Concepts of synchronization in dynamical systems

A general discussion is presented on the concepts of synchronization in finite dimensional dynamical systems with the intention of answering the question that puzzles some people of how to give a rigorous unified notion for describing the various synchronization phenomena in physical systems.     

两个不相同系统的广义同步化

采用Temam无穷维动力系统的惯性流形理论，证明了两个不相同的系统能实现广义同步化.在两个系统具有吸收集和吸引子的基础上，通过定义在一个函数类上的压缩映射的不动点得到了广义同步化流形，该流形是Lipschitz光滑流形，而且具有不变性与指数吸引性.数值仿真证实了所建立理论的正确性.     

Synchronization of moving chaotic agents

We consider a set of mobile agents in a two dimensional space, each one of them carrying a chaotic oscillator, and discuss the related synchronization issues under the framework of time-variant networks. In particular, we show that, as far as the time scale for the motion of the agents is much shorter than that of the associated dynamical systems, the global behavior can be characterized by a scaled all-to-all Laplacian matrix, and the synchronization conditions depend on the agent density on the plane.     

On <Emphasis Type="Bold">Λ</Emphasis> −<Emphasis Type="Bold">ϕ</Emphasis> generalized synchronization of chaotic dynamical systems in continuous–time

In this paper, a new type of chaos synchronization in continuous-time is proposed by combining inverse matrix projective synchronization (IMPS) and generalized synchronization (GS). This new chaos synchronization type allows us to study synchronization between different dimensional continuous-time chaotic systems in different dimensions. Based on stability property of integer-order linear continuous-time dynamical systems and Lyapunov stability theory, effective control schemes are introduced and new synchronization criterions are derived. Numerical simulations are used to validate the theoretical results and to verify the effectiveness of the proposed schemes.     

Projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks

We study projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random networks. We relax some limitations of previous work, where projective-anticipating and projective-lag synchronization can be achieved only on two coupled chaotic systems. In this paper, we realize projective-anticipating and projective-lag synchronization on complex dynamical networks composed of a large number of interconnected components. At the same time, although previous work studied projective synchronization on complex dynamical networks, the dynamics of the nodes are coupled partially linear chaotic systems. In this paper, the dynamics of the nodes of the complex networks are time-delayed chaotic systems without the limitation of the partial linearity. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective-anticipating, projective, and projective-lag synchronization of time-delayed chaotic systems on random dynamical networks, and we find both its existence and sufficient stability conditions. The validity of the proposed method is demonstrated and verified by examining specific examples using Ikeda and Mackey-Glass systems on Erdos-Renyi networks.     

Localized error bursts in estimating the state of spatiotemporal chaos

We consider the problem of estimating the current state of an evolving spatiotemporally chaotic system from noisy observations of the system state and a model of the system dynamics. Using a simple scheme for state estimation, we show the possible occurrence of temporally and spatially intermittent large bursts in the estimation error. We discuss the similarity of these bursts with those occurring at the bubbling transition in the synchronization of low dimensional chaotic dynamical systems. We characterize the spatial and temporal behavior of the bursts and investigate how the behavior changes as we vary the number and location of the observations.     

复杂网络上动力系统同步的研究进展

本文简要介绍复杂网络的基本概念并详细总结了近年来复杂网络上动力学系统的同步的研究进展，主要内容有复杂网络同步的稳定性分析，复杂网络上动力学系统同步的特点，网络的几何特征量对同步稳定性的影响，以及提高网络同步能力的方法等。最后文章提出了这一领域的几个有待解决的问题及可能的发展方向。     

