DANIEL M. ABRAMS
Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark
STEVEN H. STROGATZ
Permanent Address: 212 Kimball Hall, Department of Theoretical and Applied Mechanics, Cornell University, Ithaca, NY 14853-1503, USA.
Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark

Received March 2, 2005
Revised April 22, 2005

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.

Oscillator; synchronization