Closed orbits and their bifurcations in the crossed-field hydrogen atom

A systematic study of closed classical orbits of the hydrogen atom in crossed electric and magnetic fields is presented. We develop a local bifurcation theory for closed orbits, which is analogous to the well-known bifurcation theory for periodic orbits and allows identifying the generic closed-orbit bifurcations of codimension 1. Several bifurcation scenarios are described in detail. They are shown to have as their constituents the generic codimension-1 bifurcations, which combine into a rich variety of complicated scenarios. We propose heuristic criteria for a classification of closed orbits that can serve to systematize the complex set of orbits.