采用振幅耦合方法研究多旋转中心混沌振子的相同步

为了找到具有多个旋转中心的混沌系统的相同步与其动力学拓朴变化之间的对应关系，采用线性振幅线性耦合方法，研究了Lorenz系统和Duffing系统的相同步，首先对Lorenz系统和Duffing系统分别进行极坐标变换，在线性振幅耦合基础上计算了两个系统的平均旋转数和Lyapunov指数，发现，随耦合强度的增大，系统相同步与系统的Lyapunov指数跃变存在一一对应的关系，这表明具有多个旋转中心的混沌系统的相同步与系统动力学拓朴变化也存在着对应关系. 关键词： Lyapunov指数 振幅耦合 相同步     

Frequency locking by external force from a dynamical system with strange nonchaotic attractor

Usually, phase synchronization is studied in chaotic systems driven by either periodic force or chaotic force. In the present work, we consider frequency locking in chaotic Rössler oscillator by a special driving force from a dynamical system with a strange nonchaotic attractor. In this case, a transition from generalized marginal synchronization to frequency locking is observed. We investigate the bifurcation of the dynamical system and explain why generalized marginal synchronization can occur in this model.     

Hyperbolification of dynamical systems: The case of continuous-time systems

We present a new method to generate chaotic hyperbolic systems. The method is based on the knowledge of a chaotic hyperbolic system and the use of a synchronization technique. This procedure is called hyperbolification of dynamical systems. The aim of this process is to create or enhance the hyperbolicity of a dynamical system. In other words, hyperbolification of dynamical systems produces chaotic hyperbolic (structurally stable) behaviors in a system that would not otherwise be hyperbolic. The method of hyperbolification can be outlined as follows. We consider a known n-dimensional hyperbolic chaotic system as a drive system and another n-dimensional system as the response system plus a feedback control function to be determined in accordance with a specific synchronization criterion. We then consider the error system and apply a synchronization method, and find sufficient conditions for the errors to converge to zero and hence the synchronization between the two systems to be established. This means that we construct a 2n-dimensional continuous-time system that displays a robust hyperbolic chaotic attractor. An illustrative example is given to show the effectiveness of the proposed hyperbolification method.     

Control and synchronization of chaos in high dimensional systems: Review of some recent results

Controlling chaos and synchronization of chaos have evolved for a number of years as essentially two separate areas of research. Only recently it has been realized that both subjects share a common root in control theory. In addition, as limitations of low dimensional chaotic systems in modeling real world phenomena become increasingly apparent, investigations into the control and synchronization of high dimensional chaotic systems are beginning to attract more interest. We review some recent advances in control and synchronization of chaos in high dimensional systems. Efforts will be made to stress the common origins of the two subjects. (c) 1997 American Institute of Physics.     

Complexity and synchronization in stochastic chaotic systems

We investigate the complexity of a hyperchaotic dynamical system perturbed by noise and various nonlinear speech and music signals. The complexity is measured by the weighted recurrence entropy of the hyperchaotic and stochastic systems. The synchronization phenomenon between two stochastic systems with complex coupling is also investigated. These criteria are tested on chaotic and perturbed systems by mean conditional recurrence and normalized synchronization error. Numerical results including surface plots, normalized synchronization errors, complexity variations etc show the effectiveness of the proposed analysis.     

Synchronization processes in complex networks

During the last decades the emergence of collective dynamics in large networks of coupled units has been investigated in fields such as optics, chemistry, biology and ecology. Recently, complex networks have provided a challenging framework for the study of synchronization of dynamical units, based on the interplay between complexity in the overall topology and local dynamical properties of the coupled units. In this work, we review the constructive role played by such complex wirings for the synchronization of networks of coupled dynamical systems. We review the main techniques that have been proposed for assessing the propensity for synchronization (synchronizability) of a given networked system. We will also describe the main applications, especially in the view of selecting the optimal topology in the coupling configuration that provides enhancement of the synchronization features.     

