Nonlinear Dynamic Analysis Of The Fractional-order Delayed Memristive Chaotic Circuit System

Memristor,known as memory resistor,has a memory function with a non-linear components,which is different from the characteristics in the other three basic components of the circuit(resistor,capacitor,inductor).Not only can it remember the number of its charge and change the corresponding resistance by controlling the dynamic change of the current,but also the change characteristics of the current can be maintained at the time of power failure.Because of those characteristics in the memristor,it has a very wide range of applications in computer engineering,neural network,electronic engineering,communication engineering and other fields.Because of its new non-linear components of the circuit to show complex dynamic,it can produce complex chaotic signals to apply for the field of nonlinear dynamics.Moreover,the non-linear elements in the traditional chaotic system can be replaced by memristor,which can form a large class of chaotic circuit system based on memristor.With the development of fractional calculus and the existence of delay in the process of constructing the circuit,the theory of fractional calculus is introduced into the model of the delayed memristive system,and then the complex dynamics of these systems are analyzed,which is a subject to study the influence of the multiple parameters such as time delay,fractional order and system parameter on the dynamic behavior of the memristive chaotic system.In addition,the topic is not only has it an important theoretical research significance,but also it has a good practical application prospect for the application of the chaotic circuit in information security and confidential communication.Based on the definition of memristor and the recent theory of fractional delayed system,we propose a variety of the fractional order delayed memristive models within those theories to analyze the dynamics in the chaotic circuit system.At the same time,the stability interval of the memristive system is analyzed by studying the time-delay,fractional-order,and system parameter.Then,the transversality condition of the Hopf bifurcation theory with multiple parameters is used to obtain.Finally,the complex nonlinear dynamics of the system with multiple parameters was analyzed.The innovation of this thesis is as follows:(1)The conventional delayed memristive circuit model is extended to the fractional order system,and the model of the fractional-order delayed memristive circuit is investigated,which makes the memristive chaotic system more concise so that revealing the essential features in the chaotic system.(2)By using the Lyapunov stability theory,we study the sufficient conditions of the stability of equilibrium points based on the time delay.The stability intervals of the time delay are obtained,and the influence in multiple parameters such as time delay and fractional order is summarized.The transversality condition of the Hopf bifurcation is given.(3)The multiple parameters(time-delay,fractional-order,system parameters)of the system are firstly applied to analyze the influence on the dynamic behavior of the system.When the time delay and the fractional order of the system are changed,the complex dynamic behaviors such as bifurcation,period and chaos in the fractional order delayed memristive circuit system are revealed.Furthermore,corresponding critical values of time delay are determined,showing the orders for generating chaos,which can provides a wide range of applications for secure communication.