Bifurcation analysis on the globally coupled Kuramoto oscillators with distributed time delays

Distributed delay interactions among a group of Kuramoto phase oscillators are studied from the viewpoint of bifurcation analysis. After restricting the system on the Ott-Antonsen manifold, a simplified model consisting of delay differential equations is obtained. Hopf bifurcation diagrams are drawn on some two-parameter planes around the incoherent state when delay follows Dirac, uniform, Gamma and normal distributions, respectively, and it is illustrated that stronger coupling is needed to achieve synchrony when increasing the variance of either natural frequency or time delay. With the aid of center manifold reduction and the normal form method, the direction of Hopf bifurcation and stability of bifurcating periodic solutions are investigated, and the existence of the hysteresis loop is explained theoretically.