An approach for characterizing coupling in dynamical systems

The study of coupling in dynamical systems dates back to Christian Hyugens who, in 1665, discovered that pendulum clocks with the same length pendulum synchronize when they are near to each other. In that case the observed synchronous motion was out of phase. In this paper we propose a new approach for measuring the degree of coupling and synchronization of a dynamical system consisting of interacting subsystems. The measure is based on quantifying the active degrees of freedom (e.g. correlation dimension) of the coupled system and the constituent subsystems. The time-delay embedding scheme is extended to coupled systems and used for attractor reconstruction of the coupled dynamical system. We use the coupled Lorenz, Rossler and Hénon model systems with a coupling strength variable for evaluation of the proposed approach. Results show that we can measure the active degrees of freedom of the coupled dynamical systems and can quantify and distinguish the degree of synchronization or coupling in each of the dynamical systems studied. Furthermore, using this approach the direction of coupling can be determined.     

Synchronization of Rossler and Chen chaotic dynamical systems using active control

This Letter presents chaos synchronization of two identical Rossler and Chen systems by using active control. The proposed technique is applied to achieve chaos synchronization for the Rossler and Chen dynamical systems. We demonstrate that a coupled Rossler and Chen systems can be synchronized. Numerical simulations are used to show the effectiveness of the proposed control method.     

模态分解法在非恒同耦合系统同步研究中的推广

通过改变耦合函数将模态分解法进行了推广,应用于非恒同耦合系统同步的研究.详细研究了周期吸引子、概周期吸引子等非恒同耦合系统的同步,得到了同步的局部渐近稳定性条件.并进行了数值模拟,发现同步时动力学现象丰富.概周期吸引子耦合系统会出现稳定的周期、概周期同步解,由于耦合周期吸引子耦合系统会出现多个稳定的周期同步解,且其吸引域差别较大,均出现了同步的多值性.同时也验证了该方法的正确性.     

Synchronization of spatiotemporal chemical chaos using random signals

We report the synchronization of two uncoupled spatially extended chemical systems by superimposing identical external random signals to both of them. In one spatial dimension, under appropriate parameter conditions the model systems exhibits a transition to turbulence via backfiring of pulses. Implementing the non-vanishing random signal control to the underlying partial differential equations, synchronization is achieved not only for identical systems, but also for systems operating under unequal parameter values exhibiting a different dynamical behavior (generalized synchronization). Finally, synchronization is also achieved under the influence of a random signal superimposed globally, thus making it relevant to experimental situations.     

Coherence resonance and stochastic synchronization in a nonlinear circuit near a subcritical Hopf bifurcation

We analyze noise-induced phenomena in nonlinear dynamical systems near a subcritical Hopf bifurcation. We investigate qualitative changes of probability distributions (stochastic bifurcations), coherence resonance, and stochastic synchronization. These effects are studied in dynamical systems for which a subcritical Hopf bifurcation occurs. We perform analytical calculations, numerical simulations and experiments on an electronic circuit. For the generalized Van der Pol model we uncover the similarities between the behavior of a self-sustained oscillator characterized by a subcritical Hopf bifurcation and an excitable system. The analogy is manifested through coherence resonance and stochastic synchronization. In particular, we show both experimentally and numerically that stochastic oscillations that appear due to noise in a system with hard excitation, can be partially synchronized even outside the oscillatory regime of the deterministic system.     

Synchronization of chaotic systems: Transverse stability of trajectories in invariant manifolds

We examine synchronization of identical chaotic systems coupled in a drive/response manner. A rigorous criterion is presented which, if satisfied, guarantees that synchronization to the driving trajectory is linearly stable to perturbations. An easy to use approximate criterion for estimating linear stability is also presented. One major advantage of these criteria is that, for simple systems, many of the calculations needed to implement them can be performed analytically. Geometrical interpretations of the criterion are discussed, as well as how they may be used to investigate synchronization between mutual coupled systems and the stability of invariant manifolds within a dynamical system. Finally, the relationship between our criterion and results from control theory are discussed. Analytical and numerical results from tests of these criteria on four different dynamical systems are presented. (c) 1997 American Institute of Physics.