Design Of New Dissipative And Conservative Chaotic Systems And Its Synchronous Control

As a complex dynamic behavior peculiar to nonlinear systems,chaos exits widely in social life and nature.Since Lorenz discovered the first chaotic attractor in 1963,chaos theory has been greatly developed in many engineering fields.Therefore,the establishment of a new chaotic system model with complex dynamic behavior or new dynamic behavior has become a research hotspot in nonlinear disciplines.In recent years,scholars have found that the fractional order chaotic system also has rich dynamic behaviors,and synchronization control of the fractional order system was studied,the synchronization of fractional order of the lack of effective research method and mathematical tools,the research of fractional order control method needs further improvement.This paper first puts forward a system G < 0(the sum of the diagonal elements of the linear matrix is less than zero)of the novel of dissipative system as the research object.It researches the dynamic behaviors of the system,and the coexistence attractors in the system are identified.The phase diagram,time sequence diagram,power spectrum,Lyapunov exponents spectrum and bifurcation diagram of the system are given,and the system is analyzed by topological horseshoe theory,which proves that the system has chaotic attractors.The simulation circuit of the system is also designed,and the correctness of the analysis is verified by Multisim simulation software,and the FPGA is further verified.Based on the proposed system,the corresponding fractional order system is given.It is found that the new chaotic system and its fractional order system have coexistence double-wing attractors,and there are four-wing chaotic attractors with different structures.The parameters of the fractional order chaotic systems is completely unknown synchronization problems,so the adaptive synchronization control method is used to estimate the unknown parameters of the fractional order system.Simulation results show that the larger the control parameter k,the faster the system synchronization speed,and the larger the control parameter ?,the faster the identification of the system parameters.Secondly,the paper puts forward a dissipative system G > 0 as the research object and the dynamic behaviors of the system are analyzed.It also gives the phase diagram,Lyapunov exponents spectrum,and the topologic horseshoe and topological entropy in the system by the theory of topological horseshoe and numerical calculation are also discussed.Multisim simulation and FPGA implementation are carried out.Based on the proposed system,the corresponding fractional order system is constructed.It is found that the new dissipative system and their fractional order systems have coexistence attractors.The coexistence of the double-wing attractors and the four-wing attractors depend on the distance from initial points to unstable equilibrium points.For the parameters of the fractional order chaotic systems is completely unknown synchronization problems,the adaptive synchronization control method is used to estimate the unknown parameters of the fractional order system.Simulation results show that the larger the control parameter k or ?,the faster the system synchronization or identification of the system parameters speed.Finally,the paper puts forward a new conservative chaotic system without equilibrium points as the research object,the mathematical properties of the system are analyzed and the phase diagram,Poincaré mapping and Lyapunov exponents spectrum are given.Moreover,the topological horseshoe and entropy are obtained by the aid of the topological horseshoe theory and numerical computation.Multisim simulation and FPGA implementation are also carried out for the conservative chaotic system.Based on Barbalat theorem and Lyapunov stability theory,the adaptive synchronization control of conservative chaotic system is discussed.Simulation results show that the adaptive synchronization control method is not only applicable to the fractional order system,but also to the conservative system.