# Synchronization on Riemannian manifolds: Multiply connected implies multistable

## TL;DR

This paper investigates how the topology of Riemannian manifolds influences the stability of multi-agent synchronization, revealing that multiply connected manifolds can lead to multistability in network dynamics.

## Contribution

It extends existing synchronization results to Riemannian manifolds with complex topology, showing that large cycles induce multistability in multi-agent systems.

## Key findings

- Networks with large cycles exhibit multistability on multiply connected manifolds.
- Results generalize stability of splay and twist states from Kuramoto and quantum sync models.
- Multistability occurs when the manifold contains a closed geodesic of minimal length.

## Abstract

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).