  We investigate the dynamics of large, globally-coupled systems of Kuramoto
oscillators with heterogeneous interaction delays. For the case of
exponentially distributed time delays we derive the full stability diagram that
describes the bifurcations in the system. Of particular interest is the onset
of hysteresis where both the incoherent and partially synchronized states are
stable for a range of coupling strengths -- this occurs at a codimension-two
point at the intersection between a Hopf bifucration and saddle-node
bifurcation of cycles. By studying this codimension-two point we find the full
set of characteristic time delays and natural frequencies where bistability
exists and identify the critical time delay and critical natural frequency
below which bistability does not exist. Finally, we examine the dynamics of the
more general system where time delays are drawn from a Gamma distribution,
finding that more homogeneous time delay distributions tend to both promote the
onset of synchronization and inhibit the presence of hysteresis.
