  A Hopf bifurcation in the Kuramoto-Daido model is investigated based on the
generalized spectral theory and the center manifold reduction for a certain
class of frequency distributions. The dynamical system of the order parameter
on a four-dimensional center manifold is derived. It is shown that the
dynamical system undergoes a Hopf bifurcation as the coupling strength
increases, which proves the existence of a periodic two-cluster state of
oscillators.
