Stability criterion for synchronization of chaotic systems using linear feedback control

This Letter introduces linear balanced feedback control scheme to design controller for the synchronization of two identical chaotic systems based on Lyapunov stability theory and constrained extreme approach. The technique is applied to synchronizing two identical four-scroll chaotic systems and guiding balanced feedback gains design. In accordance with the result of the analysis, an adaptive control scheme is proposed for chaos synchronization when the parametric variations of the response system are uncertain. The feasibility and effectiveness of the proposed synchronization scheme are verified via numerical simulations.     

A new criterion for chaos and hyperchaos synchronization using linear feedback control

Based on the characteristic of the chaotic or hyperchaotic system and linear feedback control method, synchronization of the two identical chaotic or hyperchaotic systems with different initial conditions is studied. The range of the control parameter for synchronization is derived. Simulation results are provided to show the effectiveness of the proposed synchronization method.     

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。     

INVERSE SYNCHRONIZATION OF CHAOTIC SYSTEMS IN AN ERBIUM-DOPED FIBRE DUAL-RING LASER USING THE MUTUAL COUPLING METHOD

Inverse synchronization of chaos is a type of synchronization in which the dynamical variables of two chaotic systems are inversely equal. In this paper, we present a scheme for inverse synchronization of two chaotic systems in an erbium-doped fibre dual-ring laser using the mutual coupling method. For realistic values of the systems, we demonstrate two kinds of results, as follows. (1) Two independent identical chaotic systems can go into inversely synchronized chaotic oscillation for coupling greater than 0.03. (2) When some parameter of one system varies, the state of the coupled systems could go into some periodic states directly or by inverse bifurcation. Simultaneously, they will lose the synchronization as the parameter changes.     

Chaos synchronization of unified chaotic systems via LMI

This Letter focuses on the master-slave synchronization problem of the unified chaotic systems. Based on Lyapunov stability theory and linear matrix inequality (LMI) formulation, a simple linear feedback control law is obtained to make the state of two identical unified chaotic systems asymptotically synchronized. Simulation results have illustrated the effectiveness of the proposed chaos synchronization solution.     

Lorenz系统驱动声光双稳时空混沌系统并行同步的研究

研究了一个时间混沌系统驱动多个时空混沌系统的并行同步问题.以单模激光Lorenz系统和一维耦合映像格子为例,在单模激光Lorenz系统中提取一个混沌序列,通过与一维耦合映像格子中的状态变量耦合使单模激光Lorenz系统和多个同结构一维耦合映像格子同时达到广义同步,并且多个一维耦合映像格子之间实现完全并行同步.通过计算条件Lyapunov指数,可以得到并行同步所需反馈系数的取值范围.数值模拟证明了此方法的可行性和有效性.     

Synchronization and parameter identification of one class of realistic chaotic circuit

In this paper, the synchronization and the parameter identification of the chaotic Pikovsky--Rabinovich (PR) circuits are investigated. The linear error of the second corresponding variables is used to change the driven chaotic PR circuit, and the complete synchronization of the two identical chaotic PR circuits is realized with feedback intensity k increasing to a certain threshold. The Lyapunov exponents of the chaotic PR circuits are calculated by using different feedback intensities and our results are confirmed. The case where the two chaotic PR circuits are not identical is also investigated. A general positive Lyapunov function V, which consists of all the errors of the corresponding variables and parameters and changeable gain coefficient, is constructed by using the Lyapunov stability theory to study the parameter identification and complete synchronization of two non-identical chaotic circuits. The controllers and the parameter observers could be obtained analytically only by simplifying the criterion dV/dt<0 (differential coefficient of Lyapunov function V with respect to time is negative). It is confirmed that the two non-identical chaotic PR circuits could still reach complete synchronization and all the unknown parameters in the drive system are estimated exactly within a short transient period.     

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.     

