Resonances for weak coupling of the unfolding of a saddle-node periodic orbit with an oscillator

Abstract

For a family of continuous-time dynamical systems with two angle variables, a resonance is a set of parameter values such that some integer combination of the (lifted) angles remains bounded for some orbits. For weak coupling of an unfolding of a saddle-node periodic orbit with an oscillator, it is shown that the low order resonances have at least a certain amount of bifurcation structure. Furthermore, define a Chenciner bubble to be the complement of the set of parameters near a resonance for which there is an attractor-repellor pair consisting of two C1 invariant tori, or a C1 invariant torus attracting from one side and repelling from the other, or locally empty non-wandering set. The low order Chenciner bubbles are shown to have at least a certain structure. Chenciner's results for resonances in the unfolding of a degenerate Neimark-Sacker bifurcation can be seen as a special case of ours. © 2007 IOP Publishing Ltd. and London Mathematical Society.