Unfolding globally resonant homoclinic tangencies

<p style='text-indent:20px;'>Global resonance is a mechanism by which homoclinic tangency of smooth map can have infinitely many asymptotically stable, single-round periodic solutions. To understand the bifurcation structure one would expect to see near such tangency, in this paper we study one-parameter perturbations typical globally resonant tangencies. We assume tangencies are formed stable and unstable manifolds saddle fixed points two-dimensional maps. show display two infinite sequences bifurcations, saddle-node other period-doubling, between solutions stable. The distance values from global generically scales like <inline-formula><tex-math id="M1">\begin{document}$ |\lambda|^{2 k} $\end{document}</tex-math></inline-formula>, as id="M2">\begin{document}$ k \to \infty where id="M3">\begin{document}$ -1 < \lambda 1 $\end{document}</tex-math></inline-formula> eigenvalue associated with point. If perturbation taken tangent surface codimension-one tangencies, scaling instead id="M4">\begin{document}$ \frac{|\lambda|^k}{k} $\end{document}</tex-math></inline-formula>. also slower laws possible if admits further degeneracies.</p>