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Certain consensus seeking multi-agent systems can be formulated as gradientdescent flows of a disagreement function. We study how known pathologies ofgradient descent flows in Euclidean spaces carry over to consensus seekingsystems that evolve on nonlinear manifolds. In particular, we show that thenorms of agent states can diverge to infinity, but this will not happen if themanifold is the boundary of a convex set. Moreover, the system can beinitialized arbitrarily close to consensus without converging to it, but thiswill not happen if the manifold is analytic. For analytic manifolds, consensusis asymptotically stable. This last result summarizes a number of previousfindings in the literature on generalizations of the well-known Kuramoto modelto high-dimensional manifolds.

Counterexamples in synchronization: pathologies of consensus seeking gradient descent flows on surfaces