Complex Dynamics And Chaos Control In Discrete And Continuous Dynamical Systems

This thesis investigates the bifurcation of fixed points and periodic orbits, chaotic behaviors and chaos controlling of dynamics, as parameters varying, by using bifurcation theory, chaos theory, Melnikov method and second-order averag-ing method in discrete and continuous dynamical systems. For discrete dynamical systems (including the predator-prey system without Allee effect, predator-prey system with Allee effect and mathematical model for tissue inflammation), we ob-tain conditions of the existence for fold bifurcation, flip bifurcation, Hopf bifurca-tion and Marotto’s chaos by using the bifurcation theory, center manifold theorem and chaos theory. For continuous dynamical systems, on the one hand, we give the criteria for the existence of chaos under periodic perturbations by using Melnikov method to the pendulum equation with a phase shift. On the other hand, by applying the second-order averaging method and Melnikov’s method, we obtain the criteria for the existence of chaos in an averaged system under quasi-periodic perturbations. Moreover, we give numerical simulations including bifurcation di-agrams, Lyapunov exponents, phase portraits and fractal dimensions, which not only demonstrate these analytical results, but also show richer and more complex dynamic behaviors, for example, the discrete dynamical systems display different interior crisis, new chaotic attractors, non-attracting chaotic set, different routes to chaos, the symmetry-breaking of18-pairs periodic orbits and n(n>18)-pairs periodic orbits, and so on. At last, we investigate the problems of controlling chaos in pendulum systems, we use Melnikov methods to pendulum systems such that chaotic states are controlled to period-motions, and give the corresponding numerical simulations to confirm analytical results.This thesis consists of four chapters as following.In chapter1, we give the background knowledge about the bifurcation and chaos of dynamic systems. A brief review about the center manifold theorem, Melnikov method and second-order averaging method for discrete and continuous dynamical systems are showed. Some definitions and characteristics of chaos, and some routes to chaos are presented.In chapter2, we investigate the predator-prey system without Allee effect and predator-prey system with Alice effect are investigated. By using center manifold theorem and bifurcation theory, we obtain conditions for the existence of flip bi-furcation and Hopf bifurcation, respectively, and prove there are Marotto’s chaos according to the definition of Marotto’s chaos. Numerical simulations confirm ana-lytical results and give more richer dynamical behaviors. Then we get the discrete version of the mathematical model for tissue inflammation by using Euler method, and study the discrete model. Conditions of the existence for fold bifurcation, flip bifurcation and Hopf bifurcation are obtained by applying center manifold theo-rem and bifurcation theory. At the same time, we prove there are Marotto’s chaos according to the definition of Marotto’s chaos. Numerical simulations demonstrate analytical results, richer and more complex dynamical behaviors.In chapter3, the pendulum equation with a phase shift is studied. We obtain the criteria for the existence of chaos under periodic perturbations by using Mel-nikov method. At the same time, by applying the second-order averaging method and Melnikov’s method, we obtain the criteria for the existence of chaos in an averaged system under quasi-periodic perturbation for Ω=nω+εv, n=1,2,4. Numerical simulations confirm analytical results and show more complex dynam-ical behaviors.In chapter4, we investigate the chaos control for pendulum equations. The-oretically, we give conditions of the suppression of homoclinic and heteroclinic chaos by using Melnikov’s method, respectively. The numerical simulations and illustrations are given.