Every period annulus is both reversible and symmetric

TL;DR

This paper proves that all planar systems with a period annulus are both reversible and symmetric under certain involutions, revealing fundamental symmetry properties of such dynamical systems.

Contribution

It establishes the existence of an involution making the system symmetric and infinitely many involutions making it reversible, advancing understanding of symmetry in planar differential systems.

Findings

Existence of an involution making the system symmetric.

Existence of infinitely many involutions making the system reversible.

Universal symmetry properties for systems with a period annulus.

Abstract

We prove that for every planar differential system with a period annulus there exists an involution $\sigma$ such that the system is $\sigma$-symmetric. We also prove that for for every planar differential system with a period annulus there exist infinitely many involutions $\sigma$ such that the system is $\sigma$-reversible.