Complex Dynamics For Two Types Of Equations In Continuous And Discrete Dynamical Systems

In this thesis, we investigates the bifurcation of fixed points and resonant so-lutions and chaos for two types of equations in continuous and discrete dynamical systems, which are not considered yet, as the bifurcation parameters vary by ap-plying bifurcation theories, second-order averaging method, Melnikov method and chaos theory in continuous and discrete dynamical systems.For the continuous system, the complex dynamics for the physical pendulum equation with suspension axis vibrations are investigated. Firstly, we prove the conditions of existence of chaos under periodic perturbations by using Melnikov's method. By using second-order averaging method and Melinikov's method, we give the conditions of existence of chaos in averaged system under quasi-periodic perturbations forΩ=nw+(?)v, n= 1 - 4, where v is not rational to w, and can't prove the condition of existence of chaos for n= 5 - 15, and can show the chaotic behaviors for n= 5 by numerical simulations. By numerical simulations including bifurcation diagrams, phase portraits, computation of maximum Lyapunov expo-nents and Poincare map, we check up the effect of theoretical analysis and expose the complex dynamical behaviors, including the bifurcation and reverse bifurca-tion from period-one to period-two orbits; and the onset of chaos, and the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly dis-appearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parametersα,δ,f0 andΩ; and the onset of invariant torus or quasi-periodic behaviors, the entire invari-ant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to pe-riodic orbit; and the jumping behaviors which including from period-one orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors and invariant torus. In particular, the system shown the entire chaotic region or invariant torus region or entire quasi-periodic region suddenly converting to periodic orbit by adjusting the bifurcation parameters a,α,δ,f0 andΩ, which is beneficial to the control of motions of the pendulum.Secondly, we investigate the existence and the bifurcations of resonant solu-tion for w0:w:Ω≈1:1:n,1:2:n,1:3:n,2:1:n and 3:1:n by using second-order averaging method and give a criterion for the existence of resonant solution for w0:w:Ω≈1:m:n is given by using Melnikov's method and verify the theoretical analysis by numerical simulations. By numerical simulation, we expose some other interesting dynamical behaviors, including the entire invariant torus region, the cascade of invariant torus behaviors, the entire chaos region with-out periodic windows, chaotic region with complex periodic windows, the entire period-one orbits region; the jumping behaviors which including invariant torus behaviors converting tq period-one orbits, from chaos to invariant torus behaviors or from invariant torus behaviors to chaos, from period-one to chaos, from invariant torus behaviors to another invariant torus behaviors; and the interior crisis; and the different nice invariant torus attractors and chaotic attractors. The numerical results show the difference of dynamical behaviors in the physical pendulum equa-tion with suspension axis vibrations between under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations. It exhibits many nice invariant torus behaviors under the resonant conditions and we find a lot of chaotic behaviors which are different to those under the periodic/quasi-periodic perturbations.For the discrete system, the dynamical behaviors of a discreet mathematical model for respiratory process in bacterial culture are investigated. The conditions of existence for flip bifurcation and Hopf bifurcation are derived by using cen-ter manifold theorem and bifurcation theory, condition of existence of chaos in the sense of Marotto's definition of chaos is proved. The bifurcation diagrams, Lyapunov exponents and phase portraits are given for different parameters of the model, and the fractal dimension of chaotic attractor of the model is also calcu-lated. The numerical simulation results not only show the consistence with the theoretical analysis but also display the new and interesting complex dynamical behaviors compared with the continuous model, including reverse bifurcation from period-two to period-eight orbits and from period-one orbits to period-four orbits, the cascades of period-doubling bifurcations from period-one orbits to period-eight orbits and from period-three orbits to period-twelve orbits; and the onset of chaos, and the entire chaotic region without periodic windows, chaotic regions with com-plex periodic windows, the entire invariant torus without periodic windows; chaotic behaviors converting to periodic orbits; and the jumping behaviors including from chaos to invariant torus, from invariant torus to chaos and from periodic orbits to chaos; and the interleaving occurrence of periodic orbits and invariant torus behaviors; and the different nice chaotic attractors and invariant torus. The study for them is of fundamental and even practical interest.The dynamical behaviors of these systems will enrich the content of nonlinear dynamical systems and will be useful in other subjects such as chemistry, physics and biology.This thesis consists of three chapters as the following.Chapter 1 is about preparation knowledge. A brief review of center manifold theorems for continuous and discrete dynamical system is presented. At the same time, some definitions and characteristics of chaos as well as some routes to chaos are mentioned.In chapter 2, the physical pendulum equation with suspension axis vibra-tions is investigated. In section 2.2,2.3 and 2.4, the conditions of existence of chaos under periodic perturbations and under quasi-periodic perturbations are given by using Melnikov's method and second-order averaging method. By nu-merical simulations we not only check up the effect of theoretical analysis and expose the complex dynamical behaviors, but also show the chaotic behaviors as Ω=nw+(?)v, n= 7. In section 2.5, we investigate the existence and the bifurca-tions of resonant solution for w0:w:Ω≈1:1:n,1:2:n,1:3:n,2:1:n and 3:1:n by using second-order averaging method and give a criterion for the exis-tence of resonant solution for w0:w:Ω≈1:m:n is given by using Melnikov's method and verify the theoretical analysis by numerical simulations. By numerical simulation, we expose some other interesting dynamical behaviors. The numerical results show the difference of dynamical behaviors in the physical pendulum equa-tion with suspension axis vibrations between under the three frequencies resonant condition and under the periodic/quasi-periodic perturbations. It exhibits many nice invariant torus behaviors under the resonant conditions and we find a lot of chaotic behaviors which are different to those under the periodic/quasi-periodic perturbations.In chapter 3, the dynamical behaviors of a discreet mathematical model for the respiratory process in bacterial culture are investigated. The conditions of ex-istence for flip bifurcation and Hopf bifurcation are derived by using center mani-fold theorem and bifurcation theory, and we prove that there is no fold bifurcation. The chaotic existence in the sense of Marotto's definition of chaos is proved. The numerical simulation results display some new and complex dynamical behaviors.