Information encoding in homoclinic chaotic systems

We present a simple method for real-time encoding of information in the interspike intervals of a homoclinic chaotic system. The method has been experimentally tested on a CO2 laser with feedback displaying Sil'nikov chaos and synchronized with an external pulsed signal. Information is encoded by the length of the temporal intervals between consecutive pulses of the external signal. This length is varied each time a new pulse is generated.     

Periodic occurrence of chaotic behavior of homoclinic tangles

In this article, we illustrate, through numerical simulations, some important aspects of the dynamics of the periodically perturbed homoclinic solutions for a dissipative saddle. More explicitly, we demonstrate that, when homoclinic tangles are created, three different dynamical phenomena, namely, horseshoes, periodic sinks, and attractors with Sinai-Ruelle-Bowen measures, manifest themselves periodically with respect to the magnitude of the forcing function. In addition, when the stable and the unstable manifolds are pulled apart so as not to intersect, first, rank 1 attractors, then quasi-periodic attractors are added to the dynamical scene.     

In phase and antiphase synchronization of coupled homoclinic chaotic oscillators

We numerically investigate the dynamics of a closed chain of unidirectionally coupled oscillators in a regime of homoclinic chaos. The emerging synchronization regimes show analogies with the experimental behavior of a single chaotic laser subjected to a delayed feedback.     

Transversal homoclinic orbits in a transiently chaotic neural network

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

Noise enhanced phase synchronization and coherence resonance in sets of chaotic oscillators with weak global coupling

The effect of noise on phase synchronization in small sets and larger populations of weakly coupled chaotic oscillators is explored. Both independent and correlated noise are found to enhance phase synchronization of two coupled chaotic oscillators below the synchronization threshold; this is in contrast to the behavior of two coupled periodic oscillators. This constructive effect of noise results from the interplay between noise and the locking features of unstable periodic orbits. We show that in a population of nonidentical chaotic oscillators, correlated noise enhances synchronization in the weak coupling region. The interplay between noise and weak coupling induces a collective motion in which the coherence is maximal at an optimal noise intensity. Both the noise-enhanced phase synchronization and the coherence resonance numerically observed in coupled chaotic R?ssler oscillators are verified experimentally with an array of chaotic electrochemical oscillators.     

What can we learn from homoclinic orbits in chaotic dynamics?

State diagrams of two model systems involving three variables are constructed. The parameter dependence of different forms of complex nonperiodic behavior, and particularly of homoclinic orbits, is analyzed. It is shown that the onset of homoclinicity is reflected by deep changes in the qualitative behavior of the system.     

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.     

Stochastic resonance in chaotic dynamics

It is suggested that chaotic dynamical systems characterized by intermittent jumps between two preferred regions of phase space display an enhanced sensitivity to weak periodic forcings through a stochastic resonance-like mechanism. This possibility is illustrated by the study of the residence time distribution in two examples of bimodal chaos: the periodically forced Duffing oscillator and a 1-dimensional map showing intermittent behavior.     

Stochastic resonance in chaotic systems

The phenomenon of stochastic resonance (SR) is investigated for chaotic systems perturbed by white noise and a harmonic force. The bistable discrete map and the Lorenz system are considered as models. It is shown that SR in chaotic systems can be realized via both parameter variation (in the absence of noise) and by variation of the noise intensity with fixed values of the other parameters.     

Stochastic and coherence resonance in lasers: homoclinic chaos and polarization bistability

We report Stochastic Resonance and Coherence Resonance phenomena in experiments using CO₂ lasers. First, we consider a polarized laser with feedback; for suitable feedback parameters, the laser intensity undergoes homoclinic chaos consisting in the return to a saddle focus, where noise controls the permanence time around the saddle. Second, we discuss a quasi-isotropic laser where noise induces switching between two intensity components with mutually orthogonal polarization.     

Rate of escape of some chaotic Julia sets

We give a formula for the rates of escape for Julia sets with preperiodic critical points and forC endomorphisms of the interval with non-flat pre-periodic critical points outside the basin of attracting periodic points.Research supported by CNPq, Brasil     

Aperiodic stochastic resonance in chaotic maps

It is shown by means of numerical simulations that aperiodic stochastic resonance occurs in chaotic one-dimensional maps with various kinds of intermittency. The effect appears in the absence of external noise, as the system control parameter is varied. In the case of input signals slowly varying in time the analytic treatment, using the adiabatic approximation based on the expressions for the mean laminar phase duration, yields the input-output covariance function comparable with numerical results. (c) 1998 American Institute of Physics.     

