Center manifold reduction for large populations of globally coupled phase oscillators

TL;DR

This paper develops a bifurcation theory for large populations of globally coupled phase oscillators using rigged Hilbert spaces, deriving dynamics on a finite-dimensional center manifold and rigorously confirming Kuramoto's bifurcation diagram for sine coupling.

Contribution

It introduces a novel bifurcation analysis framework based on rigged Hilbert spaces and establishes the existence of a finite-dimensional center manifold for general coupling functions.

Findings

Rigorous derivation of Kuramoto's bifurcation diagram for sine coupling

Discovery of a new bifurcation phenomenon for non-sine coupling functions

Existence of a finite-dimensional center manifold in a generalized function space

Abstract

A bifurcation theory for a system of globally coupled phase oscillators is developed based on the theory of rigged Hilbert spaces. It is shown that there exists a finite-dimensional center manifold on a space of generalized functions. The dynamics on the manifold is derived for any coupling functions. When the coupling function is $\sin \theta $, a bifurcation diagram conjectured by Kuramoto is rigorously obtained. When it is not $\sin \theta $, a new type of bifurcation phenomenon is found due to the discontinuity of the projection operator to the center subspace.