Robust heterodimensional cycles in two-parameter unfolding of homoclinic tangencies

TL;DR

This paper characterizes when heterodimensional cycles emerge robustly in two-parameter unfoldings of homoclinic tangencies in high-dimensional dynamical systems, revealing conditions for persistent complex dynamics.

Contribution

It provides a necessary and sufficient condition for the emergence of $C^1$-robust heterodimensional dynamics in generic unfoldings of homoclinic tangencies, including the role of blenders.

Findings

Robust heterodimensional cycles occur under specific conditions involving central multipliers.

Heterodimensional dynamics involve blenders with robust homoclinic tangencies.

Systems with certain homoclinic tangencies are in the closure of the Newhouse domain.

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.