Center manifold reduction for large populations of globally coupled phase oscillators

Details Center manifold reduction for large populations of globally coupled phase oscillators Center manifold reduction for large populations of globally coupled phase oscillators

2011-03-02

arxiv.org/abs/1103.0363v1

Nonlinear Sciences

Adaptation and Self-Organizing Systems

Scientific paper

A bifurcation theory for a system of globally coupled phase oscillators is developed based on the theory of rigged Hilbert spaces. It is shown that there exists a finite-dimensional center manifold on a space of generalized functions. The dynamics on the manifold is derived for any coupling functions. When the coupling function is $\sin \theta $, a bifurcation diagram conjectured by Kuramoto is rigorously obtained. When it is not $\sin \theta $, a new type of bifurcation phenomenon is found due to the discontinuity of the projection operator to the center subspace.

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