Center manifold reduction for large populations of globally coupled phase oscillators

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 $\sin θ$, a bifurcation diagram conjectured by Kuramoto is rigorously obtained. When it is not $\sin θ$, a new type of bifurcation phenomenon is found due to the discontinuity of the projection operator to the center subspace.

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