Synchronization of chaotic systems with different orders

A controller is designed to realize the synchronization between chaotic systems with different orders. The structure of the controller, the error equations and the Lyapunov functions are determined based on stability theory. Hyperchaotic Chen system and Rossler system are taken for example to demonstrate the method to be effective and feasible. Simulation results show that all the state variables of Rossler system can be synchronized with those of hyperchaotic Chen system by using only one controller, and the error signals approach zero smoothly and quickly.     

Adaptive synchronization of Rossler and Chen chaotic systems

A novel adaptive synchronization method is proposed for two identical Rossler and Chen systems with uncertain parameters. Based on Lyapunov stability theory, we derive an adaptive controller without the knowledge of the system parameters, which can make the states of two identical Rossler and Chen systems globally asymptotically synchronized. Especially, when some unknown uncertain parameters are positive, we can make the controller more simple and, besides, the controller is independent of those positive uncertain parameters. All results are proved using a well-known Lyapunov stability theorem. Numerical simulations are given to validate the proposed synchronization approach.     

Adaptive synchronization with nonlinear input

This paper considers the adaptive synchronization problem of the drive—response type chaotic systems. Using a transmitted scalar signal with an unknown time-delay, a response system is constructed. By appropriately selecting the observer gain and designing the controller, synchronization can be achieved in the presence of the drive system's disturbances and unknown parameters. A well-known chaotic system, Chua's circuit, is considered as an illustrative example to demonstrate the effectiveness of the proposed approach.     

Adaptive generalized synchronization between Chen system and a multi-scroll chaotic system

Based on Lyapunov theory, the adaptive generalized synchronization between Chen system and a multi-scroll chaotic system is investigated. According to the form of target function a proper adaptive controller is designed, by which the controlled Chen system can be synchronized with a multi-scroll chaotic system including unknown parameters. The Lyapunov direct method is exploited to prove that the synchronization error and parameter identification error both converge to zero. Numerical simulation results verify the feasibility of the proposed method further.     

Synchronizing Chaotic Systems in Strict—Feedback Form Using a Single Controlle

A synchronization method is proposed for a class of chaotic systems which can be translated into the strictfeedback form.This approach needs only a single controller and turns out to be globally exponentially asymptotically stable.The effectiveness of this approach has been demonstrated with the Rossler chaotic system.     

Synchronization of Modified Chua‘s Circuit with x|x| Function

This paper considers the chaos synchronization of the modified Chua ‘s circuit with x|x| function. We firstly show that a couple of the modified Chua systems with different parameters and initial conditions can be synchronized using active control when the values of parameters both in drive system and response system are known aforehand.Furthermore, based on Lyapunov stability theory we propose an adaptive active control approach to make the states of two identical Chua systems with unknown constant parameters asymptotically synchronized. Moreover the designed controller is independent of those unknown parameters. Numerical simulations are given to validate the proposed synchronization approach.     

Synchronization of Modified Chua＇s Circuit with x｜x｜ Function

This paper considers the chaos synchronization of the modified Chua＇s circuit with x｜x｜ function. We firstly show that a couple of the modified Chua systems with different parameters and initial conditions can be synchronized using active control when the values of parameters both in drive system and response system are known aforehand.Furthermore, based on Lyapunov stability theory we propose an adaptive active control approach to make the states of two identical Chua systems with unknown constant parameters asymptotically synchronized. Moreover the designed controller is independent of those unknown parameters. Numerical simulations are given to validate the proposed synchronization approach.     

Adaptive synchronization of uncertain Liu system via nonlinear input

This paper addresses the adaptive synchronization for uncertain Liu system via a nonlinear input. By using a single nonlinear controller, the approach is utilized to implement the synchronization of Liu system with total parameters unknown. This method is simple and can be easily designed. What is more, it improves the existing conclusions in Ref [12]. Simulation results prove that the controller is effective and feasible in the end.     

Robust H∞ observer-based of a class of complex control for synchronization dynamical networks

This paper is concerned with the robust Hoo synchronization problem for a class of complex dynamical networks by applying the observer-based control. The proposed feedback control scheme is developed to ensure the asymptotic stability of the augmented system, to reconstruct the non-measurable state variables of each node and to improve the H∞ performance related to the synchronization error and observation error despite the external disturbance. Based on the Lyapunov stability theory, a synchronization criterion is obtained under which the controlled network can be robustly stabilized onto a desired state with a guaranteed H∞ performance. The controller and the observer gains can be given by the feasible solutions of a set of linear matrix inequalities （LMIs）. The effectiveness of the proposed control scheme is demonstrated by a numerical example through simulation.     

