The Generalized Synchronization Of Continuous System

The generalized synchronization may have great potential application in chaotic secure communication, and may have relation with the patterns in system science. So the problems of the generalized synchronization of chaotic is useful. But a very little work has been done to the generalized synchronization.Besides, the problems of generalized synchronization have been extended in this paper. We got the important achievements that the predecessors have not yet received. This paper studies two types of generalized synchronization. according to the method of auxiliary-system, we class the generalized synchronization into several types. The chapters are arranged as follows:In the first chapter, a brief introduction to history background, progess and typical methods for chaos and generalized synchronization.In the scond chapter, according to the method of auxiliary-system and the theory of stability, the generalized synchronization of two different unidirectional coupled Lorenz systems is studied. Then we proved the boundary of the responsed system, a sufficient criterion is rigorously proven. Furthermore, GS is classified into three types based on the modified system approach. Moreover, using the Routh-Hurwits theorem, and analyze the stability of equilibrium of the modify system, the existence of the first type,the second type of GS are strictly theoretically proved.The third chapter studies the existence of H(o|¨)lder continuity of generalized synchronization of bidirectionally coupled non-autonomous systems. Moreover, the non-smooth and multivalued property of GS manifold are strictly given. According to the modified approach, GS is classified into ten types, when the two modified systems of drive and response systems have asymptotically stable periodic cycle, asymptotically stable equilibrium or asymptotically stable quasi-periodic cycle. Based on the Schauder Fixed Point Theorem of H(o|¨)lder continuity function space under certain conditions, the existence of H(o|¨)lder continuous GS manifold are theoretically proved.The numerical simulations are given to illustrate the effectiveness in this paper.