On The Normal Form And Synchronization Of A Class Of Chaotic Systems

Since the later of 70's,the research of the chaos systems has become one of the hot problem of scientific fields, received the extensive attention of mathematics, physics, mechanics researcher. Because the remarkable characteristic of the Chaos is the sensitiveness to its initial condition, People think chaotic control and chaotic synchronization are impossible for a long time. Due to the OGY method presented by E.ott, etc. in 1990 and the pioneer works of Pecora and Carrol, The theoretical research and applied study of the chaotic control and chaotic synchronization get the development at full speed. Various kinds of methods of the chaotic control come out one after another and people find the chaotic control and chaotic synchronization can be applied to a lot of fields such as secure communications, chemical, modeling brain activity etc. Now, chaos control and synchronization already become very active research fields.Except OGY method, drive-respond method, people have proposed a lot of synchronization methods in the past ten years, for instance adaptive method, time-delay method, sample-data method, etc.. These methods have different characteristics and applied fields. The way of synchronization can be summarized: complete synchronization, phase synchronization, generalied synchronization, projective synchronization. At present, these is many research of complete synchronization, and the research of phase synchronization, generalied synchronization, projective synchronization have a lot of problems that remain to study.This paper researches on the normal form and synchronization of strict-feedback and general strict-feedback chaos systems. Firstly, we prove that, any strict-feedback chaotic systems can be rendered into a normal form with a invertible transform. The normal form and the strict-feedback are Topological isomorphism and the normal form has more succinct forms. Furthermore we present a systematic design procedure to synchronization a class of chaotic systems in the strict-feedback form. This approach needs only a single controller to realize synchronization no matter how many dimensions the chaotic system contains. Then, the Rossler chaotic system of thethree-dimensional strict-feedback system is taken as a concrete example to illustrate the procedure of designing in the normal form of strict-feedback chaotic systems and in the strict-feedback chaotic systems. Numerical simulations are also provided to show the effectiveness and feasibility of the developed methods. Finally we point out that the method does not work for general strict-feedback chaotic systems, for instance, Lorenz system. Therefore, we propose three kinds of synchronization schemes for Lorenz system, which need only a scalar driving signal to realize synchronization. These methods are enlightening to solve the synchronized problem of general strict-feedback.