Symmetry-breaking Analysis And Control For Homoclinic Bifurcations Of The Multiple Potential Oscillators

In this thesis, we at first investigate a three-well potential oscillator including a constant noise by using the theory of bifurcations and chaos. Then, we consider the symmetry-breaking analysis when the constant noise varies. Finally, we obtain the controlling and inducing conditions for chaos by adding a time-varying excitation term. The second part of this thesis is intended to study the global control condition for homoclinic bifurcation of a two-well potential oscillator, and get the global control condition that there is no longer transverse intersections between the stable and unstable manifolds in the Poincare map for the homoclinic bifurcation.The layout of this thesis is as follows.In Chapter 1, a brief review concerning the theory of local and global bifurcation, chaos of nonlinear dynamical systems, the background of this paper, and the softwares we used are introduced.In Chapter 2, a three-well potential oscillator including a constant noise is investigated. First of all, due to the effect of the constant noise, the symmetry of the oscillator is broken. The subsequent Melnikov analysis and numerical results show that the constant noise can produce a considerable effect on the dynamics behaviors no matter how small the constant noise is. Second, one of main reasons leading to a comprehensive symmetry-breaking lies in the interaction of multiple homoclinic bifurcations resulting from different sizes of homoclinic orbits of unperturbed system corresponding to the undamped and unforced three-well oscillator. Third, one of our main results shows that there coexist many interacted and intertwisted regular attractors or strange chaotic attractors corresponding to different constant noise intensities, respectively. Finally, the areas that the chaos occurred can be decreased through adding a time-varying control term, so the control or induce of chaos is feasible from the qualitative and quantitative points of view.In Chapter 3, a two-well potential oscillator including an asymmetric disturbance is investigated. The Melnikov analysis and numerical simulations show that due to the existence of the asymmetric disturbance, there exist different parameters at which the right-hand critical homoclinic bifurcation values are equal to the left-hand ones in the frequency-amplitude plane. We refer to these parameters as the global control condition for the homoclinic bifurcation. Following the variation of parameters, the fitted curves of the frequency and the amplitude corresponding to different global control conditions, are similar to the parabola-like curves, besides a special gap. The global control condition is very useful in the global suppressing or inducing of chaos, that is, by which the transversal intersections of stable and unstable manifolds of Poincare map of both sides of the saddle can be suppressed or induced, simultaneously. In addition, the common features of attractors were investigated corresponding to different parameters. The exact number and type of attractors were obtained when the critical left-hand homoclinic bifurcation is (or is not) equal to the right-hand one.In Chapter 4, we briefly conclude the thesis.