Dynamical Analysis Of Several Fractional Order Chaotic Systems Based On Logistic Models

Logistic system is one of the hot issues in the research of non-linear science.The chaotic properties of high-dimensional logistic map have practical significance for the study of ecology and other fields.Research on transition from one-dimensional chaotic map to high-dimensional map,the two-dimensional logistic system and related derivative systems have important cohesive functions.The research on period doubling bifurcation and chaos control is used to solve and control more complex high-dimensional dynamic systems.The study has important value for reference.Fractional(non-integer)chaotic systems with non-local characteristics are suitable for describing systems with characteristics of memory and heredity,showing abundant dynamic phenomena.Also it is more suitable for describing physical characteristics of real systems showing broad application prospects.The fractional-order model of Wright differential equation system and twodimensional logistic chaotic system derived from logistic equation are studied.And this dissertation extends it to the fractional order case and introduces a discretization.The dynamics characteristic of the model is fully explored.1.The dynamics behavior of Wright delay differential equations was analyzed.At present,the study on the existence and uniqueness of solutions of nonlinear fractional order delay differential equations is relatively preliminary.And the traditional Lipschitz condition has certain limitations in solving practical problems.Therefore,with the Banach fixed point theorem,this chapter proves the existence and uniqueness of solutions of system.Also,using the theory of fixed point and Jury criterion proves the existence of Neimark-Sacker bifurcation mathematically.The numerical simulation uses phase diagram,bifurcation diagram and Lyapunov exponent diagram to study the influence of fractional order ?,time delay ? and system parameters ? on the chaos of the system.The obtained results are consistent with the theoretical analysis.2.Based on the existing two-dimensional integer order logistic model,a class of two-dimensional fractional logistic chaotic dynamic systems with coupling terms is established and its dynamic characteristics are studied.It is known that the stability of the fixed point is related to the maximum eigenvalue of the Jacobian matrix of the two-dimensional logistic chaotic model in the fixed point.Therefore,the first bifurcation of the two-dimensional logistic difference model in the parameter space is derived.The boundary equation indicates that the system leads to chaos according to the Pomeau-Manneville approach,and its intermittent is related to hopf bifurcation.After that,focusing on the change of two-dimensional attractors with the change of control parameters reveals the singular characteristics of chaotic motion.Through numerical simulation,obvious period-doubling branches and various periodic windows leading to chaotic and chaotic regions can be observed.The results show that the Wright differential equation system and the two-dimensional Logistic chaotic system derived from Logistic equation are feasible to the fractional order domain.The discretization method has good applicability to the model.The related processing methods and conclusions of this dissertation can provide theoretical guidance to the study of fractional models.