  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.
