Existence of Heterodimensional Cycles near Shilnikov Loops in Systems with a $\mathbb{Z}_2$ Symmetry

TL;DR

This paper demonstrates that in symmetric dynamical systems with Shilnikov loops, heterodimensional cycles can emerge through bifurcations, leading to complex attractors with persistent homoclinic tangencies.

Contribution

It proves the existence of heterodimensional cycles near Shilnikov loops in symmetric flows, revealing new bifurcation phenomena in such systems.

Findings

Heterodimensional cycles can form at bifurcations of Shilnikov loops.

These cycles can be part of chain-transitive attractors.

Persistent homoclinic tangencies can coexist with heterodimensional cycles.

Abstract

We prove that a pair of heterodimensional cycles can be born at the bifurcations of a pair of Shilnikov loops (homoclinic loops to a saddle-focus equilibrium) having a one-dimensional unstable manifold in a volume-hyperbolic flow with a $\mathbb{Z}_2$ symmetry. We also show that these heterodimensional cycles can belong to a chain-transitive attractor of the system along with persistent homoclinic tangency.