Hovering Sets in Near Pseudo-Integrable Hamiltonian Impact Systems

The transition from rotational to discontinuous behavior of the return map of the perturbed oscillator-step system, a paradigm model for a perturbation of a pseudo-integrable Hamiltonian impact system, is studied. The form of the return map is derived, and a truncated model of this map is simulated and analyzed. For a set of parameters the existence of a hovering set, a set of nonresonant orbits that pass sometimes above the step and sometimes to its side, without ever impacting it, is established and quantified. It is further shown that reversing the sign of the perturbation destroys this set.