Robust heterodimensional dynamics in two-parameter unfoldings 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 $\dim \mathcal{M}⩾3$ for diffeomorphisms and with $\dim \mathcal{M}⩾4$ for flows, that $C^1$-robust heterodimensional dynamics of coindex one appears 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 is not sectionally dissipative. The heterodimensional dynamics also involves 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.