Homoclinic tangencies in $\mathbb{R}^n$

TL;DR

This paper investigates the occurrence of horseshoe structures near degenerate homoclinic tangencies in high-dimensional dynamical systems, extending planar results to $ $-dimensional manifolds under certain conditions.

Contribution

It extends planar homoclinic tangency results to higher dimensions, showing horseshoe existence near degenerate tangencies under $C^1$-linearizability.

Findings

Existence of horseshoes near degenerate homoclinic tangencies

Extension of planar results to $ $-dimensional manifolds

Conditions under which transverse homoclinic crossings appear

Abstract

Let $f: M \to M$ denote a diffeomorphism of a smooth manifold $M$. Let $p$ in $M$ be its hyperbolic fixed point with stable and unstable manifolds $W_S$ and $W_U$, respectively. Assume that $W_S$ is a curve. Suppose that $W_U$ and $W_S$ have a degenerate homoclinic crossing at a point $B\ne p$, i.e., they cross at $B$ tangentially with a finite order of contact. It is shown that, subject to $C^1$-linearizability and certain conditions on the invariant manifolds, a transverse homoclinic crossing will arise arbitrarily close to $B$. This proves the existence of a horseshoe structure arbitrarily close to $B$, and extends a similar planar result of Homburg and Weiss.