Adaptive projective synchronization between different chaotic systems with parametric uncertainties and external disturbances

The article deals with adaptive projective synchronization between two different chaotic systems with parametric uncertainties and external disturbances. Based on Lyapunov stability theory, the projective synchronization between a pair of different chaotic systems with fully unknown parameters are derived. An adaptive control law and a parameter update rule for uncertain parameters are designed such that the chaotic response system controls the chaotic drive system. Numerical simulation results are performed to explain the effectiveness and feasibility of the techniques.     

Adaptive synchronization of a critical chaotic system

This paper further investigates the synchronization problem of a new chaotic system with known or unknown system parameters. Based on the Lyapunov stability theory, a novel adaptive control law is derived for the synchronization of a new chaotic system with known or unknown system parameters. Theoretical analysis and numerical simulations show the effectiveness and feasibility of the proposed schemes.     

Adaptive complete synchronization of two identical or different chaotic (hyperchaotic) systems with fully unknown parameters

This paper studies the adaptive complete synchronization of chaotic and hyperchaotic systems with fully unknown parameters. In practical situations, some systems' parameters cannot be exactly known a priori, and the uncertainties often affect the stability of the process of synchronization of the chaotic oscillators. An adaptive scheme is proposed to compensate for the effects of parameters' uncertainty based on the structure of chaotic systems in this paper. Based on the Lyapunov stability theorem, an adaptive controller and a parameters update law can be designed for the synchronization of chaotic and hyperchaotic systems. The drive and response systems can be nonidentical, even with different order. Three illustrative examples are given to demonstrate the validity of this technique, and numerical simulations are also given to show the effectiveness of the proposed chaos synchronization method. In addition, this synchronization scheme is quite robust against the effect of noise.     

Adaptive generalized functional synchronization of chaotic systems with unknown parameters

A universal adaptive generalized functional synchronization approach to any two different or identical chaotic systems with unknown parameters is proposed, based on a unified mathematical expression of a large class of chaotic system. Self-adaptive parameter law and control law are given in the form of a theorem. The synchronization between the three-dimensional R6ssler chaotic system and the four-dimensional Chen＇s hyper-chaotic system is studied as an example for illustration. The computer simulation results demonstrate the feasibility of the method proposed.     

Function projective synchronization of different chaotic systems with uncertain parameters

This Letter investigates the function projective synchronization of different chaotic systems with unknown parameters. By Lyapunov stability theory, the adaptive control law and the parameter update law are derived to make the states of two different chaotic systems asymptotically synchronized up to a desired scaling function. Numerical simulations on Lorenz system and Newton-Leipnik system are presented to verify the effectiveness of the proposed scheme.     

Increasing-order Projective Synchronization of Chaotic Systems with Time Delay

This work is concerned with lag projective synchronization of chaotic systems with increasing order. The systems under consideration have unknown parameters and different structures. Combining the adaptive control method and feedback control technique, we design a suitable controller and parameter update law to achieve lag synchronization of chaotic systems with increasing order. The result is rigorously proved by the Lyapunov stability theorem. Moreover, corresponding simulation results are given to verify the effectiveness of the proposed methods.     

Adaptive generalized projective synchronization of two different chaotic systems with unknown parameters

This paper presents a general method of the generalized projective synchronization and the parameter identification between two different chaotic systems with unknown parameters. This approach is based on Lyapunov stability theory, and employs a combination of feedback control and adaptive control. With this method one can achieve the generalized projective synchronization and realize the parameter identifications between almost all chaotic （hyperchaotic） systems with unknown parameters. Numerical simulations results are presented to demonstrate the effectiveness of the method.     

