统一混沌系统的耦合同步

对线性耦合的两个统一混沌系统同步进行研究.基于线性时变连续系统的稳定性理论，得到初始值不同的两个统一混沌系统全局渐进同步的一种新的充分条件.另外，与已经提出的判定统一混沌系统同步的方法进行比较，发现这里得到的充分条件的约束关系少、缺少保守性，而且满足的耦合系数范围更广.将该方法应用于统一混沌系统，数值仿真表明了该法的有效性与可行性. 关键词： 统一混沌系统 耦合同步 线性时变连续系统     

Exponential stability of synchronization in asymmetrically coupled dynamical networks

Based on the original definition of the synchronization stability, a general framework is presented for investigating the exponential stability of synchronization in asymmetrically coupled networks. By choosing an appropriate Lyapunov function, we prove that the mechanism of the exponential synchronization stability is the asymmetrical coupling matrix with diffusive condition. We deduce the second largest eigenvalue of a symmetric matrix to govern the exponential stability of synchronization in asymmetrically coupled networks. Moreover, we have given the threshold value which can guarantee that the states of the asymmetrically coupled network achieve the exponential stability of synchronization.     

Exponential synchronization of coupled memristive neural networks via pinning control

This paper is concerned with the exponential synchronization problem of coupled memristive neural networks. In contrast to general neural networks, memristive neural networks exhibit state-dependent switching behaviors due to the physical properties of memristors. Under a mild topology condition, it is proved that a small fraction of controlled sub-systems can efficiently synchronize the coupled systems. The pinned subsystems are identified via a search algorithm. Moreover, the information exchange network needs not to be undirected or strongly connected. Finally, two numerical simulations are performed to verify the usefulness and effectiveness of our results.     

Exponential synchronization of complex networks with Markovian jump and mixed delays

In this Letter, we investigate the exponential synchronization problem for an array of N linearly coupled complex networks with Markovian jump and mixed time-delays. The complex network consists of m modes and the network switches from one mode to another according to a Markovian chain with known transition probability. The mixed time-delays are composed of discrete and distributed delays, both of which are mode-dependent. The nonlinearities imbedded with the complex networks are assumed to satisfy the sector condition that is more general than the commonly used Lipschitz condition. By making use of the Kronecker product and the stochastic analysis tool, we propose a novel Lyapunov–Krasovskii functional suitable for handling distributed delays and then show that the addressed synchronization problem is solvable if a set of linear matrix inequalities (LMIs) are feasible. Therefore, a unified LMI approach is developed to establish sufficient conditions for the coupled complex network to be globally exponentially synchronized in the mean square. Note that the LMIs can be easily solved by using the Matlab LMI toolbox and no tuning of parameters is required. A simulation example is provided to demonstrate the usefulness of the main results obtained.     

Synchronization of a network coupled with complex-variable chaotic systems

In this paper, synchronization of a network coupled with complex-variable chaotic systems is investigated. Adaptive feedback control and intermittent control schemes are adopted for achieving adaptive synchronization and exponential synchronization, respectively. Several synchronization criteria are established. In these schemes, the outer coupling matrix is not necessarily assumed to be symmetric or irreducible. Further, for a class of networks with an irreducible and balanced outer coupling matrix, a pinning control scheme is adopted for achieving synchronization. Numerical simulations are demonstrated to verify the effectiveness of the theoretical results.     

New eigenvalue based approach to synchronization in asymmetrically coupled networks

Locally and globally exponential stability of synchronization in asymmetrically nonlinear coupled networks and linear coupled networks are investigated in this paper, respectively. Some new synchronization stability criteria based on eigenvalues are derived. In these criteria, both a term that is the second largest eigenvalue of a symmetrical matrix and a term that is the largest value of the sum of the column of the asymmetrical coupling matrix play a key role. Comparing with existing results, the advantage of our synchronization stability results is that they can be analytically applied to the asymmetrically coupled networks and can overcome the complexity of calculating eigenvalues of the coupling asymmetric matrix. Therefore, these conditions are very convenient to use. Moreover, a necessary condition of globally exponential synchronization stability criterion is also given by the elements of the coupling asymmetric matrix, which can conveniently be used in judging the synchronization stability condition without calculating the eigenvalues of the coupling matrix.     

