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

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 = K_c, and a stable rotating partially synchronized state appears for K > K_c. The velocity of the rotating state is different from the average of natural frequencies of oscillators when α₁ ≠= 0.