Global Parametrization and Computation of Resonance Surfaces for Periodically Forced Oscillators

Periodically forced planar oscillators are typically studied by varying the two parameters of forcing amplitude and forcing frequency. Such differential equations can be reduced via stroboscopic sampling to a two-parameter family of diffeomorphisms of the plane. A bifurcation analysis of this family almost always includes a study of the birth and death of periodic orbits. For low forcing amplitudes, this leads to a now-classic picture of Arnold resonance tongues. Studying these resonance tongues for higher forcing amplitudes requires numerical continuation. Previous work has revealed the usefulness of considering these tongues as projections of surfaces of periodic points from the cartesian product of the phase and parameter planes to the parameter plane. Many surfaces were displayed and described in [MP 1994], but their parametrization and computation was not discussed. In this paper, we do discuss their parametrization and computation. Especially useful are global parametrizations which allow automatic computation of the surfaces. We argue that parametrization by “fμ(x) − x” is both more likely to be a global parametrization and more “dynamically natural” than two more obvious parametrizations. As a side benefit, fμ(x)− x parametrization leads to a computable way of establishing the nonorientablility of period-two surfaces.