Topology and resonances in a quasiperiodically forced oscillator

In this paper we analyze resonancebehavior in a quasiperiodicallyforced, nonlinear Mathieu equation. We develop a perturbation technique based on the method of multiple scales to find both a criterion for resonanceand approximations to solutions in the neighborhood of a resonance. We compare the perturbation results to numerical solutions to validate both the resonancecriterion and the approximate solutions. We also investigate the implications of resonancefor the topologyof attractors in the four-dimensional phase space. We show that a resonanceoccurs due to topologicaltorus bifurcations (TTBs) and that resonanttrajectories lie on topologicallyinteresting knotted tori we have recently described elsewhere ( Topologicalbifurcations of attracting 2-tori of quasiperiodicallydriven nonlinear oscillators, in review). The perturbation approximations capture both TTBs and the topologyof invariant manifoldsnear resonance.