The Study On The Theory And Application For Function Projective Synchronization In Chaotic Systems

In the past decades, chaos synchronization has become a hot subject in the field of nonlinear science due to its great potential applications in physical systems, biological networks, and secure communications etc, which has attracted much attention from scientists and engineers. Chaos synchronization means two or more chaotic systems share a common dynamical behavior, which could be induced by coupling or by external forcing. Up to now, there exist many types of synchronization, such as, complete synchronization, phase synchronization, lag synchronization, generalized synchronization, and projective synchronization, etc. Amongst all kinds of chaos synchronization, projective synchronization has been extensively investigated in recent years because it can obtain faster and securer communication with its proportional feature.At present, most of research efforts about projective synchronization have concentrated on studying the constant scaling factor, but it is seldom studying the function scaling factor. It is obvious that the unpredictability of the scaling function can additionally enhance the security of communications, which could be used to get more secure communications. So the studies of function projective synchronization are very important for theory and practical applications. This paper studies the function projective synchronization and the modified function projective synchronization problems, and gives variety control schemes in variety conditions and their applications in the field of secure communication. The main research results in this dissertation can be given as the following four parts.Firstly, we illustrate the definition of function projective synchronization in detail and prove the synchronized chaotic system more complex and chaos. Based on Lyapunov stability theory, the general methods for function projective synchronization between two identical or different chaotic systems are investigated, respectively. The function projective synchronization of a general class of chaotic systems with unknown parameters is investigated by adaptive control scheme. The adaptive control law and the parameter update law are derived to make the states of two chaotic systems asymptotically synchronized up to a desired scaling function.Secondly, we give the definition of modified function projective synchronization, which makes the responses of the synchronized dynamical states synchronizing up to a desired scaling function matrix. Modified function projective synchronization is the more general definition of function projective synchronization, which allows us to flex the scale functions of the different states independently. It is easy to see that function projective synchronization is the special cases of modified function projective synchronization, when all scaling functions of the scaling function matrix take the same scaling function. It is obvious that the unpredictability of the scaling function matrix in modified function projective synchronization can additionally enhance the security of communication. Then, based on Lyapunov stability theory, the general methods for modified function projective synchronization between two identical or different chaotic systems are investigated, respectively. Later, the modified function projective synchronization of unified chaotic systems with unknown parameters is investigated by adaptive control scheme.Thirdly, the problems of lag function projective synchronization and the lag modified function projective synchronization are investigated. Due to that delay may exist extensively in the remote communication system, the studies of the function projective synchronization and modified function projective synchronization with lag time are very important. Based on Lyapunov stability theory, the general methods for lag function projective synchronization and lag modified function projective synchronization are investigated, respectively.Finally, based on function projective synchronization and modified function projective synchronization, the schemes for secure communication are investigated. The secure communication methods using function projective synchronization and modified function projective synchronization are investigated in theory. In this case, synchronization can be completely realized and the information signal can be recovered accurately. Furthermore, these schemes offer much more flexibility in signal transmission and make the third party difficult to break the information signal because the scaling functions are arbitrary and unknown to the interceptor, which could be used to get more secure communications.