Symmetry restoration in a class of forced oscillators

This paper is concerned with a class of symmetric forced oscillators modeled by nonlinear ordinary differential equations. The forced Duffing oscillator and the forced pendulum belong to this class of oscillators. Another example is found in an electric power system and is associated with a phenomenon known as ferroresonance. Solutions for this class of systems can be symmetric or non-symmetric. Sometimes, as a physical parameter is varied, solutions lose or gain the symmetry at a bifurcation point. It will be shown that bifurcations which might otherwise be called symmetry breaking, symmetry creation via collision and symmetry creation via explosion are all the result of a collision between conjugate attractors (i.e., attractors that relate to each other by the symmetry) and a symmetric limit set. The same mechanism seems plausible for at least some bifurcations involving non-attracting sets. This point is illustrated with some examples.