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.     

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.     

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.     

Coherence of a Chain of Chaotic Oscillators in a Nonlinear Gyrotropic Medium

We investigate the generalized Kuhn's model, namely, a chain of nonlinear chaotic oscillators describing a nonlinear gyrotropic medium. It is shown that despite the chaotic behavior of separate oscillators the chain preserves some coherency as a whole. The relation between the chain synchronization and the physical properties of a random mean field is established.     

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.     

Isochronal synchronization in networks and chaos-based TDMA communication

Pairs of delay-coupled chaotic systems were shown to be able to achieve isochronal synchronization under bidirectional coupling and self-feedback. Such identical-in-time behavior was demonstrated to be stable under a set of conditions and to support simultaneous bidirectional communication between pairs of chaotic oscillators coupled with time-delay. More recently, it was shown that isochronal synchronization can emerge in networks with several hundreds of oscillators, which allows its exploitation for communication in distributed systems. In this paper, we introduce a conceptual framework for the application of isochronal synchronization to TDMA communication in networks of delay-coupled chaotic oscillators. On the basis of the stable and identical-in-time behavior of delay-coupled chaotic systems, the chaotic dynamics of distributed oscillators is used to support and sustain coordinate communication among nodes over the network. On the basis of the unique features of chaotic systems in isochronal synchronization, the chaotic signals are used to timestamp clock readings at the physical layer such that logical clock synchronization among the nodes (a prerequisite for TDMA) can be exploited using the same basic structure. The result is a standalone network communication scheme that can be advantageously applied in the context of ad-hoc networks or alike, especially short-ranged ones that yield low values of time-delay. As explored to its depths in practical implementations, this conceptual framework is argued to have potential to provide gain in simplicity, security and efficiency in communication schemes for autonomous/standalone network applications.     

混沌振子的广义旋转数和同步混沌的Hopf分岔

对应于混沌振子的各个Lyapunov指数,在切空间中定义了广义相位和广义旋转数．广义旋转数和Lyapunov指数相结合,可以更完整地描述混沌吸引子的各个运动模式的运动特征,包括伸缩与旋转．用耦合Duffing振子研究了时空混沌系统在同步混沌失稳时发生的分岔行为．结果表明,耦合振子的同步混沌态可以发生一种Hopf分岔,在Hopf分岔后,系统的功率谱中出现了一个特征频率,其值恰好等于分岔前临界横模的广义旋转数． 关键词：     

Multistability and chaotic beating of Duffing oscillators suspended on an elastic structure

We consider the dynamics of a number of externally excited chaotic oscillators suspended on an elastic structure. We show that for the given conditions of oscillations of the structure, initially uncorrelated chaotic oscillators become periodic and synchronous in clusters. In the periodic regime, we have observed multistability as two or four different attractors coexist in each cluster. A mismatch of the excitation frequency in the oscillators leads to the beating-like behaviour. We argue that the observed phenomena are generic in the parameter space and independent of the number of oscillators and their location on the elastic structure.     

First order transition to oscillation death through an environment

The emergence of dynamical abrupt transitions for the first time in an ensemble of identical limit-cycle and chaotic oscillators coupled via a common environment is reported. The transition from the oscillatory state to the death state and vice versa, in these networks of oscillators are found not only discontinuous as well as irreversible in the parameter space. This first order phase transition in these systems is termed as Explosive Death. The occurrence of such transition is studied in details by using an appropriate order parameter for both limit-cycle and chaotic oscillators, in particular, Stuart–Landau and Rössler oscillators. The backward transition point for this phenomenon is obtained analytically using linear stability analysis and is found to be consistent with the numerical results.     

Finite-time synchronization of fractional-order simplest two-component chaotic oscillators

The problem of finite-time synchronization of fractional-order simplest two-component chaotic oscillators operating at high frequency and application to digital cryptography is addressed. After the investigation of numerical chaotic behavior in the system, an adaptive feedback controller is designed to achieve the finite-time synchronization of two oscillators, based on the Lyapunov function. This controller could find application in many other fractional-order chaotic circuits. Applying synchronized fractional-order systems in digital cryptography, a well secured key system is obtained. Numerical simulations are given to illustrate and verify the analytic results.     

Studies of thermal conductivity in Fermi-Pasta-Ulam-like lattices

The pioneering computer simulations of the energy relaxation mechanisms performed by Fermi, Pasta, and Ulam (FPU) can be considered as the first attempt of understanding energy relaxation and thus heat conduction in lattices of nonlinear oscillators. In this paper we describe the most recent achievements about the divergence of heat conductivity with the system size in one-dimensional (1D) and two-dimensional FPU-like lattices. The anomalous behavior is particularly evident at low energies, where it is enhanced by the quasiharmonic character of the lattice dynamics. Remarkably, anomalies persist also in the strongly chaotic region where long-time tails develop in the current autocorrelation function. A modal analysis of the 1D case is also presented in order to gain further insight about the role played by boundary conditions.     

