  Arrays of identical limit-cycle oscillators have been used to model a wide
variety of pattern-forming systems, such as neural networks, convecting fluids,
laser arrays, and coupled biochemical oscillators. These systems are known to
exhibit rich collective behavior, from synchrony and traveling waves to
spatiotemporal chaos and incoherence. Recently, Kuramoto and his colleagues
reported a strange new mode of organization--here called the chimera state--in
which coherence and incoherence exist side by side in the same system of
oscillators. Such states have never been seen in systems with either local or
global coupling; they are apparently peculiar to the intermediate case of
nonlocal coupling. Here we give an exact solution for the chimera state, for a
one-dimensional ring of phase oscillators coupled nonlocally by a cosine
kernel. The analysis reveals that the chimera is born in a continuous
bifurcation from a spatially modulated drift state, and dies in a saddle-node
collision with an unstable version of itself.
