非线性函数耦合的Chen吸引子网络的混沌同步

利用非对称非线性函数耦合混沌同步方法,讨论了Chen吸引子的混沌同步问题,数值模拟分析初始值和耦合强度因子的选择对于实现混沌同步的影响. 将非对称非线性函数耦合同步方法进一步推广发展到完全连接网络和由星形子网络构成的复杂大网络混沌同步的研究中. 提供了确定网络中神经元之间混沌同步状态稳定性的误差发展方程,并讨论各个耦合强度因子对网络同步稳定性过程的影响,给出了相应的稳定性范围. 通过数值模拟证明利用非线性函数作为耦合函数,实现完全连接网络、星形子网络构成大网络的混沌同步是有效的. 可以预测在网络的混沌同步 关键词： 非线性耦合函数 Chen吸引子 混沌同步 网络     

Generating and enhancing lag synchronization of chaotic systems by white noise

In this paper, we study the crucial impact of white noise on lag synchronous regime in a pair of time-delay unidirectionally coupled systems. Our result demonstrates that merely via white-noise-based coupling lag synchronization could be achieved between the coupled systems (chaotic or not). And it is also demonstrated that a conventional lag synchronous regime can be enhanced by white noise. Sufficient conditions are further proved mathematically for noise-inducing and noise-enhancing lag synchronization, respectively. Additionally, the influence of parameter mismatch on the proposed lag synchronous regime is studied, by which we announce the robustness and validity of the new strategy. Two numerical examples are provided to illustrate the validity and some possible applications of the theoretical result.     

一种参数摄动的混沌异结构同步方法

研究了参数摄动情形下的混沌异结构同步问题,基于Lyapunov稳定性定理并结合范数理论给出了系统参数摄动下实现混沌异结构同步的一个充分条件,为同步控制器的设计提供了一般方法.只要两混沌系统维数相等,状态变量可测,就可利用所提方法实现系统参数摄动下的异结构同步,并能够保证在同步实现后同步控制量伴随误差变量一同收敛至零.该方法鲁棒性强,适用范围广,通过对混沌系统、超混沌系统的同步仿真,证实了该方法的有效性. 关键词： 混沌 超混沌 同步 Lyapunov函数     

Complete synchronization between two bi-directionally coupled chaotic systems via an adaptive feedback controller

In this paper, we apply a simple adaptive feedback control scheme to synchronize two bi-directionally coupled chaotic systems. Based on the invariance principle of differential equations, sufficient conditions for the global asymptotic synchronization between two bi-directionally coupled chaotic systems via an adaptive feedback controller are given. Unlike other control schemes for bi-directionally coupled systems, this scheme is very simple to implement in practice and need not consider coupling terms. As examples, the autonomous hyperchaotic Chen systems and the new non-autonomous 4D systems are illustrated. Numerical simulations show that the proposed method is effective and robust against the effect of weak noise.     

Noise-induced phase synchronization and synchronization transitions in chaotic oscillators

Whether common noise can induce complete synchronization in chaotic systems has been a topic of great relevance and long-standing controversy. We first clarify the mechanism of this phenomenon and show that the existence of a significant contraction region, where nearby trajectories converge, plays a decisive role. Second, we demonstrate that, more generally, common noise can induce phase synchronization in nonidentical chaotic systems. Such a noise-induced synchronization and synchronization transitions are of special significance for understanding neuron encoding in neurobiology.     

Fundamentals of synchronization in chaotic systems,concepts, and applications

The field of chaotic synchronization has grown considerably since its advent in 1990. Several subdisciplines and "cottage industries" have emerged that have taken on bona fide lives of their own. Our purpose in this paper is to collect results from these various areas in a review article format with a tutorial emphasis. Fundamentals of chaotic synchronization are reviewed first with emphases on the geometry of synchronization and stability criteria. Several widely used coupling configurations are examined and, when available, experimental demonstrations of their success (generally with chaotic circuit systems) are described. Particular focus is given to the recent notion of synchronous substitution-a method to synchronize chaotic systems using a larger class of scalar chaotic coupling signals than previously thought possible. Connections between this technique and well-known control theory results are also outlined. Extensions of the technique are presented that allow so-called hyperchaotic systems (systems with more than one positive Lyapunov exponent) to be synchronized. Several proposals for "secure" communication schemes have been advanced; major ones are reviewed and their strengths and weaknesses are touched upon. Arrays of coupled chaotic systems have received a great deal of attention lately and have spawned a host of interesting and, in some cases, counterintuitive phenomena including bursting above synchronization thresholds, destabilizing transitions as coupling increases (short-wavelength bifurcations), and riddled basins. In addition, a general mathematical framework for analyzing the stability of arrays with arbitrary coupling configurations is outlined. Finally, the topic of generalized synchronization is discussed, along with data analysis techniques that can be used to decide whether two systems satisfy the mathematical requirements of generalized synchronization. (c) 1997 American Institute of Physics.     

