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

Unfolding globally resonant homoclinic tangencies