Synchronization of globally coupled phase oscillators: Singularities and scaling for general couplings

DOI

10.1016/S0167-2789(98)00235-8

Document Type

Journal Article

Publication Date

1-1-1999

Publication Title

Physica D: Nonlinear Phenomena

Volume

125

Issue

2022-01-02

First Page

1

Last Page

46

ISSN

1672789

Keywords

Bifurcation, Symmetry, Oscillators

Abstract

The onset of collective behavior in a population of globally coupled oscillators with randomly distributed frequencies is studied for phase dynamical models with arbitrary coupling; the effect of a stochastic temporal variation in the frequencies is also included. The Fokker-Planck equation for the coupled Langevin system is reduced to a kinetic equation for the oscillator distribution function. Instabilities of the phase-incoherent state are studied by center manifold reduction to the amplitude dynamics of the unstable modes. Depending on the coupling, the coefficients in the normal form can be singular in the limit of weak instability when the diffusive effect of the noise is neglected. A detailed analysis of these singularities to all orders in the normal form expansion is presented. Physically, the singularities are interpreted as predicting an altered scaling of the entrained component near the onset of synchronization These predictions are verified by numerically solving the kinetic equation for various couplings and frequency distributions. © 1999 Elsevier Science B.V. All rights reserved.

Open Access

Green Accepted

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