Analytic treatment of a driven oscillator with a limit cycle

Abstract

A prototype model of driven nonlinear oscillators with a stable limit cycle is studied. In the fast-relaxation limit, dynamics can be reduced to a one-dimensional mapping parametrized by the amplitude \ensuremath{\alpha} and the phase \ensuremath{\beta} of the driving force. For a weak force, mode locking with rational winding numbers occurs. For a strong force, the parameter space may be divided into two subregions: In the unimodal region, the order of occurrence of the orbits is governed by the Metropolis-Stein-Stein U sequence of unimodal mappings; in the intermediate region, a transition between mode-locking behavior and that of the unimodal mapping takes place, and new sequences of periodic orbits occur. The systematics of the periodic orbits is investigated.