Topology and resonances in a quasiperiodically forced oscillator

Abstract

In this paper we analyze resonance behavior in a quasiperiodically forced, nonlinear Mathieu equation. We develop a perturbation technique based on the method of multiple scales to find both a criterion for resonance and approximations to solutions in the neighborhood of a resonance. We compare the perturbation results to numerical solutions to validate both the resonance criterion and the approximate solutions. We also investigate the implications of resonance for the topology of attractors in the four-dimensional phase space. We show that a resonance occurs due to topological torus bifurcations (TTBs) and that resonant trajectories lie on topologically interesting knotted tori we have recently described elsewhere (Topological bifurcations of attracting 2-tori of quasiperiodically driven nonlinear oscillators, in review). The perturbation approximations capture both TTBs and the topology of invariant manifolds near resonance.