Kuramoto Networks with Infinitely Many Stable Equilibria

TL;DR

This paper demonstrates that the Kuramoto model on graphs can have infinitely many stable equilibria, with the set of equilibria forming manifolds of arbitrary dimension, solving a conjecture about planar graphs.

Contribution

It proves the existence of infinitely many stable equilibria in the Kuramoto model on graphs and characterizes their structure using topological bifurcation theory.

Findings

Existence of manifolds of stable equilibria of arbitrary dimension.

Solution to a conjecture about the number of equilibria on planar graphs.

Application of topological bifurcation theory to stability analysis.

Abstract

We prove that the Kuramoto model on a graph can contain infinitely many non-equivalent stable equilibria. More precisely, we prove that for every positive integer d there is a connected graph such that the set of stable equilibria contains a manifold of dimension d. In particular, we solve a conjecture of R. Delabays, T. Coletta and P. Jacquod about the number of equilibria on planar graphs. Our results are based on the analysis of balanced configurations, which correspond to equilateral polygon linkages in topology. In order to analyze the stability of manifolds of equilibria we apply topological bifurcation theory.