Unfolding globally resonant homoclinic tangencies

Abstract

<p style='text-indent:20px;'>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 <inline-formula><tex-math id="M1">\begin{document}$ |\lambda|^{2 k} $\end{document}</tex-math></inline-formula>, as <inline-formula><tex-math id="M2">\begin{document}$ k \to \infty $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M3">\begin{document}$ -1 &lt; \lambda &lt; 1 $\end{document}</tex-math></inline-formula> is the stable eigenvalue associated with the fixed point. If the perturbation is taken tangent to the surface of codimension-one homoclinic tangencies, the scaling is instead like <inline-formula><tex-math id="M4">\begin{document}$ \frac{|\lambda|^k}{k} $\end{document}</tex-math></inline-formula>. We also show slower scaling laws are possible if the perturbation admits further degeneracies.</p>