Robustness of optimal synchronization in real networks

Experimental studies can provide powerful insights into the physics of complex networks. Here, we report experimental results on the influence of connection topology on synchronization in fiber-optic networks of chaotic optoelectronic oscillators. We find that the recently predicted nonmonotonic, cusplike synchronization landscape manifests itself in the rate of convergence to the synchronous state. We also observe that networks with the same number of nodes, same number of links, and identical eigenvalues of the coupling matrix can exhibit fundamentally different approaches to synchronization. This previously unnoticed difference is determined by the degeneracy of associated eigenvectors in the presence of noise and mismatches encountered in real-world conditions.     

Synchronization between fractional-order chaotic systems and integer orders chaotic systems (fractional-order chaotic systems)

Based on the idea of tracking control and stability theory of fractional-order systems, a controller is designed to synchronize the fractional-order chaotic system with chaotic systems of integer orders, and synchronize the different fractional-order chaotic systems. The proposed synchronization approach in this paper shows that the synchronization between fractional-order chaotic systems and chaotic systems of integer orders can be achieved, and the synchronization between different fractional-order chaotic systems can also be realized. Numerical experiments show that the present method works very well.     

Analytical and numerical studies of noise-induced synchronization of chaotic systems

We study the effect that the injection of a common source of noise has on the trajectories of chaotic systems, addressing some contradictory results present in the literature. We present particular examples of one-dimensional maps and the Lorenz system, both in the chaotic region, and give numerical evidence showing that the addition of a common noise to different trajectories, which start from different initial conditions, leads eventually to their perfect synchronization. When synchronization occurs, the largest Lyapunov exponent becomes negative. For a simple map we are able to show this phenomenon analytically. Finally, we analyze the structural stability of the phenomenon. (c) 2001 American Institute of Physics.     

Chaotic synchronization: a nonlinear predictive filtering approach

The synchronization of chaotic systems is a difficult task due to their sensitive dependence on the initial conditions. Perfect synchronization is almost impossible when noise is present in the system. One of the well known stochastic filtering algorithms that is used to synchronize chaotic systems in the presence of noise is the extended Kalman filter (EKF). However, for highly nonlinear systems, the approximation error introduced by the EKF has been shown to be relatively high. In this paper, a nonlinear predictive filter (NPF) is proposed for synchronizing chaotic systems. In this scheme, it is not required to approximate the underlying nonlinearity and hence there is no need to compute the Jacobian of the chaotic system. Numerical simulations are carried out to compare the performances of the NPF and EKF algorithms for synchronizing different sets of chaotic systems and/or maps. The well known Lorenz and Mackey-Glass systems as well as Ikeda map are used for numerical evaluation of the performance. Results clearly show that the NPF based approach is superior to the EKF based approach in terms of the normalized mean square error (NMSE), total NMSE, and the time taken for synchronization (measured in terms of the normalized instantaneous square error) for all the systems and/or maps considered.     

Generalized chaotic synchronization in coupled Ginzburg-Landau equations

Generalized synchronization is analyzed in unidirectionally coupled oscillatory systems exhibiting spatiotemporal chaotic behavior described by Ginzburg-Landau equations. Several types of coupling between the systems are analyzed. The largest spatial Lyapunov exponent is proposed as a new characteristic of the state of a distributed system, and its calculation is described for a distributed oscillatory system. Partial generalized synchronization is introduced as a new type of chaotic synchronization in spatially nonuniform distributed systems. The physical mechanisms responsible for the onset of generalized chaotic synchronization in spatially distributed oscillatory systems are elucidated. It is shown that the onset of generalized chaotic synchronization is described by a modified Ginzburg-Landau equation with additional dissipation irrespective of the type of coupling. The effect of noise on the onset of a generalized synchronization regime in coupled distributed systems is analyzed.     

Random delays and the synchronization of chaotic maps

We investigate the dynamics of an array of chaotic logistic maps coupled with random delay times. We report that for adequate coupling strength the array is able to synchronize, in spite of the random delays. Specifically, we find that the synchronized state is a homogeneous steady state, where the chaotic dynamics of the individual maps is suppressed. This synchronization behavior is largely independent of the connection topology and depends mainly on the average number of links per node. We carry out a statistical linear stability analysis that confirms the numerical results and provides a better understanding of the nontrivial roles of random delayed interactions.     

