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.     

A geometric theory of chaotic phase synchronization

A rigorous mathematical treatment of chaotic phase synchronization is still lacking, although it has been observed in many numerical and experimental studies. In this article we address the extension of results on phase synchronization in periodic oscillators to systems with phase coherent chaotic attractors with small phase diffusion. As models of such systems we consider special flows over diffeomorphisms in which the neutral direction is periodically perturbed. A generalization of the Averaging Theorem for periodic systems is used to extend Kuramoto's geometric theory of phase locking in periodically forced limit cycle oscillators to this class of systems. This approach results in reduced equations describing the dynamics of the phase difference between drive and response systems over long time intervals. The reduced equations are used to illustrate how the structure of a chaotic attractor is important in its response to a periodic perturbation, and to conclude that chaotic phase coherent systems may not always be treated as noisy periodic oscillators in this context. Although this approach is strictly justified for periodic perturbations affecting only the phase variable of a chaotic oscillator, we argue that these ideas are applicable much more generally.     

环形耦合Duffing振子间的同步突变

以环形耦合Duffing振子系统为研究对象,分析了耦合振子间的同步演化过程.发现在弱耦合条件下,如果所有振子受到同一周期策动力的驱动,那么系统在经历倍周期分岔、混沌态、大尺度周期态的相变时,各振子的运动轨迹之间将出现由同步到不同步再到同步的两次突变现象.利用其中任何一次同步突变现象可以实现系统相变的快速判别,并由此补充了利用倍周期分岔与混沌态的这一相变对微弱周期信号进行检测的方法. 关键词： Duffing振子 同步突变 相变 微弱信号检测     

Populations of coupled electrochemical oscillators

Experiments were carried out on arrays of chaotic electrochemical oscillators to which global coupling, periodic forcing, and feedback were applied. The global coupling converts a very weakly coupled set of chaotic oscillators to a synchronized state with sufficiently large values of coupling strength; at intermediate values both intermittent and stable chaotic cluster states occur. Cluster formation and synchronization were also obtained by applying feedback and forcing to a moderately coupled base state. The three cases differ, however, in other details. The feedback and forcing also produce periodic cluster states and more than two clusters. Configurations of two (chaotic) clusters and two, three, or four (periodic) clusters were observed. (c) 2002 American Institute of Physics.     

Experimental investigation of high-quality synchronization of coupled oscillators

We describe two experiments in which we investigate the synchronization of coupled periodic oscillators. Each experimental system consists of two identical coupled electronic periodic oscillators that display bursts of desynchronization events similar to those observed previously in coupled chaotic systems. We measure the degree of synchronization as a function of coupling strength. In the first experiment, high-quality synchronization is achieved for all coupling strengths above a critical value. In the second experiment, no high-quality synchronization is observed. We compare our results to the predictions of the several proposed criteria for synchronization. We find that none of the criteria accurately predict the range of coupling strengths over which high-quality synchronization is observed. (c) 2000 American Institute of Physics.     

Phase-flip transition in nonlinear oscillators coupled by dynamic environment

We study the dynamics of nonlinear oscillators indirectly coupled through a dynamical environment or a common medium. We observed that this form of indirect coupling leads to synchronization and phase-flip transition in periodic as well as chaotic regime of oscillators. The phase-flip transition from in- to anti-phase synchronization or vise-versa is analyzed in the parameter plane with examples of Landau-Stuart and Ro?ssler oscillators. The dynamical transitions are characterized using various indices such as average phase difference, frequency, and Lyapunov exponents. Experimental evidence of the phase-flip transition is shown using an electronic version of the van der Pol oscillators.     

Clustering zones in the turbulent phase of a system of globally coupled chaotic maps

The paper develops an approach to investigate the clustering phenomenon in the system of globally coupled chaotic maps first introduced by Kaneko in 1989. We obtain a relation between the transverse and longitudinal multipliers of the periodic clusters and prove the stability of these clusters for the case of symmetric, equally populated distributions between subclusters. Stable clusters emanate from the periodic windows of the logistic map and extend far into the turbulent phase. By numerical simulations we estimate a total basin volume of low-periodic clusters issued from the period-3 window and analyze the basin structure. The complement to the basin volume is ascribed to chaotic, very asymmetric high-dimensional clusters that are characterized by the presence of one or more leading clusters, accumulating about half of the oscillators while all the remaining oscillators do not cluster at all.     