Adaptive modified function projective synchronization of multiple time-delayed chaotic Rossler system

In this Letter we consider modified function projective synchronization of unidirectionally coupled multiple time-delayed Rossler chaotic systems using adaptive controls. Recently, delay differential equations have attracted much attention in the field of nonlinear dynamics. The high complexity of the multiple time-delayed systems can provide a new architecture for enhancing message security in chaos based encryption systems. Adaptive control can be used for synchronization when the parameters of the system are unknown. Based on Lyapunov stability theory, the adaptive control law and the parameter update law are derived to make the state of two chaotic systems are function projective synchronized. Numerical simulations are presented to demonstrate the effectiveness of the proposed adaptive controllers.     

Pragmatical generalized synchronization of chaotic systems with uncertain parameters by adaptive control

A new kind of generalized synchronization of two chaotic systems with uncertain parameters is proposed. Based on a pragmatical asymptotical stability theorem and an assumption of equal probability for ergodic initial conditions, an adaptive control law is derived so that it can be proved strictly that the common null solution of error dynamics and of parameter dynamics is actually asymptotically stable, i.e. these two identical systems are in generalized synchronization and the estimated parameters approach the uncertain values. It is called pragmatical generalized synchronization. Finally, two numerical examples are studied for two Quantum-CNN oscillator chaotic systems to show the effectiveness of the proposed generalized synchronization strategy with a double Duffing chaotic system as a goal system.     

基于自适应模糊控制的分数阶混沌系统同步

本文主要研究了带有未知外界扰动的分数阶混沌系统的同步问题.基于分数阶Lyapunov稳定性理论,构造了分数阶的参数自适应规则以及模糊自适应同步控制器.在稳定性分析中主要使用了平方Lyapunov函数.该控制方法可以实现两分数阶混沌系统的同步,使得同步误差渐近趋于0.最后,数值仿真结果验证了本文方法的有效性.     

Synchronization Control of Two Different Chaotic Systems with Known and Unknown Parameters

Chaos synchronization of two different chaotic systems with known and unknown parameters is studied. Based on the Lyapunov stability theory, two different chaotic systems with known parameters realize global synchronization via the successfully designed nonlinear controller. By employing an adaptive synchronization scheme, the synchronization of two different chaotic systems with unknown parameters is achieved. Numerical simulations validate the effectiveness of the theoretical analysis.     

不同结构混沌系统的自适应同步和反同步

针对不同结构混沌系统的同步与反同步问题进行了研究.在系统参数已知时,采用主动控制法实现混沌系统的同步与反同步,并将主动控制器的设计方法进行了推广.在参数未知时,基于Lyapunov稳定性理论和自适应控制方法,给出了自适应控制器和参数自适应律,实现了参数均未知且结构不同的驱动系统和响应系统的同步与反同步.在控制器的设计过程中,将驱动系统和响应系统进行互换,讨论了互换前后的控制器和自适应律之间的关系.数值仿真结果说明了所提出设计方法的有效性. 关键词： 混沌同步 反同步 主动控制法 自适应控制法     

A new chaotic communication scheme based on adaptive synchronization

A new chaotic communication scheme using adaptive synchronization technique of two unified chaotic systems is proposed. Different from the existing secure communication methods, the transmitted signal is modulated into the parameter of chaotic systems. The adaptive synchronization technique is used to synchronize two identical chaotic systems embedded in the transmitter and the receiver. It is assumed that the parameter of the receiver system is unknown. Based on the Lyapunov stability theory, an adaptive control law is derived to make the states of two identical unified chaotic systems with unknown system parameters asymptotically synchronized; thus the parameter of the receiver system is identified. Then the recovery of the original information signal in the receiver is successfully achieved on the basis of the estimated parameter. It is noticed that the time required for recovering the information signal and the accuracy of the recovered signal very sensitively depends on the frequency of the information signal. Numerical results have verified the effectiveness of the proposed scheme.     

时空混沌系统参量辨识律的设计与投影同步研究

研究了相互耦合的时空混沌系统的参量辨识与投影同步问题. 依据Lyapunov定理, 设计了参量辨识律和表征耦合强度的待定函数的自适应律, 对响应系统中的未知参量进行了有效辨识, 并完成了时空混沌系统的投影同步研究. 采用具有时空混沌行为的Burgers方程作为实例进行了仿真分析.     

