Persistent chimera states in nonlocally coupled phase oscillators

Abstract

Chimera states in the systems of nonlocally coupled phase oscillators are considered stable in the continuous limit of spatially distributed oscillators. However, it is reported that in the numerical simulations without taking such limit, chimera states are chaotic transient and finally collapse into the completely synchronous solution. In this Rapid Communication, we numerically study chimera states by using the coupling function different from the previous studies and obtain the result that chimera states can be stable even without taking the continuous limit, which we call the persistent chimera state. The behavior of coupled oscillator systems can describe various pattern formations in a wide range of scientific fields [1,2]. In the systems of nonlocally coupled identical oscillators, there often appears a strange phenomenon called the chimera state, which is characterized by the coexistence of coherent and incoherent domains, where the former domain consists of phase-locked oscillators and the latter domain consists of drifting oscillators with spatially changing frequencies . This interesting phenomenon was first discovered in the system of nonlocally coupled phase oscillators obeying the evolution equationwith 2π -periodic phases θ (x) on a finite interval x ∈ [0,1] under the periodic boundary condition, a smooth 2π -periodic coupling function , and the kernel G(y) = (κ/2) exp(−κ|y|), where a constant 1/κ denotes the coupling range [3]. Recently, similar spatiotemporal patterns have been found in various systems using, e.g., the logistic maps [14,15], Rössler systems [15], and FitzHugh-Nagumo oscillators [18].In the study of the chimera state, the system, Eq. (1), with the sine coupling [27]is particularly important because of its simplicity and generality. In fact, this coupling function was used also in the first discovery of the chimera state [3]. For numerical simulations, we usually discretize Eq. (1) into such form as Eq. (3). In the simulations of such discretized systems, we can confirm that chimera states are surely stable in the continuous limit N → ∞. However, the stability of chimera states in finitely discretized systems is questioned. In fact, it is reported that when N is finite, chimera states with the sine coupling are chaotic transient and finally collapse into the completely synchronous solution [12,13,26]. Recently, Ashwin and Burylko proposed the weak chimera similar to the chimera state, which is defined by the coexistence of frequency-synchronous and -asynchronous oscillators in the systems of coupled indistinguishable phase oscillators but is not necessarily spatially structured as coherent and incoherent domains [28]. They studied the weak chimera in some types of networks composed of the minimal number of oscillators with the Hansel-Mato-Meunier coupling, Eq. (4), and demonstrated that the weak chimera can be persistent (nontransient). In this Rapid Communication, we study chimera states in the systems of nonlocally coupled phase oscillators with the Hansel-MatoMeunier coupling b...

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