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.
Institute of Mathematics for Industry, Kyushu University, Fukuoka 819-0395, Japan.
Hayato
H
Chiba
Isao
I
Nishikawa
D003198:Computer Simulation; D005246:Feedback; D017711:Nonlinear Dynamics; D009991:Oscillometry / Q000379:methods*