Unfolding globally resonant homoclinic tangencies

TL;DR

This paper investigates the bifurcation structure near globally resonant homoclinic tangencies in two-dimensional maps, revealing infinite sequences of bifurcations and stability patterns with specific scaling laws.

Contribution

It provides a detailed analysis of the bifurcation sequences and scaling laws near globally resonant homoclinic tangencies, including effects of perturbation degeneracies.

Findings

Two infinite bifurcation sequences identified: saddle-node and period-doubling.

Stable single-round periodic solutions scale as |λ|^{2k} or |λ|^{k}/k depending on perturbation.

Slower scaling laws are possible with additional degeneracies.

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 $|\lambda|^{2 k}$, 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.