High-dimensional Kuramoto models on Stiefel manifolds synchronize complex networks almost globally

TL;DR

This paper proves that high-dimensional Kuramoto models on Stiefel manifolds achieve almost global synchronization under certain conditions, extending understanding beyond the classic circle case.

Contribution

It establishes conditions for almost global synchronization of high-dimensional Kuramoto models on Stiefel manifolds, a significant generalization of the classical model.

Findings

Synchronization occurs for any connected graph with equal frequencies.

Convergence is almost global under specific dimensional constraints.

Results differ from the classic circle case where network structure is critical.

Abstract

The Kuramoto model of coupled phase oscillators is often used to describe synchronization phenomena in nature. Some applications, e.g., quantum synchronization and rigid-body attitude synchronization, involve high-dimensional Kuramoto models where each oscillator lives on the n-sphere or SO(n). These manifolds are special cases of the compact, real Stiefel manifold St(p,n). Using tools from optimization and control theory, we prove that the generalized Kuramoto model on St(p,n) converges to a synchronized state for any connected graph and from almost all initial conditions provided (p,n) satisfies p<=2n/3-1 and all oscillator frequencies are equal. This result could not have been predicted based on knowledge of the Kuramoto model in complex networks over the circle. In that case, almost global synchronization is graph dependent; it applies if the network is acyclic or sufficiently dense. This paper hence identifies a property that distinguishes many high-dimensional generalizations of the Kuramoto models from the original model.