Dynamic Analysis And Application Of Lü System And Its Extended System

The Lü system acts as an intermediate system between the Lorenz system and the Chen system,realizing the transition from one system to another with the in-depth study of the whole Lorenz system family.In this paper we analyzed the dynamic behavior of extended Lü system and its application in engineering field based on the Lü system.We reviewed the overall development process of chaos and the current research results of the Lü system.Then the definition and characteristics of chaos in nonlinear system and several common concepts are introduced.Due to the expansion of the fractional order system on the integer order system,the definitions and properties of several common fractional order differential systems are described.Since the Lü system is a transitional system,its significance is self-evident.Here we started with the integer order extended Lü system,analyzed the basic characteristics such as the stability of the equilibrium point and the dissipation in the system.We expounded several common bifurcation theories and control methods in integer-order systems.Because the time delay factor has a great influence on the system,we introduced the time delay factor into the integer order extended Lü system.The situation of equilibrium point and the Hopf bifurcation characteristic in the delay system are analyzed.And the linear feedback method is adopted to control the non-time delay system and the time delay system.As the fractional order system is an extension of the integer order system,the fractional order extended Lü system model is constructed by using the definition of fractional order differential equation,the equilibrium point of the fractional order system is studied,and the chaotic characteristics of the fractional order system are analyzed by using the fractional order discretization process.There are many parameters that affect fractional-order system.The integrity of system equations is more important during the system analysis.We explored the dynamic behavior of fractional-order system in the aspects of system order and system parameters.We started form the Lyapunov index.Lyapunov control method and Lyapunov function construction method are used to determine the overall stability and boundary of the whole system.At present,the engineering application of nonlinear systems mainly embodies information communication transmission and circuit equation encryption.Therefore we designed an appropriate synchronization controller to information synchronization transmission.For the encryption application in the circuit,the circuit of Lü system is mainly designed to realize the encryption application in the circuit transfer process.We expanded the scope of the Lü system and analyzed the extended integer-order system and fractional-order system in this paper.We focused on the dynamics behaviors and application of the system,which has great theoretical significance and practical application value.