Chaos suppression through asymmetric coupling

We study pairs of identical coupled chaotic oscillators. In particular, we have used Roessler (in the funnel and no funnel regimes), Lorenz, and four-dimensional chaotic Lotka-Volterra models. In all four of these cases, a pair of identical oscillators is asymmetrically coupled. The main result of the numerical simulations is that in all cases, specific values of coupling strength and asymmetry exist that render the two oscillators periodic and synchronized. The values of the coupling strength for which this phenomenon occurs is well below the previously known value for complete synchronization. We have found that this behavior exists for all the chaotic oscillators that we have used in the analysis. We postulate that this behavior is presumably generic to all chaotic oscillators. In order to complete the study, we have tested the robustness of this phenomenon of chaos suppression versus the addition of some Gaussian noise. We found that chaos suppression is robust for the addition of finite noise level. Finally, we propose some extension to this research.     

Phase synchronization in ensembles of bursting oscillators

We study the effects of mutual and external chaotic phase synchronization in ensembles of bursting oscillators. These oscillators (used for modeling neuronal dynamics) are essentially multiple time scale systems. We show that a transition to mutual phase synchronization takes place on the bursting time scale of globally coupled oscillators, while on the spiking time scale, they behave asynchronously. We also demonstrate the effect of the onset of external chaotic phase synchronization of the bursting behavior in the studied ensemble by a periodic driving applied to one arbitrarily taken neuron. We also propose an explanation of the mechanism behind this effect. We infer that the demonstrated phenomenon can be used efficiently for controlling bursting activity in neural ensembles.     

Locking-based frequency measurement and synchronization of chaotic oscillators with complex dynamics

We propose a method for the determination of a characteristic oscillation frequency for a broad class of chaotic oscillators generating complex signals. It is based on the locking of standard periodic self-sustained oscillators by an irregular signal. The method is applied to experimental data from chaotic electrochemical oscillators, where other approaches of frequency determination (e.g., based on Hilbert transform) fail. Using the method we characterize the effects of phase synchronization for systems with ill-defined phase by external forcing and due to mutual coupling.     

Chaotic transition in a three-coupled phase-locked loop system

The chaotic transition is observed in a three-coupled phase-locked loop (PLL) system in both experiments and numerical simulations. In this system, three PLL oscillators are connected with the periodic boundary condition. Intermittency is found in partially synchronized phase, in which two of three oscillators synchronize with each other and form a pair, and the chaotic transition occurs due to the recombination of synchronized pairs so that different pair is re-formed. In this phase, on-off intermittency is also observed and statistical analyses are carried out for on-off intermittent time series. This intermittency is considered as a hybrid type of intermittency with both on-off intermittency and intermittency due to the recombination of synchronized pairs present in the same time series. We also show the chaotic transition phenomena in a three-coupled logistic map system. (c) 2001 American Institute of Physics.     

Emergence of coherence in complex networks of heterogeneous dynamical systems

We present a general theory for the onset of coherence in collections of heterogeneous maps interacting via a complex connection network. Our method allows the dynamics of the individual uncoupled systems to be either chaotic or periodic, and applies generally to networks for which the number of connections per node is large. We find that the critical coupling strength at which a transition to synchrony takes place depends separately on the dynamics of the individual uncoupled systems and on the largest eigenvalue of the adjacency matrix of the coupling network. Our theory directly generalizes the Kuramoto model of equal strength all-to-all coupled phase oscillators to the case of oscillators with more realistic dynamics coupled via a large heterogeneous network.     

Noise-enhanced phase synchronization of chaotic oscillators

The effects of noise on phase synchronization (PS) of coupled chaotic oscillators are explored. In contrast to coupled periodic oscillators, noise is found to enhance phase synchronization significantly below the threshold of PS. This constructive role of noise has been verified experimentally with chaotic electrochemical oscillators of the electrodissolution of Ni in sulfuric acid solution.     

