Integrable order parameter dynamics of globally coupled oscillators

TL;DR

This paper provides an exact analytical solution for the nonlinear dynamics of globally coupled nonidentical oscillators using a two-parameter reduction, offering a comprehensive classification of phase behaviors and bifurcations.

Contribution

It introduces an exact solution framework for oscillator dynamics with two order parameters, advancing understanding of their bifurcation structures and invariant manifolds.

Findings

Complete classification of phase portraits and bifurcations

Explicit expressions for invariant manifolds including limit cycles

Analytical solutions for arbitrary initial conditions and regimes

Abstract

We study the nonlinear dynamics of globally coupled nonidentical oscillators in the framework of two order parameter (mean field and amplitude-frequency correlator) reduction. The main result of the paper is the exact solution of the corresponding nonlinear system on an attracting manifold. We present a complete classification of phase portraits and bifurcations, obtain explicit expressions for invariant manifolds (a limit cycle among them) and derive analytical solutions for arbitrary initial data and different regimes.