Stability criterion for projective synchronization in three-dimensional chaotic systems

Projective synchronization, in which the state vectors synchronize up to a scaling factor, has recently been observed in coupled partially linear chaotic systems (Lorenz system) under certain conditions. In this Letter, we present a stability criterion that guarantees the occurrence of the projective synchronization in three-dimensional systems. By applying the criterion to two typical partially linear systems (Lorenz and disk dynamo), it shows that only some parameters play the key role in influencing the stability. Projective synchronization only happens when σ>−1 for the Lorenz and μ>0 for the disk dynamo.     

非线性耦合超混沌R(o)ssler系统和网络的同步

研究两个通过非线性函数对称耦合的超混沌Roessler系统的同步问题．通过对超混沌系统的线性项与非线性项的适当分离,构造一个特殊的非线性函数,作为耦合函数,发现在耦合强度α=0．5附近的一小段区域里存在稳定的超混沌同步现象．利用线性系统的稳定性分析准则和条件Lyapunov指数来检验同步状态的稳定性,并进一步研究了由多个超混沌Roessler系统单元通过非线性函数按照完全连接形式组成的网络的混沌同步问题。显示许多耦合单元组成的网络,满足同步稳定性的耦合强度的取值范围可以仅从2个单元组成的网络的参数取值范围估计到。此外发现耦合强度的值与耦合单元数量成反比,数值模拟结果证实所提出方法对超混沌系统和网络的混沌同步是有效的。     

Dynamic synchronization of a time-evolving optical network of chaotic oscillators

We present and experimentally demonstrate a technique for achieving and maintaining a global state of identical synchrony of an arbitrary network of chaotic oscillators even when the coupling strengths are unknown and time-varying. At each node an adaptive synchronization algorithm dynamically estimates the current strength of the net coupling signal to that node. We experimentally demonstrate this scheme in a network of three bidirectionally coupled chaotic optoelectronic feedback loops and we present numerical simulations showing its application in larger networks. The stability of the synchronous state for arbitrary coupling topologies is analyzed via a master stability function approach.     

基于状态观测器的参数调制混沌数字通信

用状态观测器构造两个与混沌系统同步的子系统，将数字信号调制发送系统的参数，两同步系统交替与发送端同步. 在接收端，利用同步误差解调出信号. 以Henon混沌为例构造基于状态观测器的参数调制与解调系统，进行数值模拟，验证了该方法的有效性. 关键词： 状态观测器 混沌同步 参数调制     

Synchronization of incommensurate non-identical fractional order chaotic systems using active control

Chaos synchronization in fractional order chaotic systems is receiving increasing attention due to its applications in secure communications. In this article we use an active control technique to synchronize incommensurate non-identical fractional order chaotic dynamical systems. The relation between system order and the synchronization time is discussed. It is observed that the synchronization can be achieved faster by increasing the system order. Further we provide an application of the proposed theory in secure communication.     

Global chaos synchronization of coupled parametrically excited pendula

In this paper, we study the synchronization behaviour of two linearly coupled parametrically excited chaotic pendula. The stability of the synchronized state is examined using Lyapunov stability theory and linear matrix inequality (LMI); and some sufficient criteria for global asymptotic synchronization are derived from which an estimated critical coupling is determined. Numerical solutions are presented to verify the theoretical analysis. We also examined the transition to stable synchronous state and show that this corresponds to a boundary crisis of the chaotic attractor.     

Lorenz系统族采样同步研究

从对连续时间混沌系统进行数字测量的角度，提出了连续时间混沌系统的采样同步问题. 利用控制理论的结果，设计了Lorenz系统族的采样同步系统. 仅需测量一个状态分量，就可在采样时刻实现同步. 数值模拟例证了同步的性能. 关键词： 混沌同步 Lorenz系统族 采样同步     

Robust pre-specified time synchronization of chaotic systems by employing time-varying switching surfaces in the sliding mode control scheme

In the conventional chaos synchronization methods, the time at which two chaotic systems are synchronized, is usually unknown and depends on initial conditions. In this work based on Lyapunov stability theory a sliding mode controller with time-varying switching surfaces is proposed to achieve chaos synchronization at a pre-specified time for the first time. The proposed controller is able to synchronize chaotic systems precisely at any time when we want. Moreover, by choosing the time-varying switching surfaces in a way that the reaching phase is eliminated, the synchronization becomes robust to uncertainties and exogenous disturbances. Simulation results are presented to show the effectiveness of the proposed method of stabilizing and synchronizing chaotic systems with complete robustness to uncertainty and disturbances exactly at a pre-specified time.