Continua and persistence of periodic orbits in ensembles of oscillators

TL;DR

This paper proves the existence of a manifold of periodic orbits in certain coupled oscillator systems, enabling analysis of their long-term behavior and explaining the persistence of splay states.

Contribution

It demonstrates the presence of a normally attracting invariant manifold foliated by periodic orbits in integrable oscillator systems, advancing understanding of their asymptotic dynamics.

Findings

Existence of a normally attracting invariant manifold in oscillator systems.

Persistence of finitely many periodic orbits under perturbations.

Identification of splay state dynamics as a persistent orbit.

Abstract

Certain systems of coupled identical oscillators like the Kuramoto-Sakaguchi or the active rotator model possess the remarkable property of being Watanabe-Strogatz integrable. We prove that such systems, which couple via a global order parameter, feature a normally attracting invariant manifold that is foliated by periodic orbits. This allows us to study the asymptotic dynamics of general ensembles of identical oscillators by applying averaging theory. For the active rotator model, perturbations result in only finitely many persisting orbits, one of them giving rise to splay state dynamics. This sheds some light on the persistence and typical behavior of splay states previously observed.