Multiple Synchronization Of Integer And Fractional Order Chaotic Systems

In this paper,the author primarily studies the multiple synchronization between the integer-order and fractional-order chaotic systems.Considering that the research above involves at least three chaotic systems,coupled with the big differences of characters between the integer and fractional order chaotic systems,it is strongly anticoded in communication.Therefore,the research achievements in the article have potential application value in secure communication field.Firstly,the concept of dual projective synchronization is described by the author.Then,the dual projective synchronization errors between the integer order and fractional order chaotic systems are shown.As for dual projective synchronization between the different chaotic systems,the condition of which can achieve inconsistent order chaotic systems is provided according to the tracking control strategy and the stability theorem relevant to fractional order differential equation.Secondly,the author respectively realize the dual projective synchronization of the pair of integer order Sprott-Lü chaotic system and their homologous fractional order chaotic systems,the complete dual projective synchronization between the pair of integer order Liu and Chen chaotic system and their corresponding fractional order chaotic systems.The tracking controllers are designed based on the tracking control strategy.The dual projective errors in numerical simulation are converged to 0,which proves the validity of the proposed controllers.Finally,the combination synchronization of integer and fractional order chaotic systems is studied.Based on the transformation method between time domain and frequency domain,the fractional order operator is converted to the complex frequency domain through the Laplace transformation.By using the integral order operators to substitute the fractional order operators,the combination synchronization of the fractional and integer order chaotic systems is shifted into the combination synchronization of the integer order systems with different dimensions.By reducing the dimension,the appropriate controller is designed for the systems on basis of the active backstepping control method and the Lyapunov stability theory.The author testifies the effectiveness of the designed nonlinear controller by discussing the difference of combination coefficient matrix and relevant numerical simulations.