Clustering of arrays of chaotic chemical oscillators by feedback and forcing

Feedback and external forcing are applied to an array of chaotic electrochemical oscillators through variations in the applied potential. We see transitions from intermittent clusters to stable chaotic clusters to stable periodic clusters to synchronized states as the feedback gain and forcing amplitude, respectively, are varied. With forcing up to four clusters are observed in stable states. The transition to synchronization with feedback occurs by the increase in the size of one cluster at the expense of the others.     

Aging transition and universal scaling in oscillator networks

Self-sustained oscillators may turn non-self-oscillatory as a result of some kind of deterioration, which we call aging for simplicity. We discuss the effect of aging on the behavior of globally and diffusively coupled oscillators which are either all periodic or chaotic. It is shown that at a certain level of aging, macroscopic oscillation stops in a way which depends on the coupling strength. A universal scaling function to describe it is analytically derived and numerically verified.     

Entanglement entropy converges to classical entropy around periodic orbits

We consider oscillators evolving subject to a periodic driving force that dynamically entangles them, and argue that this gives the linearized evolution around periodic orbits in a general chaotic Hamiltonian dynamical system. We show that the entanglement entropy, after tracing over half of the oscillators, generically asymptotes to linear growth at a rate given by the sum of the positive Lyapunov exponents of the system. These exponents give a classical entropy growth rate, in the sense of Kolmogorov, Sinai and Pesin. We also calculate the dependence of this entropy on linear mixtures of the oscillator Hilbert-space factors, to investigate the dependence of the entanglement entropy on the choice of coarse graining. We find that for almost all choices the asymptotic growth rate is the same.     

From the inverted pendulum to the periodic interface modes

The physics of the spatial propagation of monochromatic waves in periodic media is related to the temporal evolution of the parametric oscillators. We transpose the possibility that a parametric pendulum oscillates in the vicinity of its unstable equilibrium position to the case of monochromatic waves in a lossless unidimensional periodic medium. We develop this concept, that can formally applies to any kind of waves, to the case of longitudinal elastic wave. Our analysis yields us to study the propagation of monochromatic waves in a periodic structure involving two main periods. We evidence a class of phonons we refer to as periodic interface modes that propagate in these structures. These modes are similar to the optical Tamm states exhibited in photonic crystals. Our analysis is based on both a formal and an analytical approach. The application of the concept to the case of phonons in an experimentally realizable structure is given. We finally show how to control the frequencies of these phonons from the engineering of the periodic structure.     

Characterizing direction of coupling from experimental observations

We demonstrate that the direction of coupling of two interacting self-sustained electronic oscillators can be determined from the realizations of their signals. In our experiments, two electronic generators, operating in a periodic or a chaotic state, were subject to symmetrical or unidirectional coupling. In data processing, first the phases have been extracted from the observed signals and then the directionality of coupling was quantitatively estimated from the analysis of mutual dependence of the phase dynamics.     

Extraction of periodic signals in chaotic secure communication using Duffing oscillators

This paper presents a novel approach to extract the periodic signals masked by a chaotic carrier. It verifies that the driven Duffing oscillator is immune to the chaotic carrier and sensitive to certain periodic signals. A preliminary detection scenario illustrates that the frequency and amplitude of the hidden sine wave signal can be extracted from the chaotic carrier by numerical simulation. The obtained results indicate that the hidden messages in chaotic secure communication can be eavesdropped utilizing Duffing oscillators.     

Stalactite basin structure of dynamical systems with transient chaos in an invariant manifold

Dynamical systems with invariant manifolds occur in a variety of situations (e.g., identical coupled oscillators, and systems with a symmetry). We consider the case where there is both a nonchaotic attractor (e.g., a periodic orbit) and a nonattracting chaotic set (or chaotic repeller) in the invariant manifold. We consider the character of the basins for the attracting nonchaotic set in the invariant manifold and another attractor not in the invariant manifold. It is found that the boundary separating these basins has an interesting structure: The basin of the attractor not in the invariant manifold is characterized by thin cusp shaped regions ("stalactites") extending down to touch the nonattracting chaotic set in the invariant manifold. We also develop theoretical scalings applicable to these systems, and compare with numerical experiments. (c) 2000 American Institute of Physics.