  In the context of the Kuramoto model of coupled oscillators with distributed
natural frequencies interacting through a time-delayed mean-field, we derive as
a function of the delay exact results for the stability boundary between the
incoherent and the synchronized state and the nature in which the latter
bifurcates from the former at the critical point. Our results are based on an
unstable manifold expansion in the vicinity of the bifurcation, which we apply
to both the kinetic equation for the single-oscillator distribution function in
the case of a generic frequency distribution and the corresponding
Ott-Antonsen(OA)-reduced dynamics in the special case of a Lorentzian
distribution. Besides elucidating the effects of delay on the nature of
bifurcation, we show that the approach due to Ott and Antonsen, although an
ansatz, gives an amplitude dynamics of the unstable modes close to the
bifurcation that remarkably coincides with the one derived from the kinetic
equation. Further more, quite interestingly and remarkably, we show that close
to the bifurcation, the unstable manifold derived from the kinetic equation has
the same form as the OA manifold, implying thereby that the OA-ansatz form
follows also as a result of the unstable manifold expansion. We illustrate our
results by showing how delay can affect dramatically the bifurcation of a
bimodal distribution.
