Synchronization of higher-dimensional Kuramoto oscillators on networks: from scalar to matrix-weighted couplings

TL;DR

This paper generalizes the Kuramoto model to higher dimensions with vector oscillators and matrix-weighted networks, deriving conditions for synchronization and analyzing stability through eigenvalue problems.

Contribution

It introduces a d-dimensional Kuramoto model with matrix-weighted interactions and provides necessary conditions for synchronization using a Master Stability Function approach.

Findings

Synchronization requires identical frequency matrices across nodes.

Stability reduces to eigenvalue problems of scalar weighted Laplacians.

Synchronous solutions are locally stable for any positive coupling on connected networks.

Abstract

The Kuramoto model is the paradigmatic model to study synchronization in coupled oscillator systems. In its classical formulation, the oscillators move on the unit circle, each characterized by a scalar phase and a natural frequency, by interacting through a sinusoidal coupling. In this work, we propose a d-dimensional generalization in which oscillators are represented as unit vectors on the (d-1)-sphere and interact through a matrix-weighted network (MWN), a recently introduced framework where links are endowed with a matrix weight instead of a scalar one. We derive necessary conditions for global synchronization via a Master Stability Function approach: the existence of a synchronous solution requires identical frequency matrices across nodes and, in the MWN case, a coherence condition on the network structure. Through a suitable change of variables, the stability analysis reduces the full Nd-dimensional problem to a family of d-dimensional eigenvalue problems, each one parametrized by the eigenvalue of a suitable scalar weighted Laplacian, showing that the synchronous solution is locally stable for any positive coupling strength K on any connected network. Analytical results are complemented by numerical simulations.