Synchronization rate of synchronized coupled systems

Synchronization phenomena, an emergent property in networks of interacting dynamical elements, are widely observed in nature, and have become the subject of intense research. Here we will investigate the synchronization rate problem in coupled limit-cycle and chaotic oscillators. Based on the mode decomposition method and Gershgörin's discs theorem, some sufficient conditions for synchronization of coupled systems are obtained, and a synchronization rate is then derived. Such a synchronization rate indicates that the error functions between state variables of underlying individual systems tend to zero in the exponential form as time tends to the infinity. Several numerical examples are also given to validate the theoretical results.     

关于非自治系统三类广义同步存在性的研究

研究了两个单向耦合的非自治系统三类广义同步的存在性.在响应系统的修正方程具有渐近稳定平衡点、渐近稳定周期轨道或渐近稳定拟周期轨道的情况下,满足一定的条件,可将广义同步化流形存在性问题转化为Lipschitz函数族的压缩不动点问题,并且理论证明了该广义同步化流形的指数吸引性.同时,以Duffing系统为例进行了数值仿真,其结果与理论推导相一致. 关键词： 广义同步化流形 压缩不动点 指数吸引性     

Synchronization in an array of linearly stochastically coupled networks with time delays

In this paper, the complete synchronization problem is investigated in an array of linearly stochastically coupled identical networks with time delays. The stochastic coupling term, which can reflect a more realistic dynamical behavior of coupled systems in practice, is introduced to model a coupled system, and the influence from the stochastic noises on the array of coupled delayed neural networks is studied thoroughly. Based on a simple adaptive feedback control scheme and some stochastic analysis techniques, several sufficient conditions are developed to guarantee the synchronization in an array of linearly stochastically coupled neural networks with time delays. Finally, an illustrate example with numerical simulations is exploited to show the effectiveness of the theoretical results.     

Global synchronization in arrays of delayed neural networks with constant and delayed coupling

This Letter investigates the global exponential synchronization in arrays of coupled identical delayed neural networks (DNNs) with constant and delayed coupling. By referring to Lyapunov functional method and Kronecker product technique, some sufficient conditions are derived for global synchronization of such systems. These new synchronization criteria offer some adjustable matrix parameters, which is of important significance in the design and applications of such coupled DNNs, and the results improve and extend the earlier works. Finally, an example is given to illustrate the theoretical results.     

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.     

Inverse chaos synchronization between bidirectionally coupled variable multiple time delay systems

We make first report on inverse chaos synchronization between bidirectionally non-linearly and linearly coupled variable multiple time delay laser systems governed by the Ikeda model. The results are of certain importance in secure chaos-based communication systems.     

A new kind of nonlinear phenomenon in coupled fractional-order chaotic systems: coexistence of anti-phase and complete synchronization

In this paper,we have found a kind of interesting nonlinear phenomenon-hybrid synchronization in linearly coupled fractional-order chaotic systems.This new synchronization mechanism,i.e.,part of state variables are anti-phase synchronized and part completely synchronized,can be achieved using a single linear controller with only one drive variable.Based on the stability theory of the fractional-order system,we investigated the possible existence of this new synchronization mechanism.Moreover,a helpful theorem,serving as a determinant for the gain of the controller,is also presented.Solutions of coupled systems are obtained numerically by an improved Adams-Bashforth-Moulton algorithm.To support our theoretical analysis,simulation results are given.     

A Lie algebraic condition for exponential stability of discrete hybrid systems and application to hybrid synchronization

A Lie algebraic condition for global exponential stability of linear discrete switched impulsive systems is presented in this paper. By considering a Lie algebra generated by all subsystem matrices and impulsive matrices, when not all of these matrices are Schur stable, we derive new criteria for global exponential stability of linear discrete switched impulsive systems. Moreover, simple sufficient conditions in terms of Lie algebra are established for the synchronization of nonlinear discrete systems using a hybrid switching and impulsive control. As an application, discrete chaotic system's synchronization is investigated by the proposed method.     