基于非线性控制的超混沌Chen系统混沌同步

研究了基于非线性控制的超混沌Chen系统的混沌同步问题.基于Lyapunov稳定性理论，设计了非线性控制器，改进了Jiang和Huang等设计的同步误差系统的Lyapunov函数形式，理论证明了超混沌Chen系统的自同步和超混沌Chen系统与超混沌R?ssler系统的异结构同步.数值模拟进一步验证了所提出方案的有效性. 关键词： 超混沌Chen系统 自同步 异结构同步 非线性控制器     

Exponential synchronization of stochastic impulsive perturbed chaotic Lur''e systems with time-varying delay and parametric uncertainty

This paper is devoted to investigating the scheme of exponential synchronization for uncertain stochastic impulsive perturbed chaotic Lur＇e systems. The parametric uncertainty is assumed to be norm bounded. Based on the Lyapunov function method, time-varying delay feedback control technique and a modified Halanay inequality for stochastic differential equations, several sufficient conditions are presented to guarantee the exponential synchronization in mean square between two identical uncertain chaotic Lur＇e systems with stochastic and impulsive perturbations. These conditions are expressed in terms of linear matrix inequalities （LMIs）, which can easily be checked by utilizing the numerically efficient Matlab LMI toolbox. It is worth pointing out that the approach developed in this paper can provide a more general framework for the synchronization of multi-perturbation chaotic Lur＇e systems, which reflects a more realistic dynamics. Finally, a numerical example is provided to demonstrate the effectiveness of the proposed method.     

Feedback Induced Instability and Chaos in Semiconductor Lasers and Their Applications

Semiconductor laser with feedback is an excellent model for nonlinear optical system which shows chaotic dynamics. It is interesting not only from the fundamental physical study but also application standpoints of view. The dynamics of feedback induced instability and chaos, especially for optical feedback, and their applications are reviewed in this paper. The model of such a system is described by the laser rate equations. At first the dynamic behaviors of feedback induced instability and chaos in semiconductor lasers are discussed on the basis of the theory and experiments. Instability and chaos may be stabilized by the method of chaos control. Then we apply the method to suppress the noise induced by the feedback in a semiconductor laser. The synchronization of chaos between two similar systems is also an important issue in chaos applications and we discuss secure communications based on chaos synchronization. Some other examples of applications of feedback induced chaos are also described.     

Anticipation in the synchronization of chaotic semiconductor lasers with optical feedback

The synchronization of chaotic semiconductor lasers with optical feedback is studied numerically in a one-way coupling configuration, in which a small amount of the intensity of one laser (master laser) is injected coherently into the other (slave laser). A regime of anticipated synchronization is found, in which the intensity of the slave laser is synchronized to the future chaotic intensity of the master laser. Anticipation is robust to small noise and parameter mismatches, but in this case the synchronization is not complete. It is also shown that anticipated synchronization occurs in coupled time-delay systems, when the coupling has a delay that is less than the delay of the systems.     

Identifying parameter by identical synchronization between different systems

In this paper, parameters of a given (chaotic) dynamical system are estimated from time series by using identical synchronization between two different systems. This technique is based on the invariance principle of differential equations, i.e., a dynamical Lyapunov function involving synchronization error and the estimation error of parameters. The control used in this synchronization consists of feedback and adaptive control loop associated with the update law of estimation parameters. Our estimation process indicates that one may identify dynamically all unknown parameters of a given (chaotic) system as long as time series of the system are available. Lorenz and Rossler systems are used to illustrate the validity of this technique. The corresponding numerical results and analysis on the effect of noise are also given.     

A practical approach to robust impulsive lag synchronization between different chaotic systems

In this paper, a practical impulsive lag synchronization scheme for different chaotic systems with parametric uncertainties is proposed. By virtue of the new definition of synchronization and the theory of impulsive differential equations, some new and less conservative sufficient conditions are established to guarantee that the error dynamics can converge to a predetermined level. The idea and approach developed in this paper can provide a more practical framework for the synchronization between identical and different chaotic systems in parameter perturbation circumstances. Simulation results finally demonstrate the effectiveness of the method.     

