Synchronization Of Integer-order And Fractional-order Different Nonlinear Chaotic Systems

With the unceasing improving of the fractional calculus theory,chaos synchro-nization,as a significant branch of nonlinear science,has already obtained a lot of popularity.Since the fractional-order chaotic system has more regulatable variables than the integer-order one,synchronization between fractional-order chaotic system and integer-order chaotic system may produce a kind of hybrid chaotic transient sig-nal,which is conducive to ameliorate the safety of communication.Therefore,syn-chronization between fractional-order and integer-order chaotic systems with same dimensions has become a hot topic.While,most practical nonlinear chaotic models usually have non-identical dimensions,model uncertainties(such as unknown exter-nal disturbance?parameters perturbation)also have impact on the control results.In order to get a better control effect,it is necessary to attach great importance to such problems in the study.Based on Lyapunov stability theorem,to solve the above problems,this paper mainly discusses the synchronization problems between fractional-order and integer-order different nonlinear chaotic systems.The main work can be summarized as follows:1.Synchronization of integer-order and fractional-order chaotic(hyper-chaotic)systems with different dimensions is studied.By constructing two scaling matrices(i.e.,a non-identity constant,matrix and a function matrix),the synchronization error is defined.Based on the Lyapunov direct method,with respect to different forms of systems,controllers are designed respectively,and the square Lyapunov functions are used in the stability analysis.In this chapter,a synchronous approach is proposed for achieving syricironization of different nonlinear chaotic systems with non-identical dimensions,which is also fit for the synchronization problems of chaotic systems with same dimensions.2.Adaptive function projective synchronization of uncertain integer-order and fractional-order nonlinear chaotic systems with unknown external disturbances is discussed.In view of the fractional Lyapunov stability criterion,a controller with function matrix is built to realize synchronization,and a fractional-order adaptive control law is designed to adjust the related parameters of the external disturbances automatically.Without knowing the specific upper boundary value of the external disturbances,the synchronous control of uncertain nonlinear chaotic systems with different structures is achieved.The experimental results confirm the robustness of the proposed control method.