  We analyze the periodically forced Kuramoto model. This system consists of an
infinite population of phase oscillators with random intrinsic frequencies,
global sinusoidal coupling, and external sinusoidal forcing. It represents an
idealization of many phenomena in physics, chemistry and biology in which
mutual synchronization competes with forced synchronization. In other words,
the oscillators in the population try to synchronize with one another while
also trying to lock onto an external drive. Previous work on the forced
Kuramoto model uncovered two main types of attractors, called forced
entrainment and mutual entrainment, but the details of the bifurcations between
them were unclear. Here we present a complete bifurcation analysis of the model
for a special case in which the infinite-dimensional dynamics collapse to a
two-dimensional system. Exact results are obtained for the locations of Hopf,
saddle-node, and Takens-Bogdanov bifurcations. The resulting stability diagram
bears a striking resemblance to that for the weakly nonlinear forced van der
Pol oscillator.