Amplitude and phase effects on the synchronization of delay-coupled oscillators

We consider the behavior of Stuart-Landau oscillators as generic limit-cycle oscillators when they are interacting with delay. We investigate the role of amplitude and phase instabilities in producing symmetry-breaking/restoring transitions. Using analytical and numerical methods we compare the dynamics of one oscillator with delayed feedback, two oscillators mutually coupled with delay, and two delay-coupled elements with self-feedback. Taking only the phase dynamics into account, no chaotic dynamics is observed, and the stability of the identical synchronization solution is the same in each of the three studied networks of delay-coupled elements. When allowing for a variable oscillation amplitude, the delay can induce amplitude instabilities. We provide analytical proof that, in case of two mutually coupled elements, the onset of an amplitude instability always results in antiphase oscillations, leading to a leader-laggard behavior in the chaotic regime. Adding self-feedback with the same strength and delay as the coupling stabilizes the system in the transverse direction and, thus, promotes the onset of identically synchronized behavior.     

Automatic control of phase synchronization in coupled complex oscillators

We present an automatic control method for phase locking of regular and chaotic nonidentical oscillations, when all subsystems interact via feedback. This method is based on the well known principle of feedback control which takes place in nature and is successfully used in engineering. In contrast to unidirectional and bidirectional coupling, the approach presented here supposes the existence of a special controller, which allows to change the parameters of the controlled systems. First we discuss general principles of automatic phase synchronization (PS) for arbitrary coupled systems with a controller whose input is given by a special quadratic form of coordinates of the individual systems and its output is a result of the application of a linear differential operator. We demonstrate the effectiveness of our approach for controlled PS on several examples: (i) two coupled regular oscillators, (ii) coupled regular and chaotic oscillators, (iii) two coupled chaotic Rössler oscillators, (iv) two coupled foodweb models, (v) coupled chaotic Rössler and Lorenz oscillators, (vi) ensembles of locally coupled regular oscillators, (vii) ensembles of locally coupled chaotic oscillators, and (viii) ensembles of globally coupled chaotic oscillators.     

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.     

耦合混沌振子系统完全同步的动力学行为

以耦合Duffing振子为对象，研究了混沌系统进入完全同步态时的一些动力学行为. 在对称耦合情况下，随着耦合系数的变化系统达到各个混沌振子的相轨道完全相同的同步态——完全同步态. 通过计算Lyapunov指数表明，此时系统的前两个横向Lyapunov指数相等，同时系统之间的时间关联表现出明显的规律性. 关键词： Duffing振子 混沌同步 Lyapunov指数     

Spatiotemporal synchronization of coupled oscillators in a laboratory plasma

The spatiotemporal synchronization between two plasma instabilities of autonomous glow discharge tubes is observed experimentally. For this purpose, two tubes are placed separately and two chaotic waves interact with each other through a coupler. When the coupling strength is changed, the coupled oscillators exhibit synchronization in time and space. This is the first experimental evidence of spatiotemporal synchronization by mutual chaotic wave interaction in plasma.     

Experimental study of the transitions between synchronous chaos and a periodic rotating wave

In this work we characterize experimentally the transition between periodic rotating waves and synchronized chaos in a ring of unidirectionally coupled Lorenz oscillators by means of electronic circuits. The study is complemented by numerical and theoretical analysis, and the intermediate states and their transitions are identified. The route linking periodic behavior with synchronous chaos involves quasiperiodic behavior and a type of high-dimensional chaos known as chaotic rotating wave. The high-dimensional chaotic behavior is characterized, and is shown to be composed actually by three different behaviors. The experimental study confirms the robustness of this route.     

Stochastic and chaotic relaxation oscillations

For relaxation oscillators stochastic and chaotic dynamics are investigated. The effect of random perturbations upon the period is computed. For an extended system with additional state variables chaotic behavior can be expected. As an example, the Van der Pol oscillator is changed into a third-order system admitting period doubling and chaos in a certain parameter range. The distinction between chaotic oscillation and oscillation with noise is explored. Return maps, power spectra, and Lyapunov exponents are analyzed for that purpose.     

The effect of time-delay on anomalous phase synchronization

Anomalous phase synchronization in nonidentical interacting oscillators is manifest as the increase of frequency disorder prior to synchronization. We show that this effect can be enhanced when a time-delay is included in the coupling. In systems of limit-cycle and chaotic oscillators we find that the regions of phase disorder and phase synchronization can be interwoven in the parameter space such that as a function of coupling or time-delay the system shows transitions from phase ordering to disorder and back.     

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.