A center manifold reduction of the Kuramoto-Daido model with a phase-lag

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A bifurcation from the incoherent state to the partially synchronized state of the Kuramoto-Daido model with the coupling function f(θ) = sin(θ + α1) + h sin 2(θ + α2) is investigated based on the generalized spectral theory and the center manifold reduction. The dynamical equation for the order parameter on a center manifold is derived under the assumption that there exists a center manifold on the dual space of a certain test function space. It is shown that the incoherent state loses the stability at a critical coupling strength K = Kc, and a stable rotating partially synchronized state appears for K > Kc. The velocity of the rotating state is different from the average of natural frequencies of oscillators when α1 ≠= 0.

Original languageEnglish
Pages (from-to)1235-1259
Number of pages25
JournalSIAM Journal on Applied Dynamical Systems
Issue number3
Publication statusPublished - 2017


  • Center manifold reduction
  • Generalized spectral theory
  • Kuramoto model


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