The Mean-Field Ott-Antonsen Manifold is an Unstable Manifold in the Continuum Limit

TL;DR

This paper investigates the relationship between continuum and mean-field PDEs in Kuramoto-type particle systems, revealing that the Ott-Antonsen manifold in the mean-field limit corresponds to an unstable manifold in the continuum limit.

Contribution

It provides the first explicit proof that the Ott-Antonsen manifold is an unstable manifold in the continuum PDE framework, linking it to the homogeneous steady state.

Findings

The Ott-Antonsen manifold is an unstable manifold in the continuum limit.

Solutions of the PDE can be generated from mean-field solutions.

The unstable manifold of the homogeneous steady state corresponds to the Ott-Antonsen manifold.

Abstract

We study interacting particle systems of Kuramoto-type. Our focus is on the dynamical relation between the partial differential equation (PDE) arising in the continuum limit (CL) and the one obtained in the mean-field limit (MFL). Both equations arise when we are considering the limit of infinitely many interacting particles but the classes of PDEs are structurally different. The CL tracks particles effectively pointwise, while the MFL is an evolution for a typical particle. First, we briefly discuss the relation between solutions of the CL and the MFL showing how to generate solutions of the CL starting from solutions of the MFL. Our main result concerns a dynamical relation between important invariant manifolds of the CFL and the MFL. In particular, we give an explicit proof that the unstable manifold of the homogeneous steady state of the CL is the direct dynamical analogue of the famous Ott-Antonsen manifold for the MFL.