Homoclinic tangencies with infinitely many asymptotically stable single-round periodic solutions

TL;DR

This paper investigates the conditions under which a homoclinic orbit to a saddle fixed point in a smooth map leads to infinitely many stable single-round periodic solutions, revealing the role of global resonance and eigenvalue relations.

Contribution

It establishes necessary and sufficient conditions for the emergence of infinitely many stable periodic solutions near homoclinic tangencies, including eigenvalue relations and resonance conditions.

Findings

Infinite stable solutions occur under specific resonance conditions.

The phenomenon's codimension depends on the eigenvalue product sign.

Numerical examples illustrate basin structures and theoretical predictions.

Abstract

We consider a homoclinic orbit to a saddle fixed point of an arbitrary $C^\infty$ map $f$ on $\mathbb{R}^2$ and study the phenomenon that $f$ has an infinite family of asymptotically stable, single-round periodic solutions. From classical theory, this requires $f$ to have a homoclinic tangency. We show it also necessary for $f$ to satisfy a `global resonance' condition and for the eigenvalues associated with the fixed point, $\lambda$ and $\sigma$, to satisfy $|\lambda \sigma| = 1$. The phenomenon is codimension-three in the case $\lambda \sigma = -1$, but codimension-four in the case $\lambda \sigma = 1$ because here the coefficients of the leading-order resonance terms associated with $f$ at the fixed point must add to zero. We also identify conditions sufficient for the phenomenon to occur, illustrate the results for an abstract family of maps, and show numerically computed basins of attraction.