Unfolding globally resonant homoclinic tangencies

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. The distance of the bifurcation values from global resonance generically scales like \begin{document}$ |\lambda|^{2 k} $\end{document} , as \begin{document}$ k \to \infty $\end{document} , where \begin{document}$ -1 < \lambda < 1 $\end{document} is the stable eigenvalue associated with the fixed point. If the perturbation is taken tangent to the surface of codimension-one homoclinic tangencies, the scaling is instead like \begin{document}$ \frac{|\lambda|^k}{k} $\end{document} . We also show slower scaling laws are possible if the perturbation admits further degeneracies.