  We analyze a large system of globally coupled phase oscillators whose natural
frequencies are bimodally distributed. The dynamics of this system has been the
subject of long-standing interest. In 1984 Kuramoto proposed several
conjectures about its behavior; ten years later, Crawford obtained the first
analytical results by means of a local center manifold calculation.
Nevertheless, many questions have remained open, especially about the
possibility of global bifurcations. Here we derive the system's complete
stability diagram for the special case where the bimodal distribution consists
of two equally weighted Lorentzians. Using an ansatz recently discovered by Ott
and Antonsen, we show that in this case the infinite-dimensional problem
reduces exactly to a flow in four dimensions. Depending on the parameters and
initial conditions, the long-term dynamics evolves to one of three states:
incoherence, where all the oscillators are desynchronized; partial synchrony,
where a macroscopic group of phase-locked oscillators coexists with a sea of
desynchronized ones; and a standing wave state, where two counter-rotating
groups of phase-locked oscillators emerge. Analytical results are presented for
the bifurcation boundaries between these states. Similar results are also
obtained for the case in which the bimodal distribution is given by the sum of
two Gaussians.
