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The Kuramoto model of a network of coupled phase oscillators exhibits afirst-order phase transition when the distribution of natural frequencies has afinite flat region at its maximum. First-order phase transitions includinghysteresis and bistability are also present if the frequency distribution of asingle network is bimodal. In this study we are interested in the interplay ofthese two configurations and analyze the Kuramoto model with compact bimodalfrequency distributions in the continuum limit. As of yet, a rigorous analytictreatment has been elusive. By combining Kuramoto's self-consistency approach,Crawford's symmetry considerations, and exploiting the Ott-Antonsen ansatzapplied to a family of rational distribution functions that converge towardsthe compact distribution, we derive a full bifurcation diagram for the system'sorder parameter dynamics. We show that the route to synchronization alwayspasses through a standing wave regime when the bimodal distribution iscompounded by two unimodal distributions with compact support. This is incontrast to a possible transition across a region of bistability when the twocompounding unimodal distributions have infinite support.

First-order phase transitions in the Kuramoto model with compact bimodal frequency distributions