Bifurcations And Chaos Of Two Classes Of Dynamical Systems

This thesis discusses the dynamics of the Josephson system and the Tin-dcrbell map as the parameters varying.For the Josephson system with parametric excitation, it is analyzed that we propose two types of the unperturbed systems: and Firstly, by applying Melnikov method, the unperturbed heterocilinic and ho-moclinic bifurcations result in chaos under periodic perturbations. Secondly, by the second-order averaging method and the subharmonic Melnikov func-tion, we analyze bifurcations and the existence of harmonic,(2,3, n-order) subharmonic and (2,3-order) superharmonic solutions near the unperturbed centers. But, for the first unperturbed system, there are no2-order subhar-monic and supcrharmonic solutions. Thirdly, by using numerical simulations, including bifurcation diagrams in two-and three-dimensional spaces, the maxi-mum Lyapunov exponents, phase portraits and Poincare map, we demonstrate our theoretical analysis and further study the effect of the parameters on dy-namical behavior. We find the complex and interesting dynamics, such as the jumping behaviors, symmetry-breaking of periodic orbits, transient chaos, on-set of chaos, chaos suddenly converting to period orbits, interior crisis, chaotic attractors, some strange non-chaotic motions, non-attracting chaotic set, inter-locking period-doubling bifurcations from period-n(n=6,8,9, etc) orbits and interlocking reverse period-doubling bifurcations from period-n(n=4,8,12, etc) orbits in chaotic regions, and so on. In particular, we observe the process from interlocking period-doubling bifurcations of periodic orbits to chaos after some strange non-chaotic motions as the parameter β increases.For the Tinkerbell map, it is the first time systematically to discuss the existence of fold bifurcation, flip bifurcation and Hopf bifurcation and chaos in the sense of Marotto by both analytical and numerical methods. More pre-cisely, for the Tinderbell map, this thesis reports the findings of chaos in the sense of Marotto, a route from an invariant circle to transient chaos with a great abundance of periodic windows, including period-2,7,8,9,10,13,17,19,23,26, and so on, and suddenly appearing or disappearing chaos, an invariant circle turning to a period-one orbit., symmetry-breaking of periodic orbits, inter-locking period-doubling bifurcations in chaotic regions, interior crisis, chaotic attractors, coexisting (2,10,13) chaotic sets, two coexisting invariant circles, two attracting chaotic sets coexisting with a non-attracting chaotic set, and so on. In particular, it is found that there is no obvious route from period-doubling bifurcations to chaos, but there is a route from a period-one orbit to an invari-ant circle and then to transient chaos as the parameters vary. Combining the existing results in the literature with the new results reported in this thesis, a more complete understanding of the Tinkerbell map is obtained.This thesis consists of six chapters as following:In Chapter1, we introduce the background and the present situation of the related research field, and the content, methods and significance of my thesis.Chapter2presents the preparatory knowledge of bifurcations and chaos on continuous and discrete systems, including the center manifold theorem, the second-order averaging method, the Melnikov method, definitions and charac-teristics of chaos, fractal dimension, and the route to chaos.In Chapter3, by applying the Melnikov method, we give the conditions resulting in chaos under periodic perturbation in Josephson system with para-metric excitation Moreover using numerical simulations, we demonstrate the theoretical results, discuss the influence of the parameters on dynamics, and find more complex dynamics.In chapter4, we analyze bifurcations of periodic solutions in the above Josephson system with parametric excitation. By using second-order averaging method and subharinonic Mclnikov function, we analyze bifurcations and the existence of harmonic,(2.3.n-order) subharmonic and (2,3-order) superhar-monic solutions near the unperturbed centers. The numerical simulations show the consistence with the theoretical analysis and exhibit some new properties.In chapter5, dynamical behavior of the Tinkerbell map are investigated in detail. Some sufficient conditions are derived on the existence of fold bifurcation, flip bifurcation and Hopf bifurcation, and chaos in the sense of Marotto is verified by both analytical and numerical methods. Numerical simulations exhibit new and interesting dynamical behavior.In chapter6, we introduce the observed route to chaos in the thesis.