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We investigate the dynamics of large, globally-coupled systems of Kuramotooscillators with heterogeneous interaction delays. For the case ofexponentially distributed time delays we derive the full stability diagram thatdescribes the bifurcations in the system. Of particular interest is the onsetof hysteresis where both the incoherent and partially synchronized states arestable for a range of coupling strengths -- this occurs at a codimension-twopoint at the intersection between a Hopf bifucration and saddle-nodebifurcation of cycles. By studying this codimension-two point we find the fullset of characteristic time delays and natural frequencies where bistabilityexists and identify the critical time delay and critical natural frequencybelow which bistability does not exist. Finally, we examine the dynamics of themore general system where time delays are drawn from a Gamma distribution,finding that more homogeneous time delay distributions tend to both promote theonset of synchronization and inhibit the presence of hysteresis.

Stability Diagram, Hysteresis, and Critical Time Delay and Frequency for the Kuramoto Model with Heterogeneous Interaction Delays