We consider a planar periodic orbit subject to pulsatile periodic perturbations in a fixed direction with a particular frequency and magnitude. This yields a map from an initial point to the next point after a perturbation and just before the next perturbation is applied. Depending on the frequency and magnitude of the perturbation, the map exhibits different dynamics. For example, after a finite number of iterations the map returns back to the initial point which is typically referred to as phase locking. If points away from the periodic orbit contracts back to it immediately following a perturbation, then this map is a one-dimensional phase map that maps the phase of a point on the periodic orbit to the phase of the point on the periodic orbit following a perturbation and just before the next perturbation is applied.
We look at the case where the contraction back to the periodic orbit is not immediate. In this case, the map is a two-dimensional map, but it can be approximated well if at least one of the following is true:
We study the periodically forced poincare oscillator, which we choose because there is an explicit parameter that controls the contraction to the periodic orbit. We compute the two-parameter perturbation-frequency and -magnitude bifurcation diagram and investigate its evolution as the contraction parameter increases to the infinite limit, where the dynamics is governed by the one-dimensional phase map. We give a comprehensive overview of the evolution, and observe an unreported sequence of co-dimension three bifurcations.