Homoclinic Bifurcations that Give Rise to Heterodimensional Cycles near A Saddle-Focus Equilibrium

TL;DR

This paper demonstrates how homoclinic bifurcations near a saddle-focus equilibrium in higher-dimensional flows can generate heterodimensional cycles, revealing new types of heterodimensional connections beyond classical cases.

Contribution

It introduces two novel heterodimensional connection types arising from homoclinic bifurcations in flows of dimension 4 and higher.

Findings

Heterodimensional cycles can originate from homoclinic bifurcations near saddle-focus equilibria.

Two new heterodimensional connection types are identified: between a periodic orbit and a homoclinic loop, and between a periodic orbit and the saddle-focus.

These findings extend understanding of bifurcation structures in higher-dimensional dynamical systems.

Abstract

We show that heterodimensional cycles can be born at the bifurcations of a pair of homoclinic loops to a saddle-focus equilibrium for flows in dimension 4 and higher. In addition to the classical heterodimensional connection between two periodic orbits, we found two new types of heterodimensional connections: one is a heteroclinic between a periodic orbit of index 2 and a homoclinic loop, and the other connects a periodic orbit of index 3 to the saddle-focus equilibrium.