Synchronizing multiple chaotic maps with a randomized scalar coupling

Synchronization between chaotic systems has recently been a topic of great interest among physicists and engineers. In addition to theoretical results, a number of applications in communications and control have also been proposed. We have previously shown that identical chaotic maps can, under certain conditions, be synchronized by a common noise-like input. This raises the question whether the chaotic output from a map can synchronize other maps to itself. In this paper, we present results on the synchronization of two identical maps where the output of one map is used to drive both maps. We then apply this method to synchronizing identical but internally inhomogeneous populations of chaotic oscillators using a randomly constructed scalar coupling signal, which tends to white noise as the number of oscillators increases. Such a system may have applications in secure communication.     

基于函数矩阵的一类混沌系统同步

研究了一类混沌系统的同步问题、基于稳定性理论和极点配置技术,设计了两个混沌系统之间的同步方案,实现两个混沌系统之间的同步,通过函数矩阵,实现驱动系统和响应系统的状态变量按给定的函数矩阵同步,同时证明了该方法同样适用于两个混沌系统之间的滞后同步,通过对Lorenz混沌系统和Lorenz超混沌系统的数值模拟,进一步验证了所提方案的有效性。     

Perfect synchronization of chaotic systems： a controllability perspective

This paper presents a synchronization method, motivated from the constructive controllability analysis, for two identical chaotic systems. This technique is applied to achieve perfect synchronization for Lorenz systems and coupled dynamo systems. It turns out that states of the drive system and the response system are synchronized within finite time, and the reaching time is independent of initial conditions, which can be specified in advance. In addition to the simultaneous synchronization, the response system is synchronized un-simultaneously to the drive system with different reaching time for each state. The performance of the resulting system is analytically quantified in the face of initial condition error, and with numerical experiments the proposed method is demonstrated to perform well.     

On multi switching compound synchronization of non identical chaotic systems

Multi switching compound synchronization (MSCOS) between four non identical chaotic systems is investigated. Chen system is considered as scaling drive system. Lorenz and Lü systems are taken as base drive systems. The compound of the multi drive system is then synchronized with Rössler response system using multi switching synchronization scheme. In MSCOS, the state variables of three drive systems synchronize with different state variables of response system, simultaneously. For suitable choice of scaling factors, switched modified function projective synchronization is obtained as a special case of MSCOS among chaotic systems. To achieve the desired synchronization among four non identical chaotic systems, sufficient condition is obtained using a nonlinear controller and Lyapunov stability theory, and corresponding theoretical proofs are given. The effectiveness of the proposed controllers is verified by numerical simulations using MATLAB.     

基于积分观测器实现一类受扰混沌系统的同步

在系统参数和系统输出同时受扰的情形下,基于积分观测器的方法,实现了一类不确定混沌系统的同步.首先利用Lyapunov 稳定性理论,给出了混沌系统同步所需的充分条件,此条件可以快速地实现驱动系统与响应系统的同步,然后结合Schur补定理,将此条件转化为求解线性矩阵不等式组.最后,借助Matlab软件包中的LMI工具箱,通过对受扰MLC电路系统的同步数值仿真,证实了该方法的有效性.     

Synchronization of time delayed semiconductor lasers and its applications in digital cryptography

This work aims to demonstrate the effect of synchronization phenomena in chaotic laser systems described by modified Lang-Kobayashi's (L-K) delay differential equations. The synchronized system considered for numerical simulations and for cryptography consists of identical semiconductor lasers operating in single longitudinal mode. The two lasers are bidirectionally coupled by linear optical feedback. As this is essential in simultaneous transmission of messages, we have applied the corresponding coupled chaotic dynamics to secure communications. An investigation of the system together with a novel scheme for digital cryptography and visual recurrence analysis (VRA) of the chaotic time series are presented. Extended statistical tests with the proposed two phase scheme demonstrate the efficiency of these infinite dimensional systems in being tolerant to different types of statistical attacks. The result emphasizes the merits of the uncertainty and high dimensionality of optical chaos system in duplex high speed secure communications.     

Using chaos to produce synchronized stochastic dynamics in non-homogeneous map arrays with a random scalar coupling

Unidirectionally coupled chaotic systems hold great interest from the information processing and communications perspective. In this Letter, we report on a novel method for synchronizing two identical but internally non-homogeneous populations of chaotic maps using a scalar random coupling between them. The resulting synchronized dynamics is stochastic, and can be used in secure multi-user communication applications.     

Adaptive projective synchronization of different chaotic systems with nonlinearity inputs

We investigate the projective synchronization of different chaotic systems with nonlinearity inputs. Based on the adaptive technique, sliding mode control method and pole assignment technique, a novel adaptive projective synchronization scheme is proposed to ensure the drive system and the response system with nonlinearity inputs can be rapidly synchronized up to the given scaling factor.     

Adaptive projective synchronization of different chaotic systems with nonlinearity inputs

We investigate the projective synchronization of different chaotic systems with nonlinearity inputs.Based on the adaptive technique,sliding mode control method and pole assignment technique,a novel adaptive projective synchronization scheme is proposed to ensure the drive system and the response system with nonlinearity inputs can be rapidly synchronized up to the given scaling factor.     

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     

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.     

