Robust heterodimensional cycles in two-parameter unfolding of homoclinic tangencies

Abstract

We establish a necessary and sufficient condition for the birth of heterodimensional cycles from a generic homoclinic tangency to a hyperbolic periodic orbit. We prove for $C^r$ ($r=3,\dots,\infty,\omega$) dynamical systems on a manifold $\mathcal{M}$, with $\dim \mathcal{M}\geqslant 3$ for diffeomorphisms and with $\dim \mathcal{M}\geqslant 4$ for flows, that $C^1$-robust heterodimensional dynamics of coindex one appear in any generic two-parameter $C^r$ unfolding of a homoclinic tangency to a periodic orbit such that at least one central multiplier is not real and the central dynamics are not sectionally dissipative. The heterodimensional dynamics also involve a blender exhibiting $C^1$-robust homoclinic tangencies. As a corollary, any system with a homoclinic tangency of the class described above belongs to the $C^r$ closure of the $C^1$-open Newhouse domain.