Research On Universal Function Projective Synchronization Control Of Chaotic Systems

Dynamic characteristics of chaotic systems are complicated and unique. There exist great potential applications about chaotic systems in many fields of science. Researches on chaotic synchronization have been by far an international hotspots, in which chaotic projective syn-chronization is an important one. As the projective synchronization occurs, the drive system and the response system evolve up to some scaling factor. Many domestic and international scholars generalize the scaling factor to a diagonal matrix, to a function or a function matrix. Various kinds of projective synchronization are presented. The generalization enlarges the range of application and enhances the flexibility of application of projective synchronization. In this dissertation, we investigate that there is some "displacement" between the drive system and the response system. The definition of universal function projective synchronization(UFPS) is first proposed. An adaptive control method is presented for universal function projective synchro-nization of chaotic systems with unknown parameters. Then an active variable universe fuzzy adaptive control method and a fast terminal sliding mode control method are presented for u-niversal function projective synchronization of chaotic system with uncertainty. Moreover, a new kind of chaotic system—HX-type chaotic system is defined. Parallel distribution compen-sation method is applicable to universal function projective synchronization of the systems. At last, considering the time delay, the definition of universal function projective lag synchroniza-tion is introduced. An integral sliding mode control method is presented for universal function projective lag synchronization of chaotic systems.The details of this dissertation are as follows: 1. Considering some "displacement" may exist between the drive system and the response system, universal function projective synchronization of two chaotic systems is defined, which further generalized the definition of projective synchronization. The "displacement" is described by the state of a reference system. The reference system can be a constant vector, a periodic sys-tem, a quasi-periodic system, a chaotic system, a hyperchaotic system or their combination. By setting the reference system and the scaling function matrix, universal function projective syn-chronization can degenerate to complete synchronization, anti-synchronization, projective syn-chronization, function projective synchronization, modified function projective synchronization. Based on the Lyapunov stability theory, an adaptive control method is derived such that universal function projective synchronization of two different hyperchaotic systems with unknown param-eters is realized, which is up to scaling function factor matrix and four kinds of reference systems, respectively. Simulation results illustrated the effectiveness of the proposed control method.2. Based on three necessary conditions of hyperchaotic system, a new hyperchaotic sys-tem, which has a convergent line, is introduced by adding an additional state variable to Lorenz system. The dissipativity, equilibriums, bifurcation diagram, Lyapunov exponents, Lyapunov dimension, Poincare section are studied. With proper parameters, the system can be periodic, quasi-periodic, chaotic or hyperchaotic. In a large parameter range, the system is hyperchaot-ic. With an active variable universe adaptive fuzzy control method, universal function projective synchronization of the hyperchaotic systems with uncertainty is realized. Universal function pro-jective synchronization of the new hyperchaotic system in the sense of hyperchaotic Liu system illustrates the effectiveness of the control scheme.3. A four-dimension system with three equilibriums is constructed by adding a feedback term to the second equation of Lorenz system. The nonlinear characteristics, such as dissipa-tivity, the existence of attractor, equilibriums, bifurcation diagram and Lyapunov exponents, are analyzed. The change of system parameter causes change of dynamic behavior. Lyapunov ex-ponents spectrum suggests that the system has rich dynamic characteristics. In a broad range of system parameter, the system is hyperchaotic. Fast terminal sliding mode control method is presented for universal function projective synchronization of different hyperchaotic systems with uncertainty. UFPS between the new hyperchaotic system and hyperchaotic Chen system illustrates the effectiveness of the proposed control method.4. The HX equations of chaotic system of which right side is polynomial is achieved by us-ing termwise fuzzy inference modelling and linear superposition method. Not all the coefficient of each term of the HX equations varies. Some HX equations are identical to the original chaotic system. Because of universal approximation of fuzzy system, the HX equations with variable coefficients are chaotic(hyperchaotic). We call it HX-type chaotic (hyperchaotic) system. We can get a family of HX-type chaotic (hyperchaotic) system by simply changing the fuzzy parti-tion. The exact T-S fuzzy models of three kinds of HX-type hyperchaotic systems are achieved by using exact T-S modelling method. Three HX-type systems illustrated the modelling method. Based T-S models, UFPS of HX-type hyperchaotic is accomplished via piecewise parallel dis-tributed compensation technology. 5. There always exists time delay in control engineering. Based on this fact, universal func-tion projective lag synchronization is proposed. The synchronization scheme is a generalization of universal function projective synchronization and modified function projective lag synchro-nization. Because of involving the time delay, this synchronization scheme can be used widely. Active sliding mode control and active fuzzy sliding mode control are introduced to universal function projective lag synchronization of chaotic systems.