Synchronization on Riemannian Manifolds: Multiply Connected Implies Multistable

This article concerns the evolution of multiagent 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 that, roughly speaking, are dominated by a large cycle yield 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
S
¹
and its generalization, the quantum sync model on SO(n). It also extends them to the Lohe model on U(n).