Unfolding globally resonant homoclinic tangencies

Abstract

Global resonance is a mechanism by which a homoclinic tangency of a smooth map can have infinitely many asymptotically stable, single-round periodic solutions. To understand the bifurcation structure one would expect to see near such a tangency, in this paper we study one-parameter perturbations of typical globally resonant homoclinic tangencies. We assume the tangencies are formed by the stable and unstable manifolds of saddle fixed points of two-dimensional maps. We show the perturbations display two infinite sequences of bifurcations, one saddle-node the other period-doubling, between which single-round periodic solutions are asymptotically stable. Generically these scale like $|\lambda {|}^{2k}$, as $k\to \infty$, where $-1<\lambda <1$ is the stable eigenvalue associated with the fixed point. If the perturbation is taken tangent to the surface of codimension-one homoclinic tangencies, they instead scale like $\frac{|\lambda {|}^k}{k}$. We also show slower scaling laws are possible if the perturbation admits further degeneracies.

19 pages, 7 figures