Synchronization on Riemannian manifolds: Multiply connected implies multistable

This note concerns the evolution of multi-agent systems on networks over Riemannian manifolds. The motion of each agent is governed by the gradient descent flow of a disagreement function that is a sum of (squared) distances between pairs of communicating agents. Two metrics are considered: geodesic distances and chordal distances for manifolds that are embedded in an ambient Euclidean space. We show that networks which, roughly speaking, are dominated by a large cycle yield a multistable systems if the manifold is multiply connected or contains a closed geodesic that is of locally minimum length in a space of closed curves. This result summarizes previous results on the stability of splay or twist state equilibria of the Kuramoto model on the circle and its generalization, the quantum sync model on SO(n). It also extends them to the Lohe model on U(n).