Study For The Complexity Of Generalized Synchronization

This dissertation mainly studies the complexity of generalized synchronization(GS) of two unidirectional coupled Lorenz systems when the response system has parametric excitation or external excitation. The sufficient criterion is rigorously proven when two Lorenz systems come to generalized synchronization which is near to chaotic orbits and kinds of complicated periodic orbits. Main research work and contributions are as follows:(1) In the paper the concept, research background and trend in development refer to chaos and generalized synchronization are briefly introduced, some theoretical knowledge and methods used in the dissertation are surveyed.(2) The generalized synchronization (GS) of two unidirectional coupled Lorenz systems is studied when the response system has parametric excitation. Based on the theories of stability, by using method of auxiliary-system, the sufficient criterion is proved when two Lorenz systems collapse to generalized synchronization which is near to chaotic orbits and kinds of complicated periodic orbits. In addition, for the modified response system, by using the Hamilton theorem and Melnikov approach, the sufficient criterions are proved when the modified system collapses to chaotic orbits and asymptotically stable limit cycles of different periodic orbits. Therefore, the existence of generalized synchronization of two unidirectional coupled Lorenz systems under parametric excitation is strictly theoretically proved. All of these have shown that generalized synchronization is more complicated than complete synchronization. Numerical simulations illustrate the correctness of the present theory.(3) The complexity of generalized synchronization (GS) of two unidirectional coupled Lorenz systems is studied when the response system collapses to external periodic perturbation(external excitation). Using the Hamilton theorem, Melnikov approach and method of auxiliary-system to analyze the modified system, the proofs are proposed when the modified system collapses to chaotic orbits and asymptotically stable limit cycles of different periodic orbits. Further, the existence of generalized synchronization of two unidirectional coupled Lorenz systems under external excitation is strictly theoretically proved. The corresponding numerical are presented to verify the effectiveness of this method.