Towards Almost Global Synchronization on the Stiefel Manifold

TL;DR

This paper extends synchronization results from spheres to the more general Stiefel manifold, showing that all connected graphs synchronize under certain conditions on the parameters p and n.

Contribution

It generalizes the concept of graph synchronization from spheres to the Stiefel manifold, establishing new conditions for almost global synchronization.

Findings

All connected graphs are $ ext{St}(p,n)$-synchronizing if $p \

The synchronization condition depends on the inequality $p \, ext{leq} \, \frac{2n}{3} - 1$.

The work broadens the understanding of synchronization beyond the sphere case to more complex manifolds.

Abstract

A graph $\mathcal{G}$ is referred to as $\mathsf{S}^1$-synchronizing if, roughly speaking, the Kuramoto-like model whose interaction topology is given by $\mathcal{G}$ synchronizes almost globally. The Kuramoto model evolves on the unit circle, \ie the $1$-sphere $\mathsf{S}^1$. This paper concerns generalizations of the Kuramoto-like model and the concept of synchronizing graphs on the Stiefel manifold $\mathsf{St}(p,n)$. Previous work on state-space oscillators have largely been influenced by results and techniques that pertain to the $\mathsf{S}^1$-case. It has recently been shown that all connected graphs are $\mathsf{S}^n$-synchronizing for all $n\geq2$. The previous point of departure may thus have been overly conservative. The $n$-sphere is a special case of the Stiefel manifold, namely $\mathsf{St}(1,n+1)$. As such, it is natural to ask for the extent to which the results on $\mathsf{S}^{n}$ can be extended to the Stiefel manifold. This paper shows that all connected graphs are $\mathsf{St}(p,n)$-synchronizing provided the pair $(p,n)$ satisfies $p\leq \tfrac{2n}{3}-1$.