Kuramoto Networks with Infinitely Many Stable Equilibria

Abstract

SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3267-3283, December 2023.
Abstract. We prove that the Kuramoto model on a graph can contain infinitely many nonequivalent stable equilibria. More precisely, we prove that for every [math] there is a connected graph such that the set of stable equilibria contains a manifold of dimension [math]. In particular, we solve a conjecture of Delabays, Coletta, and 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.