Research On Torus Bifurcations And Chaos Of Multi-Degree-of-Freedom Vibro-Impact Systems

This dissertation systematically presents a study on the tori bifurcation of multi-degree-of-freedom vibro-impact systems. By theoretical analysis and numerical simulations, the two-parameter bifurcations and unfoldings of vibro-impact systems in five resonant cases are investigated. Some attractors, coexisting attractors and homoclinic and heteroclinic orbits that appear in the systems are analyzed. Moreover, tori losing stability and leading to chaos is investigated. The main contributions of this dissertation are as following:Chapter 1 surveys the some recent achievements, developments and unsolved problems of stability, bifurcations and chaos of vibro-impact systems, and the advance in the corresponding differential equation and discrete map dynamics. The main contents and results of the dissertation are introduced.Chapter 2, establishing the four-dimensional Poincare map of a two-degree-of-freedom vibro-impact system, investigates two-parameter dynamic behaviors under weak resonant conditions. The map is reduced to a two-dimensional normal form by the center manifold theorem and theory of normal forms. Condition there exists the Hopf bifurcation or subharmonic bifurcation is derived, which is called Arnold tongue. Numerical simulations with regard to five-order resonance show that there exist the invariant T1 circle and periodic 5-5 motions. It is also discovered that there exist simultaneously other attractors and local ones. As control parameters vary further, bifurcations and chaos on the resonant tongue are more complex than out of it. Moreover, simulations on a so called "resonant peak" of Model 3 reveal the interactions between phase lock and weak (strong) resonance in the bifurcations of quasi-periodic motions and routes to chaos, and exhibit the complicated and interesting dynamic behaviors of the vibro-impact systems.Chapter 3 focuses on the two-parameter bifurcations of vibro-impact system under strong resonant conditions . By the center manifold theorem and theory of normal forms, the map is reduced to a two-dimensional normal form. We use the approximation of maps near their fixed points by shifts along the orbits of certain systems of autonomous ordinary differential equations and predict global bifurcations of closed invariant curves happening in the maps near homo- and heteroclinic bifurcations of the approximating systems. Conditions there exist the Hopf bifurcation or subharmonic bifurcation are derived and the local dynamic behaviors with two-parameter unfoldings are analyzed, which develop the one-parameter bifurcation theory in strong resonant cases. Numerical simulations show that there exist the invariant T1 circle and subharmonic motions. Moreover, routes from invariant T1 circle or subharmonic fixed points to chaos are reported.In Chapter 4, an analysis method of Hopf-Flip bifurcations of multi-degree-of-freedom vibro-impact systems is established. After the four (or more)-dimensional Poincare1 map of vibro-impact system is reduced into a three-dimensional normal form by virtue of the center manifold theorem and theory of normal forms, two-parameter unfoldings and their phase trajectories near critical points are analyzed. Based on Matlab and the projection technique, the programming and numerical computations indicate that there exist curve doubling bifurcation (a torus doubling bifurcation), Hopf bifurcation of fixed points of order two as well as period doubling bifurcation and Hopf bifurcation of fixed points near the critical point. As the control parameters in the region of Hopf bifurcations increase gradually, there are three types of routes from periodic motion to chaos and two of them are of nontypical. Moreover, when a pair of conjugate complex eigenvalue satisfy 1:4 resonant condition, rich dynamic behaviors of vibro-impact system are revealed numerically, which involve invariant T1 torus, 2XT1 tori, 4XT1 tori and fixed points of order four. As the parameter vector changes in different directions, the routes leading to chaos are observed.Chapter 5 investigates the Hopf-Hopf bif...