Symmetric homoclinic tangles in reversible dynamical systems have positive topological entropy

TL;DR

This paper proves that in certain reversible dynamical systems, the presence of strongly transverse intersections of stable and unstable manifolds of symmetric periodic orbits guarantees positive topological entropy, indicating complex chaotic behavior.

Contribution

It establishes a new link between symmetric homoclinic tangles and positive entropy in reversible systems with specific fixed point structures.

Findings

Positive topological entropy when stable and unstable manifolds intersect strongly transversely.

Symmetric periodic orbits of saddle type lead to complex dynamics.

Reversible vector fields with certain fixed point sets exhibit chaotic behavior under these conditions.

Abstract

We consider reversible vector fields in $\mathbb{R}^{2n}$ such that the set of fixed points of the involutory reversing symmetry is $n$-dimensional. Let such system have a smooth one-parameter family of symmetric periodic orbits which is of saddle type in normal directions. We establish that topological entropy is positive when the stable and unstable manifolds of this family of periodic orbits have a strongly-transverse intersection.