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 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.
Highlights
Coupled phase-oscillators are important models related to synchronization
The famous Kuramoto model for the process of synchronization has been attracting many attentions upon it is proposed and has been developed for decades. This model consists of a population of N coupled phase oscillators {φj} with natural frequencies {ωj}, and the dynamics are described by the following equations of motion: 1College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China. 2Department of Physics and the Beijing-Hong Kong-Singapore Joint Center for Nonlinear and Complex Systems (Beijing), Beijing Normal University, Beijing 100875, China
We focused on theoretical descriptions of low-dimensional order-parameter dynamics of coupled phase oscillators
Summary
Apart from parameters K and ωj, the dynamics of each phase variable φj depends only on itself and the order parameter α, which is an important characteristic of this mean-field model. A more general form of this mean-field phase-oscillator model can be written as φ j (t) = F (α, φj, β, γj), j = 1, ..., N,. As the coupling function F(α, φ, β) is always 2π-periodic for φ, it can be written as the Fourier expansion,. Substituting the expansion of F(α, φ, β) (9) into Eq (10), we have the dynamical equations of order parameters as. The complexity of equation (11) depends on the coupling function F, or explicitly, on the Fourier coefficients {fj(α, β)}.
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