Hopf bifurcation in two groups of delay-coupled kuramoto oscillators

Hopf bifurcation in two groups of Kuramoto's phase oscillators with delay-coupled interactions is investigated on the Ott-Antonsen's manifold. We find that the reduced delay differential system undergoes Hopf bifurcations when the coupling strength between two groups exceeds some critical values. With the increasing of time delay, stability switches are observed which leads to the synchrony switches for the Kuramoto system. The direction of Hopf bifurcation and the stability of bifurcating periodic solutions are investigated by deriving the normal forms on the center manifold. With respect to the Kuramoto system, simulations are performed to support our analytic results.