Synchronization of chaotic oscillator time scales

We consider chaotic oscillator synchronization and propose a new approach for detecting the synchronized behavior of chaotic oscillators. This approach is based on analysis of different time scales in the time series generated by coupled chaotic oscillators. We show that complete synchronization, phase synchronization, lag synchronization, and generalized synchronization are particular cases of the synchronized behavior called time-scale synchronization. A quantitative measure of chaotic oscillator synchronous behavior is proposed. This approach is applied to coupled Rössler systems.     

An approach to chaotic synchronization

This paper deals with the chaotic oscillator synchronization. An approach to the synchronization of chaotic oscillators has been proposed. This approach is based on the analysis of different time scales in the time series generated by the coupled chaotic oscillators. It has been shown that complete synchronization, phase synchronization, lag synchronization, and generalized synchronization are the particular cases of the synchronized behavior called "time-scale synchronization." The quantitative measure of chaotic oscillator synchronous behavior has been proposed. This approach has been applied for the coupled R?ssler systems and two coupled Chua's circuits.     

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.     

Phase synchronization in two coupled chaotic neurons

Chaotically-spiking dynamics of Hindmarsh–Rose neurons are discussed based on a flexible definition of phase for chaotic flow. The phase synchronization of two coupled chaotic neurons is in fact the spike synchronization. As a multiple time-scale model, the coupled HR neurons have quite different behaviors from the Rössler oscillators only having single time-scale mechanism. Using such a multiple time-scale model, the phase function can detect synchronization dynamics that cannot be distinguished by cross-correlation. Moreover, simulation results show that the Lyapunov exponents cannot be used as a definite criterion for the occurrence of chaotic phase synchronization for such a system. Evaluation of the phase function shows its utility in analyzing nonlinear neural systems.     

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.     

Using white noise to enhance synchronization of coupled chaotic systems

In the paper, complete synchronization of two chaotic oscillators via unidirectional coupling determined by white noise distribution is investigated. It is analytically proved that chaos synchronization could be achieved with probability one merely via white-noise-based coupling. The established theoretical result supports the observation of an interesting phenomenon that a certain kind of white noise could enhance chaos synchronization between two chaotic oscillators. Furthermore, numerical examples are provided to illustrate some possible applications of the theoretical result.     

Analysis of phase synchronization of chaotic oscillations in terms of symbolic CTQ-analysis

The application of symbolic CTQ-analysis for studying synchronization of chaotic oscillations is considered. This approach differs substantially from its analogs since it makes it possible to diagnose and measure quantitatively the characteristics of intermittency regimes in synchronization of chaotic systems and, hence, to analyzer the temporal structure of synchronization. The application of the symbolic analysis apparatus based on the T alphabet to systems with phase locking and synchronization of time scales is demonstrated for the first time. As an example, a complex system of two mutually coupled nonidentical Rössler oscillators in the helical chaos regime with attractors having an ill-conditioned phase is considered. The results show that the method considered here makes it possible to reliably diagnose synchronism sooner than a phase locking and/or time-scale synchronization threshold is detected.     

Persistent clusters in lattices of coupled nonidentical chaotic systems

Two-dimensional (2D) lattices of diffusively coupled chaotic oscillators are studied. In previous work, it was shown that various cluster synchronization regimes exist when the oscillators are identical. Here, analytical and numerical studies allow us to conclude that these cluster synchronization regimes persist when the chaotic oscillators have slightly different parameters. In the analytical approach, the stability of almost-perfect synchronization regimes is proved via the Lyapunov function method for a wide class of systems, and the synchronization error is estimated. Examples include a 2D lattice of nonidentical Lorenz systems with scalar diffusive coupling. In the numerical study, it is shown that in lattices of Lorenz and Rossler systems the cluster synchronization regimes are stable and robust against up to 10%-15% parameter mismatch and against small noise.     

Intermittency of intermittencies at the phase synchronization boundary in the presence of noise

The intermittent behavior at the boundary of phase synchronization in the presence of noise is investigated. It is shown that in a certain range of the coupling parameter and noise intensity, the system experiences the intermittency of needle’s eye- and ring-type intermittencies. The basic results are demonstrated with two unidirectionally coupled Ressler chaotic oscillators.     

Observability of lag synchronization of coupled chaotic oscillators

Lag synchronization is a recently discovered theoretical phenomenon where the dynamical variables of two coupled, nonidentical chaotic oscillators are synchronized with a time delay relative to each other. We investigate experimentally and numerically to what extent lag synchronization can be observed in physical systems where noise is inevitable. Our measurements and numerical computation suggest that lag synchronization is typically destroyed when the noise level is comparable to the amount of average system mismatch. At small noise levels, lag synchronization occurs in an intermittent fashion.     

Hidden data transmission using generalized synchronization in the presence of noise

A way of data transmission using generalized chaotic synchronization is suggested. Since the characteristics of the valid signal are corrected by a noise generator with desired parameters, this way of data transmission offers a high degree of reliability. It is shown that noise in the method of generalized synchronization improves the privacy of the transmitted data. The basic ideas of the method are illustrated with unidirectionally coupled Ressler systems used as oscillators on the transmitter and receiver sides.     

