Order parameter analysis for low-dimensional behaviors of coupled phase-oscillators

TL;DR

This paper introduces an order parameter analysis approach to understand low-dimensional behaviors in coupled phase-oscillators, clarifying the scope of the Ott-Antonsen ansatz and proposing new approximation methods.

Contribution

It reveals that the Ott-Antonsen ansatz is based on system size and coupling function conditions, and develops two approximation methods for systems outside its scope.

Findings

The OA ansatz applies under infinite system size and specific Fourier coefficient conditions.

The order parameter analysis links the OA ansatz to dynamical symmetry.

Two approximation methods, ensemble approach and dominating-term assumption, are proposed.

Abstract

Coupled phase-oscillators are important models related to synchronization. Recently, Ott-Antonsen(OA) ansatz is developed and used to get low-dimensional collective behaviors in coupled oscillator systems. In this paper, we develop a simple and concise approach based on the equations of order parameters, namely, order parameter analysis, with which we point out that the OA ansatz is rooted in the dynamical symmetry of the order parameters. With our approach the scope of the OA ansatz is identified as two conditions, i.e., infinite size of the system and only three nonzero Fourier coefficients of the coupling function. Coinciding with each of the conditions, a distinctive system out of the scope is taken into account and discussed with the order parameter analysis. Two approximation methods are introduced respectively, namely the ensemble approach and the dominating-term assumption.