不确定混沌系统的回归神经网络同步

将高阶连接的神经元融合到分布式回归神经网络，研究了出现非模型动态性时不确定混沌系统的辨识和同步问题。采用李雅谱诺夫稳定理论对高阶神经网络回归模型的权值进行学习更新，同时，获取整个系统稳定特性的分析结果，而且通过李雅谱诺夫方法设计出消除不确定混沌系统的同步误差的自适应控制律。最后将所提出的方法应用到不确定Rossler混沌系统的建模与同步     

Adaptive synchronization between two different chaotic systems with unknown parameters

A unified mathematical expression describing a class of chaotic systems is presented, for which the problem of adaptive synchronization between two different chaotic systems with unknown parameters has been studied. Based on Lyapunov stability theory, an adaptive synchronization controller is designed and analytic expression of the controller and the adaptive laws of parameters are developed. The adaptive synchronizations between Lorenz and Chen systems, a modified Chua's circuit and Rössler systems are taken as two illustrative examples to show the effectiveness of the proposed method.     

参数未知的不同阶数混沌系统广义同步及参数估计

采用扩阶方法(使响应系统和驱动系统的维数相同)，把不同阶数混沌系统的广义同步问题转化为相同阶数混沌系统之间的广义同步，基于Lyapunov稳定性定理和自适应控制方法(用于相同阶数混沌系统的同步)，给出了自适应控制器和参数自适应律，进而实现了不同阶数混沌系统的广义同步.将该方法应用于参数未知的超Lü，Lorenz，广义Lorenz和Liu等系统之间的广义混沌同步，理论证明了该方法可以使这些系统达到渐近广义同步，并且可以辨识驱动系统和响应系统的所有参数，数值模拟进一步证明了该方法的有效性.     

Adaptive synchronization of a switching system and its applications to secure communications

This paper studies the adaptive synchronization of a switching system with unknown parameters which switches between the R?ssler system and a unified chaotic system. Using the Lyapunov stability theory and adaptive control method, the receiver system will achieve synchronization with the drive system and the unknown parameters would be estimated by the receiver. Then the proposed switching system is used for secure communications based on the communication schemes including chaotic masking, chaotic modulation, and chaotic shift key strategies. Since the system switches between two chaotic systems and the parameters are almost unknown, it is more difficult for the intruder to extract the useful message from the transmission channel. In addition, two new schemes in which the chaotic signal used to mask (or modulate) the transmitted signal switches between two components of a chaotic system are also presented. Finally, some simulation results are given to show the effectiveness of the proposed communication schemes.     

一类带有未知参数的受扰混沌系统的观测器同步

在系统参数未知的情形下,观测器方法和自适应方法相结合,实现了一类受扰混沌系统的同步.借助于Lyapunov稳定性理论和Barbalat引理,给出了观测器的设计方法.此方法约束条件较少,适应于大部分常见的混沌系统.最后,通过对典型混沌系统的同步数值仿真,证实了方法的有效性和正确性. 关键词： 混沌系统 外界干扰 同步 观测器法     

不确定分数阶时滞混沌系统自适应神经网络同步控制

针对带有完全未知的非线性不确定项和外界扰动的异结构分数阶时滞混沌系统的同步问题,基于Lyapunov稳定性理论,设计了自适应径向基函数(radial basis function,RBF)神经网络控制器以及整数阶的参数自适应律.该控制器结合了RBF神经网络和自适应控制技术,RBF神经网络用来逼近未知非线性函数,自适应律用于调整控制器中相应的参数.构造平方Lyapunov函数进行稳定性分析,基于Barbalat引理证明了同步误差渐近趋于零.数值仿真结果表明了该控制器的有效性.