  Hysteresis phenomena and multistability play crucial roles in the dynamics of
coupled oscillators, which are now interpreted from the point of view of
codimension-two bifurcations. On the Ott-Antonsen's manifold, complete
bifurcation sets of delay-coupled Kuramoto model are derived regarding coupling
strength and delay as bifurcation parameters. It is rigorously proved that the
system must undergo Bautin bifurcations for some critical values, thus there
always exists saddle-node bifurcation of periodic solutions inducing hysteresis
loop. With the aid of center manifold reduction method and the Matlab Package
DDE-Biftool, the location of Bautin and double Hopf points and detailed
dynamics are theoretically determined. We find that, near these critical
points, at most four coherent states (two of which are stable) and a stable
incoherent state may coexist, and that the system undergoes Neimark-Sacker
bifurcation of periodic solutions. Finally, the clear scenarios about the
synchronous transition in delayed Kuramoto model are depicted.
