A Study of Symmetric Forced Oscillators

Abstract

In this thesis we study a class of symmetric forced oscillators modeled by non-linear ordinary differential equations. Solutions for this class of systems can be symmetric or non-symmetric. When a symmetric periodic solution loses its stability as a physical parameter is varied, and two non-symmetric periodic solutions appear, this is called a symmetry breaking bifurcation. In a symmetry increasing bifurcation two conjugate chaotic attractors (i.e.,attractors which are related to each other by the symmetry) collide and form a larger symmetric chaotic attractor. Symmetry can also be restored via explosions where, as a physical parameter is varied, two conjugate attractors (chaotic or periodic) which do not intersect are suddenly embedded in one symmetric attractor. In this thesis we show that all these apparently distinct bifurcations can be realized by a single mechanism in which two conjugate attractors collide with a symmetric limit set. The same mechanism seems to operate for at least some bifurcations involving non-attracting limit sets. We illustrate this point with examples of symmetry restoration in attracting and non-attracting sets found in the forced Duffing oscillator and in a power system. Symmetry restoration in the power system is associated with a phenomenon known as ferroresonance. The study of the ferroresonance phenomenon motivated this thesis. Part of this thesis is devoted to studying one aspect of the ferroresonance phenomenon the appearance of a strange attractor with a band-like structure. This attractor was called previously a 'pseudo-periodic' attractor. Some methods for analyzing the non-autonomous systems under study are shown. We construct three different maps which highlight different features of symmetry restoring bifurcations. One map in particular captures the symmetry of a solution by sampling it every half the period of the forcing. We describe a numerical method to construct a bifurcation diagram of periodic solutions and present a non-standard approach for converting the forced oscillator to an autonomous system.