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Networks of identical, symmetrically coupled oscillators can spontaneously split into synchronized and desynchronized sub-populations. Such chimera states were discovered in 2002, but are not well understood theoretically. Here we obtain the first exact results about the stability, dynamics, and bifurcations of chimera states by analyzing a minimal model consisting of two interacting populations of oscillators. Along with a completely synchronous state, the system displays stable chimeras, breathing chimeras, and saddle-node, Hopf and homoclinic bifurcations of chimeras. [2]. When brain waves are recorded, the awake side of the brain shows desynchronized electrical activity, corresponding to millions of neurons oscillating out of phase, whereas the sleeping side is highly synchronized.From a physicist's perspective, unihemispheric sleep suggests the following (admittedly, extremely idealized) problem: What's the simplest system of two oscillator populations, loosely analogous to the two hemispheres, such that one synchronizes while the other does not?Our work in this direction was motivated by a series of recent findings in nonlinear dynamics [3,4,5,6,7,8]. In 2002, Kuramoto and Battogtokh reported that arrays of nonlocally coupled oscillators could spontaneously split into synchronized and desynchronized subpopulations [3]. The existence of such "chimera states" came as a surprise, given that the oscillators were identical and symmetrically coupled. On a one-dimensional ring [3,4] the chimera took the form of synchronized domain next to a desynchronized one. In two dimensions, it appeared as a strange new kind of spiral wave [5], with phase-locked oscillators in its arms coexisting with phaserandomized oscillators in its core-a circumstance made possible only by the nonlocality of the coupling. These phenomena were unprecedented in studies of pattern formation [9] and synchronization [10] in physics, chemistry, and biology, and remain poorly understood.Previous mathematical studies of chimera states have assumed that they are statistically stationary [3,4,5,6,7]. What has been lacking is an analysis of their dynamics, stability, and bifurcations.In this Letter we obtain the first such results by considering the simplest model that supports chimera states: a pair of oscillator populations in which each oscillator is coupled equally to all the others in its group, and less strongly to those in the other group. For this model we solve for the stationary chimeras and delineate where they exist in parameter space. An unexpected finding is that chimeras need not be stationary. They can breathe. Then the phase coherence in the desynchronized population waxes and wanes, while the phase difference between the two populations begins to wobble.The governing equations for the model arewhere σ = 1, 2 and N σ is the number of oscillators in population σ. The oscillators are assumed identical, so the frequency ω and phase lag α are the same for all of them. The strength of the coupling from oscillators in σ ′ onto those in σ...

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Systems of N identical phase oscillators with global sinusoidal coupling are known to display low-dimensional dynamics. Although this phenomenon was first observed about 20 years ago, its underlying cause has remained a puzzle. Here we expose the structure working behind the scenes of these systems, by proving that the governing equations are generated by the action of the Möbius group, a three-parameter subgroup of fractional linear transformations that map the unit disc to itself. When there are no auxiliary state variables, the group action partitions the N -dimensional state space into three-dimensional invariant manifolds (the group orbits). The N − 3 constants of motion associated with this foliation are the N − 3 functionally independent cross ratios of the oscillator phases. No further reduction is possible, in general; numerical experiments on models of Josephson junction arrays suggest that the invariant manifolds often contain three-dimensional regions of neutrally stable chaos.Large arrays of coupled limit-cycle oscillators have been used to model diverse systems in physics, biology, chemistry, engineering and social science. The special case of phase oscillators coupled all-to-all through sinusoidal interactions has attracted mathematical interest because of its analytical tractability. About 20 years ago, numerical experiments revealed that these systems display an exceptionally simple form of collective behavior: for all N ≥ 3, where N is the number of oscillators, all trajectories are confined to manifolds with N − 3 fewer dimensions than the state space itself. Several insights have been obtained over the past two decades, but it has remained an open problem to pinpoint the symmetry or other structure that causes this non-generic behavior. Here we show that group theory provides the explanation: the governing equations for these systems arise naturally from the action of the group of conformal mappings of the unit disc to itself. This link unifies and explains the previous numerical and analytical results, and yields new constants of motion for this class of dynamical systems.

We analyze a model of globally coupled nonlinear oscillators with randomly distributed frequencies.Twenty-five years ago it was conjectured that, for coupling strengths below a certain threshold, this system would always relax to an incoherent state. We prove this conjecture for the system linearized about the incoherent state, for frequency distributions with compact support. The relaxation is exponentially fast at intermediate times but slower than exponential at long times. The decay mechanism is remarkably similar to Landau damping in plasmas, even though the model was originally inspired by biological rhythms.PACS numbers: 05.45.+b, 52.35.Fp, 87.10.+e Nonlinear oscillators are among the oldest and best understood types of dynamical systems, yet little is known about their collective behavior. Recently there has been a great deal of interest in coupled oscillators, in part because they arise in many branches of science, and also because of a broader interest in high-dimensional dynamical systems [1][2][3][4][5][6][7][8][9].The problem studied in this Letter was originally inspired by the biological phenomenon of mutual synchronization [l]. In some parts of southeast Asia, thousands of male fireflies gather in trees at night and flash on and off in unison. Other examples include chorusing of crickets, synchronous firing of cardiac pacemaker cells, and metabolic synchrony in yeast cell suspensions [1]. A simple model of such systems consists of a population of coupled phase-only oscillators with distributed natural frequencies. The governing equations [2] are K N e/-a>/ + -£-Esin(0 y -0/)for j -1, . . . , N^> 1. Here 0, is the phase of oscillator /, to, is its natural frequency, and K >: 0 is the coupling strength. The frequencies are randomly chosen from a probability density g((o), assumed to be one-humped and symmetric about its mean. By choosing a rotating frame at the mean frequency, we may assume that g(co) has mean zero. For simplicity, the coupling in (l) is all-toall, corresponding to mean-field theory as N~* <». Early studies [1,2] of Eq. (1) revealed a beautiful connection to equilibrium statistical mechanics: A phase transition occurs at a critical coupling given by K c = 2//rg(0). For K K c , mutual synchronization occurs spontaneously in a small group of oscillators. This transition can be described by a complex order parameter re ,¥ = N -, Z/V y , where r measures the phase coherence of the population. In the limit of infinite TV, Kuramoto [2] determined the steady values of r self-consistently.He showed that incoherence (r=0) is always a steady solution, but a branch of partially locked solutions (r > 0) bifurcates supercritically from incoherence at K c .It has turned out to be a much more delicate matter to analyze the stability of the steady states. In particular, there has been a controversy about the relaxation to incoherence for K