Heterodimensional cycles derived from homoclinic tangencies via Hopf bifurcations

TL;DR

This paper investigates how homoclinic tangencies in three-dimensional diffeomorphisms can lead to heterodimensional cycles through Hopf bifurcations, revealing new pathways for complex dynamics.

Contribution

It demonstrates that homoclinic tangencies can generate heterodimensional cycles via Hopf bifurcations in three-dimensional systems.

Findings

Hopf bifurcation occurs on the unfolding curve of the tangency.

A homoclinic point to a bifurcating periodic orbit exists.

The original map can be approximated by one with a heterodimensional cycle.

Abstract

We analyze three-dimensional $C^{r}$ diffeomorphisms ($r\ge 5$) exhibiting a quadratic focus-saddle homoclinic tangency whose multipliers satisfy $|\lambda\gamma| = 1$. For a proper three-parameter unfolding that splits the tangency, varies the argument of the stable multipliers, and controls the modulus $|\lambda\gamma|$, we show that a Hopf bifurcation occurs on this curve and that a homoclinic point to the bifurcating periodic orbit is present. As a consequence, the original map $f$ can be $C^{r}$-approximated by a diffeomorphism exhibiting a coindex-one heterodimensional cycle in the saddle case.