Synchronization in arrays of coupled self-induced friction oscillators

We investigate synchronization phenomena in systems of self-induced dry friction oscillators with kinematic excitation coupled by linear springs. Friction force is modelled according to exponential model. Initially, a single degree of freedom mass-spring system on a moving belt is considered to check the type of motion of the system (periodic, non-periodic). Then the system is coupled in chain of identical oscillators starting from two, up to four oscillators. A reference probe of two coupled oscillators is applied in order to detect synchronization thresholds for both periodic and non-periodic motion of the system. The master stability function is applied to predict the synchronization thresholds for longer chains of oscillators basing on two oscillator probe. It is shown that synchronization is possible both for three and four coupled oscillators under certain circumstances. Our results confirmed that this technique can be also applied for the systems with discontinuities.     

Stochastic synchronization of coupled neural networks with intermittent control

In this Letter, we study the exponential stochastic synchronization problem for coupled neural networks with stochastic noise perturbations. Based on Lyapunov stability theory, inequality techniques, the properties of Weiner process, and adding different intermittent controllers, several sufficient conditions are obtained to ensure exponential stochastic synchronization of coupled neural networks with or without coupling delays under stochastic perturbations. These stochastic synchronization criteria are expressed in terms of several lower-dimensional linear matrix inequalities (LMIs) and can be easily verified. Moreover, the results of this Letter are applicable to both directed and undirected weighted networks. A numerical example and its simulations are offered to show the effectiveness of our new results.     

Experimental and numerical study on the basin stability of the coupled metronomes

The basin stability is an effective parameter to measure the stability of multistable system under perturbations. In this paper, we try to explore the effects of the coupling strength on the basin stability of the coupled metronomes. In two coupled non-identical metronomes, the coupling strength linearly decreases the basin stability of in-phase synchronization while increases that of the anti-phase synchronization. In three coupled metronomes, there are rich coexisting collectively dynamics as in-phase, anti-phase synchronization, quasi-period states and period 4 states. The coupling strength may still change the basin stability of these coexisting dynamics states. The results are observed in experimental systems and numerical models. Our findings are significant on understanding the multistable dynamics under noisy environment.     

Complete and generalized synchronization in a class of noise perturbed chaotic systems

In the paper, in light of the LaSalle-type invariance principle for stochastic differential equations, chaos synchronization is investigated for a class of chaotic systems dissatisfying a globally Lipschitz condition with noise perturbation. Sufficient criteria for both complete synchronization and generalized synchronization are rigorously established and thus successfully applied to realize chaos synchronization in the coupled unified chaotic systems. Furthermore, concrete examples as well as their numerical simulations are provided to illustrate the possible application of the established criteria.     

Synchronization of stochastically hybrid coupled neural networks with coupling discrete and distributed time-varying delays

A general model of linearly stochastically coupled identical connected neural networks with hybrid coupling is proposed, which is composed of constant coupling, coupling discrete time-varying delay and coupling distributed timevarying delay. All the coupling terms are subjected to stochastic disturbances described in terms of Brownian motion, which reflects a more realistic dynamical behaviour of coupled systems in practice. Based on a simple adaptive feedback controller and stochastic stability theory, several sufficient criteria are presented to ensure the synchronization of linearly stochastically coupled complex networks with coupling mixed time-varying delays. Finally, numerical simulations illustrated by scale-free complex networks verify the effectiveness of the proposed controllers.     

Synchronization analysis of singular hybrid coupled networks

In this Letter, singular hybrid coupled systems are introduced to describe complex networks with a special class of constraints. The synchronization problem of singular hybrid coupled systems with time-varying nonlinear perturbation is investigated. A sufficient condition for global synchronization is derived based on the Lyapunov stability theory. The singular system is regular and impulse free. Finally, a numerical example is provided to illustrate the effectiveness of the proposal conditions.