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

The Kuramoto model of a system of coupled phase oscillators describe synchronization phenomena in nature. We propose a generalization of the Kuramoto model where each oscillator state lives on the compact, real Stiefel manifold St(p,n). Previous work on high-dimensional Kuramoto models have largely been influenced by results and techniques that pertain to the original model. This paper uses optimization and control theory to prove that the generalized Kuramoto model on St(p,n) converges to a completely synchronized state for any connected graph 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 on the circle with homogeneous oscillator frequencies. In that case, almost global synchronization is graph dependent; it applies if the network is acyclic or sufficiently dense. The problem of characterizing all such graphs is still open. This paper hence identifies a property that distinguishes many high-dimensional generalizations of the Kuramoto model from the original model. It should therefore have important implications for modeling of synchronization phenomena in physics and control of multi-agent systems in engineering applications.