The Application Of The Small Gain Theorem To Chaos Synchronization

As a complex nonlinear dynamical behavior, chaos is a general phenomenon in nature. Because of its extreme sensitivity to initial conditions, and its high randomness generated by deterministic equations, the birth of chaos has been a source of great excitement in scientific and engineering communities. At the beginning of 1990's, chaos synchronization was presented. Since then, due to its great applications in secure communication, information science and biotic science, chaos synchronization has become a hot spot of research.In this dissertation, firstly, the conceptions of chaos and chaos synchronization are reviewed and some methods on chaos synchronization such as, drive-response synchronization,coupling synchronization,adaptive synchronization,out-put feedback synchronization are recalled.Secondly, the small gain theorem is introduced in chapter 4. This theorem is an important but difficult theorem in nonlinear systems and it is very useful in judging input-to-state stability. Based on this theorem, a new method is proposed for chaos synchronization. By this method we divide error dynamical system into two subsystems and use a suitable linear state feedback controller to make the two subsystems be input-to-state stable, hence we can get the range of the controller's parameter by the small gain theorem, and thus the error dynamical system becomes globally asymptotically stable. Therefore the chaotic system can be synchronized.Finally, we discuss the applications of this method in chaos synchronization. Three examples and a numerical simulation are shown to verify the feasibility of this method.