Extended Controller Designs On The Synchronization Of Different Chaotic Systems

Chaos forms a chaos theory gradually since it has been found. With the development of chaos, study has been carried on the chaos by more and more people. People began to find that chaos exits in every corner of our lives, so chaos has a wide range of purposes such as the life sciences, confidential communications and some other fields. Chaos control and synchronization in the recent years has been a hot topic that people research in. Synchronization of the chaotic systems and its applications have attracted more and more attentions in the recent years, and different kinds of synchronization have been proposed such as complete synchronization, anti-synchronization, phase synchronization. From the control theory's point of view, we can design a controller for the response system and the response system's variables need to be rebuilded so that it can make them realize the synchronization between the drive system and the response system. The main content is as follows:In this article, we firstly summarize the relevant contents about chaos theory, including the definition of chaos, the applications of chaos in real life, the origin of chaos theory, the characteristics of chaos, the methods and meaning of chaotic synchronization and so on. How to apply the study and the achievement to server people has become an important topic in the nonlinear science development. Besides, we also introduce the Lyapunov characteristic. This chapter mainly lays the groundwork for the next chapters.Then we make use of the stability condition based on the phase angle which was proposed by Matignon in 1996 to develop a new method on the controller design, and we do not need to calculate Lyapunov index, besides, we combined the linear, matrix and other theories of knowledge the studied in class and propose a new method to gain the complete synchronization of the different chaotic systems. This method is simple and fast, reliable, and it greatly reduce the time complexity. At last, we carried out a Numerical simulation to demonstrate the proposed method.With the development of the study on generalized projective synchronization (GPS) in the chaotic system, various approaches have been proposed in the literatures for generalized projective synchronization(GPS) of chaotic systems. We develop an extended method on the generalized projected synchronization and prove the method. We need the constant parameters of the drive system to realize the synchronization of the two different systems, at the same time, we don't need to calculate Lyapunov index. Based on the method, we design a nonlinear controller according to the scale factor of generalized projective synchronization (GPS). At last, we make use of some different typedef chaos systems to do the Numerical simulations, and we demonstrate the method is effective.