Semiclassical structure of chaotic resonance eigenfunctions

We study the resonance (or Gamow) eigenstates of open chaotic systems in the semiclassical limit, distinguishing between left and right eigenstates of the nonunitary quantum propagator and also between short-lived and long-lived states. The long-lived left (right) eigenstates are shown to concentrate as variant Planck's over 2pi-->0 on the forward (backward) trapped set of the classical dynamics. The limit of a sequence of eigenstates [psi(variant Planck's over)] 2pi-->0 is found to exhibit a remarkably rich structure in phase space that depends on the corresponding limiting decay rate. These results are illustrated for the open baker's map, for which the probability density in position space is observed to have self-similarity properties.     

Coherence resonance in coupled chaotic oscillators

Existing works on coherence resonance, i.e., the phenomenon of noise-enhanced temporal regularity, focus on excitable dynamical systems such as those described by the FitzHugh-Nagumo equations. We extend the scope of coherence resonance to an important class of dynamical systems: coupled chaotic oscillators. In particular, we show that, when a system of coupled chaotic oscillators is under the influence of noise, the degree of temporal regularity of dynamical variables characterizing the difference among the oscillators can increase and reach a maximum value at some optimal noise level. We present numerical results illustrating the phenomenon and give a physical theory to explain it.     

Probabilistic approach to homoclinic chaos

Three-dimensional systems possessing a homoclinic orbit associated to a saddle focus with eigenvalues ±i, – and giving rise to homoclinic chaos when the Shil'nikov condition < is satisfied are studied. The 2D Poincaré map and its 1D contractions capturing the essential features of the flow are given. At homoclinicity, these 1D maps are found to be piecewise linear. This property allows one to reduce the Frobenius—Perron equation to a master equation whose solution is analytically known. The probabilistic properties such as the time autocorrelation function of the state variablex are explicitly derived.     

Multi-pulse homoclinic orbits and chaotic dynamics of a parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities

In this paper,the complicated dynamics and multi-pulse homoclinic orbits of a two-degree-of-freedom parametrically excited nonlinear nano-oscillator with coupled cubic nonlinearities are studied.The damping,parametrical excitation and the nonlinearities are regarded as weak.The averaged equation depicting the fast and slow dynamics is derived through the method of multiple scales.The dynamics near the resonance band is revealed by doing a singular perturbation analysis and combining the extended Melnikov method.We are able to determine the criterion for the existence of the multi-pulse homoclinic orbits which can form the Shilnikov orbits and give rise to chaos.At last,numerical results are also given to illustrate the nonlinear behaviors and chaotic motions in the nonlinear nano-oscillator.     

From generalized synchrony to topological decoherence: emergent sets in coupled chaotic systems

We consider the evolution of the unstable periodic orbit structure of coupled chaotic systems. This involves the creation of a complicated set outside of the synchronization manifold (the emergent set). We quantitatively identify a critical transition point in its development (the decoherence transition). For asymmetric systems we also describe a migration of unstable periodic orbits that is of central importance in understanding these systems. Our framework provides an experimentally measurable transition, even in situations where previously described bifurcation structures are inapplicable.     

Synchronization and coherence resonance in chaotic neural networks

Synchronization and coherence of chaotic Morris--Lecar (ML) neural networks have been investigated by numerical methods. The synchronization of the neurons can be enhanced by increasing the number of the shortcuts, even though all neurons are chaotic when uncoupled. Moreover, the coherence of the neurons exhibits a non-monotonic dependence on the density of shortcuts. There is an optimal number of shortcuts at which the neurons' motion is most ordered, i.e. the order parameter (the characteristic correlation time) that is introduced to measure the coherence of the neurons has a maximum. These phenomena imply that stochastic shortcuts can tame spatiotemporal chaos. The effects of the coupling strength have also been studied. The value of the optimal number of shortcuts goes down as the coupling strength increases.     

Stochastic resonance and chaotic resonance in bimodal maps: A case study

We present the results of an extensive numerical study on the phenomenon of stochastic resonance in a bimodal cubic map. Both Gaussian random noise as well as deterministic chaos are used as input to drive the system between the basins. Our main result is that when two identical systems capable of stochastic resonance are coupled, the SNR of either system is enhanced at an optimum coupling strength. Our results may be relevant for the study of stochastic resonance in biological systems.