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

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 equations of order parameters, namely, order parameter analysis, with which we point out that OA ansatz is rooted in the dynamical symmetry of order parameters. With our approach the scope of OA ansatz is identified as two conditions, i.e., the limit of infinitely many oscillators 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 expectation assumption and the dominating-term assumption.