Center manifold reduction for large populations of globally coupled phase oscillators

Hayato Chiba, Isao Nishikawa

Research output: Contribution to journalArticle

37 Citations (Scopus)

Abstract

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.

Original languageEnglish
Article number043103
JournalChaos
Volume21
Issue number4
DOIs
Publication statusPublished - Jan 1 2011

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Center Manifold Reduction
oscillators
Center Manifold
Bifurcation Theory
Projection Operator
Bifurcation Diagram
Generalized Functions
Discontinuity
Bifurcation (mathematics)
Bifurcation
Hilbert space
Subspace
Hilbert spaces
discontinuity
projection
diagrams
operators

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Physics and Astronomy(all)
  • Applied Mathematics

Cite this

Center manifold reduction for large populations of globally coupled phase oscillators. / Chiba, Hayato; Nishikawa, Isao.

In: Chaos, Vol. 21, No. 4, 043103, 01.01.2011.

Research output: Contribution to journalArticle

Chiba, Hayato ; Nishikawa, Isao. / Center manifold reduction for large populations of globally coupled phase oscillators. In: Chaos. 2011 ; Vol. 21, No. 4.
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