Delayed Feedback Control And Synchronization Of Fractional-order Delayed Memristor-based Chaotic Systems

Chaos,which exists widely in nature,originates from deterministic systems and it is sensitive to initial conditions.Therefore,effective control is a key step in the study of chaos theory.The continuous maturity of chaos control and synchronization theory provides a solid theoretical basis for the application of chaos in communication.As a kind of nonlinear two-terminal element with memory function,Its memory and non-linearity are not possessed by the other three basic circuit elements(resistors,inductors and capacitors).With the development of the mathematical and circuit model of the memristor,the memristor is used to replace the non-linear circuit elements in the traditional chaotic circuit.which shows complex dynamic phenomena.With the development of fractional calculus theory and considering the existence of time delay factor,it can produce more complex and changeable chaotic signals by introducing the fractional order theory into the memristive chaotic system.In this paper,the application of fractional calculus theory in time-delay memristor chaotic circuits,and the related controller is studied to control the chaotic state of the system,analyze its dynamic behavior,and achieve the synchronization effect.With the help of synchronization of the chaotic system in the field of communication,its application value can be significantly improved.In this paper,the control and synchronization of fractional delayed memristive chaotic systems are studied.Firstly,a linear delayed feedback controller is proposed to control chaos and Hopf bifurcation in a fractional active delayed memristive chaotic systems.At the same time,the active control strategy is used to synchronize and anti-synchronize the drive system and the response system.The main innovations of this paper are as follows:(1)On the basis of fractional active delayed memristivechaotic system,the dynamic equation of fractional order delayed memristive chaotic system is derived.Due to the lack of research on the control and synchronization of fractional-order delayed memristi've chaotic systems,a linear delayed feedback controller is proposed in this paper to control the chaotic state of the system.(2)By using Lyapunov stability theorem,the sufficient conditions for the stability of the equilibrium point of the memristive chaotic system are studied,and the influence of feedback gain parameters on the critical value of the fractional-order and delay parameter in the controlled system is discussed.Through theoretical analysis and numerical simulation,the correctness of the numerical solutions of bifurcation points of time delay parameters and fractional-order parameters is verified under different feedback gain conditions.(3)Based on fractional active delayed memristive chaotic system,active control strategy is further used to achieve synchronization and anti-synchronization effect between drive system and response system.At the same time,based on the Lemma of stability theory,the effects of time delay and fractional-order parameters on the synchronization speed of the system are analyzed.Finally,the simulation results show the validity of synchronization and the correctness of theoretical analysis,which provides a broad application prospect for the later application in the field of secure communication.