An algebraic approach to the Kuramoto model

TL;DR

This paper introduces a complex-valued matrix formulation of the Kuramoto model, enabling exact solutions and analytical insights into its dynamics, demonstrating the potential of higher-order number fields for nonlinear systems analysis.

Contribution

It presents a novel complex-valued reformulation of the Kuramoto model that allows for exact solutions and deeper analytical understanding of the system's behavior.

Findings

Exact solutions for the complex-valued Kuramoto model derived

Reformulation in higher-order number fields offers tractable analysis

Provides analytical insights into individual system realizations

Abstract

We study the Kuramoto model with attractive sine coupling. We introduce a complex-valued matrix formulation whose argument coincides with the original Kuramoto dynamics. We derive an exact solution for the complex-valued model, which permits analytical insight into individual realizations of the Kuramoto model. The existence of a complex-valued form of the Kuramoto model provides a key demonstration that, in some cases, re-formulations of nonlinear dynamics in higher-order number fields may provide tractable analytical approaches.