Dynamics Of Fractional-order Chaotic Systems And Its Application To Synchronization

Chaotic system is a highly complex nonlinear system with inherent randomness,the sensitivity of initial values and boundedness.It is widely used in physics,economy,engineering and other fields.Fractional calculus has a history of more than 300 years.Compared with integral calculus,fractional calculus can describe objective physical phenomena more accurately.Therefore,researchers introduce fractional calculus into chaos theory,so that fractional-order chaotic systems have more complex dynamical characteristics,and have broader prospects in secure communication,information processing and other applications.In recent years,the research progress of fractional-order chaotic systems is rapid.Due to its complex characteristics,fractional-order chaotic systems have attracted wide attention in the field of synchronization control.With the rapid development of computer technology,scholars are interested in the synchronization and control of fractional-order chaotic systems.At present,a variety of fractional-order synchronization methods have been proposed,but many methods are not perfect,so it is very meaningful to study the finite time synchronization of fractional-order chaotic systems.The specific work content of this paper is as follows:(1)A four-dimensional switching chaotic system is constructed.The equilibrium point and its stability of the switching chaotic systems are analyzed based on theoretical calculation,and the abundant dynamics of each subsystem are studied by using nonlinear dynamical tools such as coexisting phase diagram,coexisting Lyapunov exponents and coexisting bifurcation diagram.Through the complexity analysis and the comparison of the maximum complexity value,the most complex integer-order subsystem is obtained.The Adomain decomposition method is used to analyze the fractional-order chaotic system and solve the equations.The basic dynamics of the system are presented.After that,the phase diagram is used to observe the special dynamics of the system with different parameters.Finally,Multisim and field programmable gate array(FPGA)are used to implement the circuit of the system,and the obtained results are consistent,verifying the feasibility of the system.(2)A five-dimensional fractional-order chaotic system is constructed.Giving the solution formula according to Adomain decomposition theory,the attractor phase diagram is drawn according to the solution.After that,the equilibrium points of the system are calculated by theoretical method,and we know that there is no equilibrium point.Furthermore,the system parameters are analyzed and the dynamic behavior of the system is studied.The initial value of the system is analyzed and the multi-stability of the system with the change of the initial value is studied.Finally,the results of the analog circuit realized on Multisim are consistent with those of the experimental circuit based on FPGA and the numerical analysis,which proves the correctness of the circuit design and the feasibility of the system.(3)For the fractional-order chaotic system constructed above,an appropriate controller is designed according to the finite-time synchronization method to realize the synchronization of the two systems respectively.The correctness and feasibility of this method is proved by MATLAB simulation results.