Projective Synchronization Of Fractional And Integer Order Chaotic Systems

In this paper,we focuses on the dislocated projective synchronization and the mixed function projection synchronization between fractional order and integer order chaotic systems.Considering the complexity of the hyper-chaotic system itself and the uncertainty of the system parameters,this paper takes the hyper-chaotic system as the research object in the mixed function projection synchronization.The theoretical results have potential application value in the field of secure communication.Firstly,the concept of dislocated projective synchronization is described,and the dislocated projective synchronization error of the fractional order Liu system and the integer order Chen system is shown.Based on the mentioned above,a nonlinear controller is designed depended on the conclusion of the small gain theorem.The effectiveness of the controller is proved theoretically.The numerical simulation verifies the rationality of the designed controller.Secondly,the dislocated projective synchronization between the fractional order MAVPD system and the integer order single scroll system is realized.The controller is designed based on the small gain theorem.The dislocated projective synchronization error in the numerical simulation converges to 0,which suggests the effectiveness of the controller.Finally,we study the mixed function projective synchronization and identification of uncertain parameters between hyper fractional and integer order chaotic systems.According to the transformation method between time-domain and frequency-domain,the fractional order operator is converted to the complex frequency domain.By using the integral order operators to approximate the fractional order operators,the synchronization of the fractional and integer order chaotic systems is transformed into the synchronization of the integer order systems with different dimensions.By reducing the dimension,and based on the Lyapunov stability theory,the appropriate controller and parameter identification rules are designed for the system.The numerical simulations verify the validity of the designed nonlinear controller.