuring global exponential stability of the unique equilibrium point of the networks is found. The result obtained holds not only for constant delay but also for time-varying delays.     

Time scales in evolutionary dynamics

Evolutionary game theory has traditionally assumed that all individuals in a population interact with each other between reproduction events. We show that eliminating this restriction by explicitly considering the time scales of interaction and selection leads to dramatic changes in the outcome of evolution. Examples include the selection of the inefficient strategy in the Harmony and Stag-Hunt games, and the disappearance of the coexistence state in the Snowdrift game. Our results hold for any population size and in more general situations with additional factors influencing fitness.     

Chaos in discrete maps,deterministic scattering,and nondifferentiable functions

Arguments in favor of the nondifferentiability with respect to initial data of some functions associated with deterministic discrete-time dynamical systems are presented. A correspondence between a discrete-time dynamical system and a deterministic scattering model is found and used to interpret nondifferentiability conditions. A connection with random walks is also found.     

Chaos and dynamics of spinning particles in Kerr spacetime

We study chaos dynamics of spinning particles in Kerr spacetime of rotating black holes use the Papapetrou equations by numerical integration. Because of spin, this system exists many chaos solutions, and exhibits some exceptional dynamic character. We investigate the relations between the orbits chaos and the spin magnitude S, pericenter, polar angle and Kerr rotation parameter a by means of a kind of brand new Fast Lyapulov Indicator (FLI) which is defined in general relativity. The classical definition of Lyapulov exponent (LE) perhaps fails in curve spacetime. And we emphasize that the Poincaré sections cannot be used to detect chaos for this case. Via calculations, some new interesting conclusions are found: though chaos is easier to emerge with bigger S, but not always depends on S monotonically; the Kerr parameter a has a contrary action on the chaos occurrence. Furthermore, the spin of particles can destroy the symmetry of the orbits about the equatorial plane. And for some special initial conditions, the orbits have equilibrium points.     

Chaos-induced diffusion in nonlinear discrete dynamics

The large-scale motion of one-dimensional discrete systemX _t+1=X _t+f _a(X _t), (t=0, 1,2,...;f _a(X+1)=f _a(X)) is studied. This motion can be asymptotically decoposed into a drift and a diffusion (chaos-induced diffustion). We derive the formulae for the drift velocityv as the average jump number per step and for the diffusion coefficientD in terms of the jump number's time correlation function. It is shown that the coarsegrained probability distribution is asymptotically gaussian. Considering piecewise linear models and the sinusoidal one, we study the onset behavior of diffusion and its gross behavior. It is found thatD is proportional to the length of the region with a non-zero jump number, if the critical dynamics is well-defined. Otherwise we have logarithmic or inverse power corrections to the simple law originating from an intermittency type behavior or from band splitting phenomena. Maps with a smooth maximum (as e.g. the sinusoidal) exhibit several additional types of trajectories: running modes with broken symmetry, localized trajectories, regular periodic or chaotic, non-diffusive or diffusive ones. These dynamical states appear in a nested window structure, which is described.     

Species abundance patterns in complex evolutionary dynamics

An analytic theory of species abundance patterns (SAPs) in biological networks is presented. The theory is based on multispecies replicator dynamics equivalent to the Lotka-Volterra equation, with diverse interspecies interactions. Various SAPs observed in nature are derived from a single parameter. The abundance distribution is formed like a widely observed left-skewed lognormal distribution. As the model has a general form, the result can be applied to similar patterns in other complex biological networks, e.g., gene expression.     

Chaos dynamics in semiconductor lasers with polarization-rotated optical feedback

By using polarization-rotated optical feedback from the transverse-electric (TE) mode to the transverse-magnetic (TM) mode, chaotic oscillations for both polarization modes are excited in a semiconductor laser. We find different correlations between these chaotic oscillations than those found in previous studies. In this study, the dynamics are strongly dependent on their radio-frequency (RF) components and they are divided into three RF regions. For low-pass filtered signals lower than the laser relaxation oscillation, there is an antiphase correlation between the two polarization modes. On the other hand, the two polarization modes have an in-phase correlation for the RF components of the high-pass filtered signals, which are higher than the relaxation oscillation. However, no correlations were observed between the two modes for the intermediate RF components that include the relaxation oscillation frequency. We also perform numerical calculations for the model and obtain good agreement between the theoretical and experimental results.     

Stochastic dynamics of adaptive evolutionary search

In this paper the stochastic dynamics of adaptive evolutionary search, as performed by the optimization algorithm Population-Based Incremental Learning, is analyzed with physicists' methods for stochastic processes. The master equation of the process is approximated by van Kampen's small fluctuations assumption. It results in an elegant formalism which allows for an understanding of the macroscopic behaviour of the algorithm together with its fluctuations. We consider the search process to be adaptive since the algorithm iteratively reduces its mutation rate while approaching an optimum. On the one hand, it is this feature which allows the algorithm to quickly converge towards an optimum. On the other hand it results in the possibility to get trapped by a local optimum only. To arrive at a detailed understanding we discuss the influence of fluctuations, as caused by mutation, on this behaviour. We study the algorithm for rather small sytem sizes in order to gain an intuitive understanding of the algorithm's performance.     

Arcs of discrete dynamics and constants of motion

A theorem is proved which may relate the attractors of families of dissipative discrete-time dynamical systems to certain closed orbits of conservative systems. The result is illustrated by an example taken from dynamics defined by mappings.     

Kink dynamics in the highly discrete sine-Gordon system

We study kink dynamics in a very discrete sine-Gordon system where the kink width is of the order of the lattice spacing. Numerical simulations exhibit new properties of kinks in this case: they lose the memory of their initial velocity and propagate preferentially at well-defined velocities which correspond to quasi-steady states, while a kink moving at other velocities suffers relatively high rates of radiation of small amplitude oscillations. When a small external driving force is applied to the system, the same velocities appear as plateus in the strongly nonlinear mobility of the kink. The energy radiated by the kink is calculated for a simple model that preserves the discrete character of the system, and the preferential velocities for the kink are obtained to good accuracy. Similar results may be expected to be valid for other discrete systems manifesting topological solitons. The numerical simulations reveal also new stable “multiple-kink” excitations which can propagate almost freely in extremely discrete systems where “ordinary” simple kinks are pinned to the lattice by discreteness. The stability of the “multiple-kinks” is discussed.