Towards Almost Global Synchronization on the Stiefel Manifold

Abstract

The Kuramoto model evolves on the circle, i.e. the 1-sphere S^1. A graph G is referred to as S^1-synchronizing if the Kuramoto model on G synchronizes almost globally. This paper generalizes the Kuramoto model and the concept of synchronizing graphs to the Stiefel manifold St(p,n). Previous work on generalizations of the Kuramoto model have largely been influenced by results and techniques that pertain to the original model. It was recently shown that all connected graphs are S^n-synchronizing for all n>=2. However, that does not hold for n=1. Previous results on generalized models may thus have been overly conservative. The n-sphere is a special case of the Stiefel manifold, namely St(1,n+1). As such, it is natural to ask for the extent to which the results on S^n can be extended to the Stiefel manifold. This paper shows that all connected graphs are St(p,n)-synchronizing provided the pair (p,n) satisfies p=