Effect of colored noise on chains of chaotic oscillators

Equations that describe a ring system consisting of a closed circuit of n (n = 2, 3, 4, ...) unidirectionally coupled self-oscillation systems that exhibit chaotic dynamics are analyzed in the presence of external colored noise. For simplicity, detailed results of numerical calculations are presented for three oscillators. It is demonstrated that the external colored noise that is exerted upon partial oscillators of the ring system may facilitate the development of synchronous oscillations and reduce transient processes related to stabilization of chaotic synchronization. The effect is qualitatively interpreted. For comparison, numerical methods are employed to analyze the effect of external colored noise on an open circuit consisting of three oscillators.     

耦合分数阶混沌振荡器的主-从同步

近年来对分数阶系统的动力学研究得到了较为广泛的关注。本文研究了基于主-从耦合同步法的同步技术并实现了两个耦合的分数阶振荡器的混沌同步。仿真结果表明：在适当的耦合强度的调节下，该方法可实现两个耦合分数阶混沌振荡器的准确同步，且分数阶混沌振荡器的同步率明显慢于整数阶混沌振荡器的同步率；而耦合分数阶混沌振荡器在实现同步的过程中，随着阶数的提高，同步误差曲线变得平滑，这表明，系统阶数的提高改善了耦合混沌振荡器实现同步的平稳性。     

Hierarchical synchronization in complex networks with heterogeneous degrees

We study synchronization behavior in networks of coupled chaotic oscillators with heterogeneous connection degrees. Our focus is on regimes away from the complete synchronization state, when the coupling is not strong enough, when the oscillators are under the influence of noise or when the oscillators are nonidentical. We have found a hierarchical organization of the synchronization behavior with respect to the collective dynamics of the network. Oscillators with more connections (hubs) are synchronized more closely by the collective dynamics and constitute the dynamical core of the network. The numerical observation of this hierarchical synchronization is supported with an analysis based on a mean field approximation and the master stability function.     

Influence of noise on the behavior of oscillators near the synchronization boundary

Nonautonomous behavior of oscillators in the presence of noise is considered. The influence of noise on the dynamics of local zero Lyapunov exponents for nonautonomous dynamic systems that are near the synchronization boundary is considered. It is shown that the action of noise on a nonautonomous dynamic system that is near the synchronization boundary produces domains of synchronous motion in the series realization, which alternate with asynchronous domains. In accordance with this, the distribution of local zero Lyapunov exponents corresponding to laminar phases shift toward negative values. This effect is demonstrated with a discrete-time system (map of a circle onto itself) that is a reference model to describe the synchronization phenomenon and also with a reference system exhibiting chaotic dynamics (Ressler system).     

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.     

Effect of common noise on phase synchronization in coupled chaotic oscillators

We report a general phenomenon concerning the effect of noise on phase synchronization in coupled chaotic oscillators: the average phase-synchronization time exhibits a nonmonotonic behavior with the noise amplitude. In particular, we find that the time exhibits a local minimum for relatively small noise amplitude but a local maximum for stronger noise. We provide numerical results, experimental evidence from coupled chaotic circuits, and a heuristic argument to establish the generality of this phenomenon.     

Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters

This paper studies the adaptive complete synchronization of chaotic and hyperchaotic systems with fully unknown parameters. In practical situations, some systems' parameters cannot be exactly known a priori, and the uncertainties often affect the stability of the process of synchronization of the chaotic oscillators. An adaptive scheme is proposed to compensate for the effects of parameters' uncertainty based on the structure of chaotic systems in this paper. Based on the Lyapunov stability theorem, an adaptive controller and a parameters update law can be designed for the synchronization of chaotic and hyperchaotic systems. The drive and response systems can be nonidentical, even with different order. Three illustrative examples are given to demonstrate the validity of this technique, and numerical simulations are also given to show the effectiveness of the proposed chaos synchronization method. In addition, this synchronization scheme is quite robust against the effect of noise.     

Chaotic synchronization of unidirectionally coupled electron-wave media with interacting counterpropagating waves

Chaotic synchronization of two electron-wave media with interacting counterpropagating waves and cubic phase nonlinearity (transverse-field backward-wave oscillators) is studied. Analysis is based on considering a continuous set of the phases of a chaotic signal. The parameters of chaotic synchronization in a system of unidirectionally coupled backward-wave oscillators are found, and the complex dynamics of establishing the chaotic synchronization conditions in an active medium is investigated.     

Communicating with noise: How chaos and noise combine to generate secure encryption keys

An approach for the secure transmission of encrypted messages using chaos and noise is presented in this paper. The method is based on the synchronization of certain types of chaotic oscillators in response to a common noise input. This allows two distant oscillators to generate identical output which can be used as a key for encryption and decryption of a message signal. The noiselike synchronizing input-which contains no message information-is communicated to identical oscillators in the transmitter and the receiver over a public channel. The encrypted message is also sent over a public channel, while the key is never transmitted at all. The chaotic nature of the oscillators which generate the key and the randomness of the signal driving the process combine to make the recovery of the key by an eavesdropper extremely difficult. We evaluate system performance with respect to security and robustness and show that a robust and secure system can be obtained. (c) 1998 American Institute of Physics.