Generalized chaos synchronization of a weighted complex network with different nodes

This paper proposes a method of realizing generalized chaos synchronization of a weighted complex network with different nodes. Chaotic systems with diverse structures are taken as the nodes of the complex dynamical network, the nonlinear terms of the systems are taken as coupling functions, and the relations among the nodes are built through weighted connections. The structure of the coupling functions between the connected nodes is obtained based on Lyapunov stability theory. A complex network with nodes of Lorenz system, Coullet system, Rõssler system and the New system is taken as an example for simulation study and the results show that generalized chaos synchronization exists in the whole weighted complex network with different nodes when the coupling strength among the nodes is given with any weight value. The method can be used in realizing generalized chaos synchronization of a weighted complex network with different nodes. Furthermore, both the weight value of the coupling strength among the nodes and the number of the nodes have no effect on the stability of synchronization in the whole complex network.     

不确定混沌系统的参数识别与同步控制器的backstepping设计

设计了一种参数观测器，对不确定Rossler系统中的未知参数进行了识别，采用backstepping方法实现了异结构不确定Rossler系统和Coullet系统之间的混沌同步.根据稳定性原理，确定了同步控制器的结构和控制增益的取值范围.仿真模拟结果表明，在参数观测器和backstepping控制器的共同作用下， Coullet系统所有的状态变量严格地跟踪了不确定Rossler系统的混沌轨迹，从而证明了该观测器和控制器的有效性.     

Q-S (complete or anticipated) synchronization backstepping scheme in a class of discrete-time chaotic (hyperchaotic) systems: a symbolic-numeric computation approach

First, a type of Q-S (complete or anticipated) synchronization is defined in discrete-time dynamical systems. Second, based on backstepping design with a scalar controller, a systematic, concrete and automatic scheme is presented to investigate Q-S (complete or anticipated) synchronization between the discrete-time drive system and response system with strict-feedback form. Finally, the proposed scheme is used to illustrate Q-S (complete or anticipated) synchronization between the two-dimensional discrete-time Lorenz system and Fold system, as well as the three-dimensional hyperchaotic discrete-time Rossler system and generalized discrete-time Rossler system. Moreover numerical simulations are used to verify the effectiveness of the proposed scheme. Our scheme can be also extended to investigate Q-S (complete or anticipated) synchronization between other discrete-time dynamical systems with strict-feedback forms. With the aid of symbolic-numeric computation, the scheme can be performed to yield automatically the scalar controller and to verify its effectiveness in computer.     

含不确定性混沌系统的模糊自适应同步

研究了含不确定性混沌系统的同步问题.基于Takagi-Sugeno(T-S)模糊动态模型，给出了一个新的自适应模糊同步控制设计方法.该方法同时适用于相同结构混沌系统的同步以及异构混沌系统的同步.为说明问题，给出了Lorenz混沌系统和Rossler混沌系统的同步控制设计和仿真结果. 关键词： 混沌系统 模糊控制 同步     

异结构不确定混沌系统的同步控制与参数识别

设计了一种同步控制器和参数识别器来实现异结构不确定混沌系统的同步控制。以单模激光Lorenz系统和不确定Genesio系统为例,验证了这种控制器的有效性。根据Lyapunov稳定性理论,分步构造出Lyapunov函数,从而给出同步控制器和参数识别器的结构。仿真结果表明：利用本文提出的同步控制方案,仅用一个控制器就能使不确定Genesio系统的所有状态变量严格地跟踪Lorenz系统的混沌轨迹,而且误差变量经过短暂的时间序列以后始终平稳地趋于零,实现了异结构混沌系统的混沌同步,且不确定Genesio系统的参数全部可以得到识别。     

Chen系统的自适应追踪控制

利用非线性单输入控制器实现混沌Chen系统的自适应追踪控制.根据Chen系统的结构特点选取合适的反馈方式，设计单输入自适应控制器，使Chen系统自适应追踪参数未知的Rossler系统的某一变量，并由Lyapunov直接方法证明误差信号渐近稳定于零.数值仿真结果表明该控制方法可行，且可以实现未知参数的辨识.     

Q-S (lag or anticipated) synchronization backstepping scheme in a class of continuous-time hyperchaotic systems--a symbolic-numeric computation approach

First, a Q-S (lag or anticipated) synchronization of continuous-time dynamical systems is defined. Second, based on a backstepping design with one controller, a systematic, concrete, and automatic scheme is developed to investigate the Q-S (lag or anticipated) synchronization between the drive system and response system with a strict-feedback form. Two identical hyperchaotic Tamasevicius-Namajunas-Cenys(TNC) systems as well as the hyperchaotic TNC system and hyperchaotic Rossler system are chosen to illustrate the proposed scheme. Numerical simulations are used to verify the effectiveness of the proposed scheme. The scheme can also be extended to study Q-S (lag or anticipated) synchronization between other dynamical systems with strict-feedback forms. With the aid of symbolic-numeric computation, the scheme can be performed to yield automatically the scalar controller in computer.     

Genesio-Tesi和Coullet混沌系统之间的非线性反馈同步

分析了Genesio-Tesi 系统和Coullet系统的特性，表明两系统拓扑不等价但奇异吸引子结构具有一定相似性，而且采用非线性反馈控制方法实现了两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.数值仿真的结 果表明理论分析的正确性. 关键词： 非线性反馈控制 Genesio-Tesi系统 Coullet系统 混沌同步