Mechanism Of Synchronization Types And Generalized Synchronization Of Coupled Smooth Chaotic Dynamical Systems

Mechanism of synchronization types of coupled smooth chaotic dynamical systems and the generalized synchronization of near-identical chaotic systems are the important and open problems in the research of nonlinear dynamics in recent years. Supported by the National Natural Science Foundation of China under Grant NO.10702065, stability of the various types of synchronization of coupled smooth chaotic dynamical systems and the generalized synchronization of near-identical chaotic systems are investigated in this dissertation. In Chapter 1, the preparative knowledges are introduced, including the basic concepts of stability and the theory of exponential dichotomy. In Chapter 2, the necessary conditions of the existence of the various types of synchronization of the coupled smooth chaotic dynamical systems are obtained by the invariance of the synchronization mani-fold. The results show that if the synchronizations with time difference (i.e., anticipatory, lag and complete synchronization) exist, then they should be complete synchronization for the linearly unidirectional and bidirectional coupled smooth chaotic identical dynam-ical systems without coupling time delays and perhaps be lag, anticipatory or complete synchronization for the linearly uni-/bi-directional coupled smooth chaotic identical dy-namical systems with coupling time delays. And for the latter case the time difference must respectively be the coupling delay or half of the difference between the coupling time delays. Then by constructing the Lyapunov functional, and the Lipschitz property of the smooth chaotic systems and the L2-Cauchy inequality, the stability of the various syn-chronization of the uni-/bi-directional linearly coupled smooth chaotic identical dynamical systems with or without coupling time delays are deduced. The results show that if the coupling strengths are large enough, the synchronization is asymptotically globally stable for the uni-/bi-directional linearly coupled smooth chaotic identical dynamical systems without coupling delays and the delayed unidirectional linearly coupled systems. But for the stability of the synchronization of the delayed bidirectional linearly coupled systems, not only the above conditions but also the condition that the coupling delays are small enough must be met. At last, the numerical simulations are executed by the nonlinear dy-namics software, WinPP. In chapter 3, the generalized synchronization of the master-slave near-identical chaotic systems is considered by the exponential dichotomy approach. The considered systems include Lu system, Chen system and Genesio system, etc. The syn-chronization of the master-slave systems with the minor perturbation of the parameters in the slave is analyzed. The results show that if the coupling strengths satisfy some certain conditions, the master is synchronized by the slave. Then they are numerically simulated by employing WinPP. In chapter 4, We present the conclusion.