Cellular automaton for chimera states

Abstract

A minimalistic model for chimera states is presented. The model is a cellular automaton (CA) which depends on only one adjustable parameter, the range of the nonlocal coupling, and is built from elementary cellular automata and the majority (voting) rule. This suggests the universality of chimera-like behavior from a new point of view: Already simple CA rules based on the majority rule exhibit this behavior. After a short transient, we find chimera states for arbitrary initial conditions, the system spontaneously splitting into stable domains separated by static boundaries, ones synchronously oscillating and the others incoherent. When the coupling range is local, nontrivial coherent structures with different periodicities are formed. This motivates the need of simple models, reduced to the barest essentials, to describe the underlying mechanisms behind their formation. Cellular automata (CAs) [22,[50][51][52][53][54][55][56][57] hold promise for that goal. For example, chimera states were found in a three-level CA of ZykovMikhailov type [58], and Boolean phase oscillators, realized with electronic logic circuits, were also shown to support transient chimeras [44].In this letter we regard chimera states as an experimental fact of nature rather than a feature of certain systems of differential equations or maps. We then formulate a simple CA model for chimera states, describing a possible universal mechanism behind their spontaneous emergence out of any initial condition. We sketch a general mathematical approach to show how CAs can be regarded as approximations (shadowings) of maps of coupled oscillators. Although we do not attempt here to connect our specific CA model to any such map, we hypothesize that the latter should exist [59,60]. Chimeras are here modelled as specific instances of domain formation in spatially extended systems, a behavior that is statistically robust to small perturbations and which is ubiquitously found in nature. The chimera states encountered here are of the weak type [30,31] and are stable and coexist with synchronously oscillating domains separated by static walls. The model depends on one free parameter only, ξ ∈ N (ξ ≥ 1), whose physical meaning is the neighborhood radius (nonlocal coupling range). When ξ is small, nontrivial coherent structures are formed. However, when ξ is sufficiently large, incoherent domains of thickness d ≥ ξ + 1 arise.We first show how any map on a ring of N spatially coupled oscillators can be approximated by a CA. Let ϕ j t ∈ [0, 2π) denote the phase of the oscillator at location j, j ∈ [0, N − 1] and discrete time t. We assume that the evolution of the phases in the torus T

The changing notion of chimera states, a critical review

“…3. Some of the cellular-automaton chimeras published by García-Morales in 2016 also have a well-known periodicity; and while he found no periodicity for some of the others during the tested simulation time, it is of course fundamentally true that "because of the finiteness of the dynamics, the periodicity of any structure is bounded" 66 , that is, because the states of the system are discrete, it is bound to repeat itself eventually. García-Morales was possibly also the first to recognize how the community's chimera definition had broadened, mentioning how he would "regard chimera states as an experimental fact of nature rather than a feature of certain systems of differential equations or maps" 66 .…”

“…They did this by defining a "weak chimera" to be a state in which the average phase velocities of at least two oscillators converge in the limit of infinite time T → ∞, while it remains different for at least one other oscillator 78 . This definition was subsequently used to classify states in several later works 47,66,[79][80][81][82][83] . One may assume that it was inspired by the observation that the effective average frequencies of the incoherent oscillators differ from the average frequency of the synchronized cluster in both Kuramoto and Battogtokh's 2002 chimera 1 as well as several later chimera states 18,42,67,71,84,85 .…”

Diagrammatic approach to cellular automata and the emergence of form with inner structure

“…1 b). The following is a CA model for chimera states that has been introduced in a recent article [13] x j t+1 = y (1,0),j t+1…”

“…A good deal of effort has been devoted to the identification of appropriate CA models in the huge computational space for specific needs [5]. Building on very recent work [12,13] this article advances a general and systematic method as a pathway to tackle this problem.…”

From deterministic cellular automata to coupled map lattices

“…The problem of how to preserve certain continuum properties as isotropy has been considered by Nishiyama and Tokihiro [44] who construct an isotropic CA model for reaction diffusion systems by exploiting the fact that the continuum limit of a random walk yields a diffusion equation. Very recently [45], we have also proposed a general method to derive CA approximations (shadowings) from CMLs of nonlinearly coupled oscillators.…”