Chaos Generalized Synchronization And Its Control

Chaos is one of the main contents in nonlinear science and become an active research subjectin the face of nonlinear researcher of21st century. People have had a well acknowledge of chaoticdynamical behavior in many scientific fields.Chaos synchronization is the important chaotic dynamical behavior. At present, differentschemes of chaos synchronization have been studied, which include complete synchronization,projective synchronization, lag synchronization, functional synchronization, generalizedsynchronization (GS), etc. Among them, generalized synchronization (GS), which is a ubiquitousphenomenon in nature, may have much more complexity and mystery. The theoretical studies ofgeneral synchronization, however, is not enough comparatively. Since Nikolai. F.Rulkov etc. firstlydetected and described the phenomenon of generalized synchronization in1995, GS has attractedmuch attention of researchers and become a hot issue. GS also has been applied to secretcommunication, communication encryption, biology project, etc. So the study of generalizedsynchronization has the important theoretical significance and applicable value.This paper studies chaotic generalized synchronization and its control. The fundamentalacknowledges of chaos and basic principles concerned in this paper are introduced briefly in theexordium. Then the problems of GS on chaotic systems are mainly discussed, including GS ofbi-directionally coupled chaos systems by impulsive control, GS of a given manifold and adaptiveGS. Some conclusions and the prospects of research in the future are given in the last section.Details are as follows:1. Bi-directionally coupled chaos systems with impulsive control are proposed. Sufficientconditions for realizing generalized synchronization are obtained by using the auxiliary system.Numerical simulations about chaotic systems and hyper-chaotic systems illustrate the theoreticalresults effectively.2. The studies of generalized synchronization often assume that the manifold of GS is linear ornonlinear. Because there is no a united consideration on all kinds of GS manifolds, so new coupledchaotic systems are given to make manifolds of GS more general in this part. The sufficientconditions are obtained when the systems reach the GS. The effective natures of theoretical resultsare supported by numerical simulations.3. Based on Lyapunov stability theory, the adaptive generalized synchronization problems oftwo types of bi-directionally coupled chaos systems with proposed manifolds are considered. Thesufficient and less conservative conditions are theoretically presented when the systems reach the GS. The message signal is covered in chaotic system with a given manifold for transmission by theuse of GS scheme. Because the required manifold can be changed, there is the correspondingchaotic state. So this approach can effectively improve information security.