Chaos near a reversible homoclinic bifocus

TL;DR

This paper demonstrates that neighborhoods of a reversible bifocal homoclinic orbit contain complex chaotic invariant sets and super-homoclinic orbits, revealing intricate dynamics near such bifurcations.

Contribution

It establishes the existence of chaotic sets and super-homoclinic orbits near reversible bifocal homoclinic orbits, highlighting the rich dynamics in these systems.

Findings

Chaotic invariant sets on N-symbols exist near the bifocal homoclinic orbit.

Super-homoclinic orbits are present, indicating complex trajectory structures.

Switching dynamics are associated with networks of secondary homoclinic orbits.

Abstract

We show that any neighborhood of a non-degenerate reversible bifocal homoclinic orbit contains chaotic suspended invariant sets on $N$-symbols for all $N\geq 2$. This will be achieved by showing switching associated with networks of secondary homoclinic orbits. We also prove the existence of super-homoclinic orbits (trajectories homoclinic to a network of homoclinic orbits), whose presence leads to a particularly rich structure.