Dynamic Bifurcation in Hamiltonian Systems with One Degree of Freedom

Hamiltonian systems of one degree of freedom depending on a parameter are considered. It is assumed that the equilibrium states have a known and simple bifurcation structure with respect to the parameter. When the parametric dependence is replaced by a slow time dependence the question arises whether solutions of the fully time-dependent system follow branches of equilibrium solutions that are stable on the short timescale, and, when they do, whether they follow them promptly. This is the problem of dynamic bifurcation. We consider paradigms for supercritical and transcritical bifurcation, and find that for supercritical pitchfork bifurcation, most orbits follow the stable branches; for transcritical bifurcation most orbits do not. Asymptotic and numerical estimates of the probability of the rare “hovering” orbits, which fail to follow stable branches in the supercritical case, are given.