Synchronization of diffusively coupled systems on compact Riemannian manifolds in the presence of drift

Recently, it has been shown that the synchronization manifold is an asymptotically stable invariant set of diffusively coupled systems on Riemannian manifolds. We regionally investigate the stability properties of the synchronization manifold when the systems are subject to drift. When the drift vector field is quad (i.e. satisfies a certain quadratic inequality) and the underlying Riemannian manifold is compact, we prove that a sufficiently large algebraic connectivity of the underlying graph is sufficient for the synchronization manifold to remain asymptotically stable. For drift vector fields which are quad or contracting, we explicitly characterize the rate at which the solution converges to the synchronization manifold. Our main result is that the synchronization manifold is asymptotically stable even for drift vector fields which are only locally Lipschitz continuous, as long as the algebraic connectivity of the underlying graph is sufficiently large.