# A linear reformulation of the Kuramoto model of self-synchronizing oscillators

## TL;DR

This paper presents a linear reformulation of the Kuramoto model, enabling explicit solutions for synchronization and critical points in finite and continuum oscillator systems, extending analysis beyond traditional mean-field approaches.

## Contribution

The paper introduces a linear reformulation of the Kuramoto model that allows explicit solutions for synchronization phenomena in finite and continuum systems, surpassing the limitations of the original model.

## Key findings

- Explicit solution for the order parameter and critical point in finite systems.
- Analytical treatment of the full phase-locking transition.
- Extension potential to locally or asymmetrically coupled systems.

## Abstract

The present paper introduces a linear reformulation of the Kuramoto model describing a self-synchronizing phase transition in a system of globally coupled oscillators that in general have different characteristic frequencies. The reformulated model provides an alternative coherent framework through which one can analytically tackle synchronization problems that are not amenable to the original Kuramoto analysis. It allows one to solve explicitly for the synchronization order parameter and the critical point of 1) the full phase-locking transition for a system with a finite number of oscillators (unlike the original Kuramoto model, which is solvable implicitly only in the mean-field limit) and 2) a new class of continuum systems. It also makes it possible to probe the system's dynamics as it moves towards a steady state. While discussion in this paper is restricted to systems with global coupling, the new formalism introduced by the linear reformulation also lends itself to solving systems that exhibit local or asymmetric coupling.