Stages of chaotic synchronization

In an experimental investigation of the response of a chaotic system to a chaotic driving force, we have observed synchronization of chaos of the response system in the forms of generalized synchronization, phase synchronization, and lag synchronization to the driving signal. In this paper we compare the features of these forms of synchronized chaos and study their relations and physical origins. We found that different forms of chaotic synchronization could be interpreted as different stages of nonlinear interaction between the coupled chaotic systems. (c) 1998 American Institute of Physics.     

Multiple attractors and periodic transients in synchronized nonlinear circuits

I study a pair of synchronized nonlinear circuits which may be periodic or chaotic. The circuits are synchronized by a one-way driving signal from the drive circuit to the response circuit. Because the nonlinearities are symmetric about zero, the drive circuit has two periodic attractors. When the value of a bifurcation parameter is above a certain threshold, the response circuit also has two periodic attractors, one in-sync with the drive and one out-of-sync. Below the threshold, the drive circuit still has two attractors but the response circuit has only one attractor, the in-sync attractor. If the response circuit is started in the basin of attraction of the former out-of-sync attractor, a long periodic transient (many cycles long) is seen.     

基于参数自适应控制的混沌同步

讨论了驱动系统和响应系统都是相同混沌映射但其中参数不同时的同步问题.采用参数自适应控制算法,当混沌映射为logistic映射时,得到两系统同步得一个充分条件,而为一般混沌映射时,得到两系统同步的一个必要条件.还讨论了存在相同噪音时的同步问题. 关键词：     

声光双稳系统的混沌同步

首先给出布拉格型声光双稳系统耦合驱动的混沌同步化方案，用最大条件Ｌｙａｐｕｎｏｖ指数分析方法得出耦合驱动下系统混沌输出同步化条件，发现通过适当比例的耦合驱动可以使两组混沌系统达到同步的混沌输出。分析此混沌同步化方案可以抵抗噪声的干扰，并且在两系统出现偏差时仍可以实现混沌同步，找到了实用的单变量延时微分系统非Ｐｅｃｏｒａ－Ｃａｒｒｏｌｌ规则的混沌同步化方案。最后做了实验验证。     

Generalized Projective Synchronization Between Rössler System and New Unified Chaotic System

Based on symbolic computation system Maple and Lyapunov stability theory, an active control method is used to projectively synchronize two different chaotic systems — Lorenz-Chen-Lü system (LCL) and Rössler system, which belong to different dynamic systems. In this paper, we achieve generalized projective synchronization between the two different chaotic systems by directing the scaling factor onto the desired value arbitrarily. To illustrate our result, numerical simulations are used to perform the process of the synchronization and successfully put the orbits of drive system (LCL) and orbits of the response system (Rössler system) in the same plot for understanding intuitively.     

利用混沌信号驱动实现各子系统间的同步

通过对非同参数Lorenz系统驱动模型的讨论,发现对于非同参数情况,驱动系统与响应系统间严格的混沌同步关系是不存在的,同步只具有相对的意义.同时,探讨了在混沌信号驱动下稳定系统的动力学行为,说明其输出同样是混沌的.指出利用混沌信号驱动两个非同参数的子系统,可以实现两子系统之间的混沌同步输出.给出同步条件,得出的结论在实际应用中可能具有指导意义. 关键词：     

Multi-switching dual compound synchronization of chaotic systems

A novel multi-switching dual compound synchronization scheme is proposed for chaotic systems which has not been investigated so far. The goal is to design appropriate controllers by using Lyapunov stability theory and nonlinear control to establish the asymptotically stable synchronized state for six drive and two response systems. Multi-switching dual compound synchronization can be considered as an extension of multi-switching dual combination synchronization. An example is presented to elaborate the proposed scheme where Lorenz, Chen, Lu systems are considered as master systems and T system is considered as slave system. The results obtained by theoretical and graphical analysis are in excellent agreement.     

Continuous-time chaotic systems： Arbitrary full-state hybrid projective synchronization via a scalar signal

Relerrlng to contlnuous-Ume claaotlc systems, tills paper presents a new projective syncnromzatlon scheme, wnlcn enables each drive system state to be synchronized with a linear combination of response system states for any arbitrary scaling matrix. The proposed method, based on a structural condition related to the uncontrollable eigenvalues of the error system, can be applied to a wide class of continuous-time chaotic （hyperchaotic） systems and represents a general framework that includes any type of synchronization defined to date. An example involving a hyperchaotic oscillator is reported, with the aim of showing how a response system attractor is arbitrarily shaped using a scalar synchronizing signal only. Finally, it is shown that the recently introduced dislocated synchronization can be readily achieved using the conceived scheme.     

Manipulating the scaling factor of projective synchronization in three-dimensional chaotic systems

Scaling factor characterizes the synchronized dynamics of projective synchronization in partially linear chaotic systems but it is difficult to be estimated. To manipulate projective synchronization of chaotic systems in a favored way, a control algorithm is introduced to direct the scaling factor onto a desired value. The control approach is derived from the Lyapunov stability theory. It allows us to arbitrarily amplify or reduce the scale of the response of the slave system via a feedback control on the master system. In numerical experiments, we illustrate the application to the Lorenz system. (c) 2001 American Institute of Physics.