Global Analytical Solution of the Identical Kuramoto Model for N=3 via Koopman Eigenfunctions

TL;DR

This paper derives a global analytical solution for the phase trajectories of a three-oscillator Kuramoto model with identical oscillators using Koopman eigenfunctions, advancing understanding of nonlinear synchronization dynamics.

Contribution

It introduces a novel method to explicitly solve the phase dynamics of the N=3 Kuramoto model through Koopman eigenfunctions, extending beyond the N=2 case.

Findings

Explicit algebraic reconstruction of phase trajectories for N=3

Reduction of phase dynamics to time-dependent quartic equations

Use of Koopman eigenfunctions to relate phases to time

Abstract

The Kuramoto model is a paradigmatic model of collective synchronization in coupled oscillator systems. Although its mathematical properties have been extensively investigated, exact phase trajectories from arbitrary initial conditions have been available only for the simplest case, N=2. In this study, we provide a global analytical solution for the phase trajectories of the all-to-all coupled Kuramoto model with identical oscillators for N=3. This solution is obtained by constructing Koopman eigenfunctions that relate the phases to time and reducing the phase dynamics to time-dependent quartic equations. The algebraic branch corresponding to the initial condition is then selected to recover the corresponding phase trajectory. This gives an explicit algebraic reconstruction of the nonlinear phase dynamics from Koopman eigenfunctions.