Equilibria in Kuramoto oscillator networks: An algebraic approach

TL;DR

This paper introduces an algebraic method to analytically find and classify equilibrium points in Kuramoto oscillator networks, including complex and various network topologies, advancing understanding of collective synchronization phenomena.

Contribution

It presents a novel algebraic approach to determine equilibria in Kuramoto networks without relying on traditional approximations, enabling exact solutions for diverse network structures.

Findings

Classified all equilibria in complete graphs.

Derived equilibrium solutions for networks with phase lag.

Extended analysis to circulant, multi-layer, and random networks.

Abstract

Kuramoto networks constitute a paradigmatic model for the investigation of collective behavior in networked systems. Despite many advances in recent years, many open questions remain on the solutions for systems composed of coupled Kuramoto oscillators on complex networks. In this article, we describe an algebraic method to find equilibrium points for this kind of system without using standard approximations in the limit of infinite system size or the continuum limit. To do this, we use a recently introduced algebraic approach to the Kuramoto dynamics, which results in an explicitly solvable complex-valued equation that captures the dynamics of the original Kuramoto model. Using this new approach, we obtain equilibria for both the nonlinear original Kuramoto and complex-valued systems. We then completely classify all equilibria in the case of complete graphs originally studied by Kuramoto. Finally, we go on to study equilibria in networks of coupled oscillators with phase lag, in generalized circulant networks, multi-layer networks, and also random networks.