Bifurcations And Chaos In Two Continual And Discrete Dynamical Systems

This thesis discuss the dynamics of one continuous and one discrete dy-namical system as the parameters varying.For the Duffing equation with damping excitation and one external forcing and fifth nonlinear-restoring force, we prove the criterion of existence of chaos under periodic perturbation by applying Melnikov method, and by second-order averaging method and Melnikov method give the criterion of existence of chaos of averaged system under quasi-periodic perturbation forΩ=nw+(?)σ,n= 2,4,6, and prove that we can't give the criterion of existence of chaos for n= 1,3,5,7 - 20 by applying averaging method, whereσis not rational to w, and show the existence of chaotic dynamics in averaged equation and original equation by numerical simulations. And under the three frequencies satisfyingΩ:w:W0= n:m:1 (n, m= 1,2,3,1/2), the resonances and bifurca-tions of the system are investigated by using qualitative theory, bifurcation and chaos theories, and by applying Melnikov method and numerical simulations discuss the resonances under the conditionsΩ:w:w0= n:m:1 (m> 3). For generalized Henon, the conditions of existence for fold bifurcation, flip bi-furcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, chaotic behavior in the sense of Marotto's definition is proved under parameters satisfying some conditions. Meanwhile, the numerical simulations, including bifurcation diagram of fixed points, homoclinic and hete-roclinic bifurcation surface, bifurcation diagrams in three-and two-dimensional spaces, the maximum Lyapunov exponent, phase portraits, Poincare maps, not only show the consistence with the theoretical analysis, but also exhibit the rich and new dynamical behaviors, including the transient chaos with com-plex windows, jumping behaviors of period-orbits, the bubble from period-2(4) to period-4(12), chaotic behavior and periodic motion (or quasi-periodic mo-tions) occurs alternately, onset of chaos which occurs more than once, chaotic behaviors and invariant cycles suddenly appear or disappear, chaos suddenly disappears to period orbits, period-doubling bifurcations and reverse period- doubling bifurcations to chaos, the symmetry-breaking of period-orbits, initial crisis, quasi-periodic route to chaos and many quasi-periodic attractors, chaotic attractors, non-attracting chaotic sets, the strange chaotic attractors and the strange non-chaotic attractors. The computation of maximum Lyapunov expo-nents confirm the chaotic behaviors.This thesis consists of four chapters as following:Chapter 1 is the preparation knowledge. A brief review of center maniold theorem, second-order averaging method and Melnikov method for continuous and discrete dynamical system are presented. Some definitions and character-istics of chaos as well as some routes to chaos are mentioned.In Chapter 2, by using qualitative theory, bifurcation and chaos theory and numerical simulations, the complex dynamics of the Duffing equation with damping excitation and one external forcing is investigated. Applying Melnikov method, the criterions of existence of chaos under periodic perturbation and applying second-order averaging method and Melnikov method, the criterions of existence of chaos of averaged system under quasi-periodic perturbationΩ= nw+(?)σ,n= 2,4,6 are given, but the criterion of existence of chaos for n= 1,3,5,7 - 20 can't been proven. And using the numerical simulations to prove the theoretical analysis and to study the influence of the parameters in original equation on dynamics, and find more complex dynamics.In chapter 3, Duffing equation with damping excitation and one external periodic excitation is also investigated. The dynamics of the system under the three frequencies satisfying some resonances conditions are given. It is easy to obtain the resonances and bifurcations under the conditionsΩ:w:w0= n:m: 1 (m,n= 1,2,3,1/2) by second-order averaging method and bifurcation and chaos theories. The Melnikov's method and numerical simulations are applied to take into account the resonances when some specific resonant conditions among these three different frequenciesΩ:w:w0= n:m:1 (m> 3). The numerical simulations show the consistence with the theoretical analyses and exhibit some specific properties. In chapter 4, the bifurcations and chaos phenomenons of generalized Henon map are investigated. The conditions of existence for fold bifurcation, flip bi-furcation and Hopf bifurcation are derived by using center manifold theorem and bifurcation theory, chaotic behavior in the sense of Marotto's definition is proved. And numerical simulation results show the consistence with the theo-retical analyses and display the new and interesting dynamical behaviors.