单模激光Lorenz系统与3D混沌系统之间的混沌同步

研究了异结构混沌系统之间的同步控制问题.采用非线性反馈控制方法实现了3D混沌系统和单模激光Lorenz混沌系统之间的混沌同步.根据系统的稳定性理论,得到了非线性反馈控制器的结构和反馈控制增益的取值范围.仿真模拟的结果表明:目标系统和响应系统达到完全同步,两系统状态变量随时间的演化轨迹完全一致,并且误差变量经过短暂的时间序列以后始终平稳地趋于零.仿真模拟的结果证明了这种方法的有效性.     

光电反馈混沌系统脉冲同步特性研究

提出光电反馈混沌状态下的混沌脉冲同步.首先, 对激光器的混沌动力学方程进行了数值分析, 通过改变反馈系数和脉冲系数来观测混沌状态的变化, 实现了对混沌激光器的控制;然后, 对脉冲同步方程进行数值分析, 给出了同步的模拟图.实验结果表明此方法能实现混沌同步.     

利用单模激光Lorenz系统实现异结构混沌系统混沌同步

对混沌同步问题进行了研究。设计了一种同步控制器来实现异结构混沌系统之间的混沌同步。减少了控制器的数量,仅用一个控制器就能实现这一同步控制。同时,控制器只需在目标和响应系统中提取一个易提取信号即可实现,从而大大降低了同步代价,便于实际应用。以单模激光Lorenz系统和coullet系统为例,验证了这种控制器的有效性。仿真模拟结果表明,在控制器的作用下,coullet系统所有状态变量严格地跟踪了单模激光Lorenz系统的混沌轨迹,两者迅速地达到完全同步。该设计的控制方法可用于任意混沌系统,具有一定的普适性。     

Synchronization of memristor-based chaotic systems by a simplified control and its application to image en-/decryption using DNA encoding

In this paper, a simplified control is proposed to realize chaotic synchronization based on two identical memristor-based chaotic systems. Firstly, by using a cubic function model of memristor, a memristor-based chaotic system is realized by circuit emulation. Secondly, based on the minimum phase system and Routh–Hurwitz criterion, the state trajectories of two identical chaotic systems can be asymptotically synchronized by the proposed simplified control. The simulation results illustrate that the designed simplified control is effective to chaotic synchronization. Finally, an encryption scheme of color image using DNA encoding is proposed based on chaotic synchronization, and the results of statistical analysis are given to show the effectiveness of the proposed scheme in image processing.     

General methods for modified projective synchronization of hyperchaotic systems with known or unknown parameters

This work is concerned with the general methods for modified projective synchronization of hyperchaotic systems. A systematic method of active control is developed to synchronize two hyperchaotic systems with known parameters. Moreover, by combining the adaptive control and linear feedback methods, general sufficient conditions for the modified projective synchronization of identical or different chaotic systems with fully unknown or partially unknown parameters are presented. Meanwhile, the speed of parameters identification can be regulated by adjusting adaptive gain matrix. Numerical simulations verify the effectiveness of the proposed methods.     

忆阻混沌系统的脉冲同步与初值影响研究

以光滑三次型磁控忆阻器的蔡氏电路为例, 研究了两个同构忆阻混沌系统的脉冲控制同步方法.基于Lyapunov稳定性理论, 给出了忆阻混沌系统的脉冲同步渐近稳定条件; 结合误差系统的最大条件Lyapunov指数谱, 讨论了系统初值对脉冲同步性能的影响, 并进行了相应的数值仿真实验.结果表明, 在合适的脉冲控制参数条件下, 同构的忆阻混沌系统的脉冲同步是可行而有效的; 忆阻混沌系统的初值对脉冲同步性能存在一定的影响, 但可通过增大脉冲耦合强度来抑制系统初值的影响.