Emergent behaviors of high-dimensional Kuramoto models on Stiefel manifolds

Abstract

In this paper, we study emergent dynamics for first and second-order high-dimensional Kuramoto type models on Stiefel manifolds. For the first-order consensus model on the Stiefel manifold proposed in Markdahl et al. (2018), we show that a homogeneous ensemble relaxes to completely consensus state exponentially fast. On the other hand for a heterogeneous ensemble, we provide sufficient conditions leading to the locked state in which relative distances between two states converge to definite values in a large coupling regime. We also propose a second-order extension of the first-order one, and study their emergent behaviors using Lyapunov functionals such as an energy functional and an averaged distance functional.