Every Period Annulus is Both Reversible and Symmetric

Abstract

We prove that for every planar differential system with a period annulus there exists a unique involution \(\sigma \) such that the system is \(\sigma \)-symmetric. We also prove that, given a system with a period annulus and a global section \(\delta \), there exist a unique involution \(\sigma \) such that the system is \(\sigma \)-reversible and \(\delta \) is the fixed points curve of \(\sigma \). As a consequence, every system with a period annulus admits infinitely many reversibilities.