Stability And Chaos Synchronization For Several Nonlinear Systems

Nonlinear science is a frontier discipline which studies the common nonlinear phe-nomena in di?erent disciplines. It is also a comprehensive discipline which developedon the basis of subjects characterized by nonlinear behavior. Since the study on thenonlinear dynamics has not only theoretical significance but also practical significance,it is very important in the research of nonlinear science. Stability is the preconditionfor the control system design, while chaos is one of common characteristics of nonlinearsystems. Recently, the stability and chaos synchronization of nonlinear systems receivemuch attention in nonlinear dynamics. After a brief introduction to the background andresearch progress in the filed, the research work focus mainly on two parts. The first partdiscusses the stability of delayed neural networks (DNNs), since DNN is an importantclass of nonlinear systems. The second part investigates chaos synchronization of severaldi?erent nonlinear systems.The main contributions on the stability of delayed neural networks are consideringinterference factors and extending neural network models. On the one hand, there mayexist some interference factors in delayed neural networks, for instance, parameter un-certainties, stochastic disturbance, impulse e?ects, etc. By using Lyapunov functional,Young and Halanay inequalities, the robust exponential stability for uncertain BAM neu-ral networks and the exponential stability for impulsive cellular neural networks are stud-ied. On the other hand, the thesis establishes some new delayed neural networks models,such as Markov jump neural networks, T-S fuzzy neural networks and higher-order neuralnetworks. Based on Lyapunov stability theory, stochastic analysis approaches and decom-position of the state space, the robust stability of Markov jump Cohen-Grossberg neuralnetworks, the robust stability of T-S fuzzy cellular neural networks and the multiperiod-icity of higher-order Cohen-Grossberg neural networks are investigated.The main contributions on chaos synchronization of several di?erent nonlinear sys-tems in the thesis include the following three points. Firstly, the control methods of chaossynchronization are improved. The methods in modern control theory, such as adaptivecontrol, robust H∞control and pinning control are applied to the chaos synchroniza-tion. Moreover, the adaptive synchronization of chaotic delayed neural networks, theH∞synchronization of delayed chaotic neural networks and the pinning control synchro-nization of generalized complex dynamical networks are analyzed. Secondly, the chaossynchronization notations are extended. In addition to complete synchronization, thelag synchronization and projective synchronization of chaotic systems are studied. Theadaptive hybrid lag projective synchronization (AHLPS) problem of a class of unifiedchaotic systems with channel time-delay and parameter uncertainty is investigated. TheAHLPS is a new type of chaos synchronization, which includes complete synchroniza-tion, anti-synchronization, lag synchronization and projective synchronization. Thirdly,the external conditions of chaos synchronization are relaxed. The synchronization of aclass of neural networks with impulsive noise is investigated by analyzing the exponential stability of error system. Furthermore, designing laws for the controlling gain matrix inthe synchronization of neural networks are proposed via output or state coupling.