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 $f(\theta ) = \sin (\theta +\alpha _1) + h\sin 2(\theta +\alpha _2)$ is investigated based on the generalized spectral theory and the center manifold reduction. The dynamical system of 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 $\alpha _1 \neq 0$.