Travelling chimeras in oscillator lattices with advective–diffusive coupling

Abstract

We consider a one-dimensional array of phase oscillators coupled via an auxiliary complex field. While in the seminal chimera studies by Kumamoto and Battogtokh only diffusion of the field was considered, we include advection which makes the coupling left–right asymmetric. Chimera starts to move and we demonstrate that a weakly turbulent moving pattern appears. It possesses a relatively large synchronous domain where the phases are nearly equal, and a more disordered domain where the local driving field is small. For a dense system with a large number of oscillators, there are strong local correlations in the disordered domain, which at most places looks like a smooth phase profile. We find also exact regular travelling wave chimera-like solutions of different complexity, but only some of them are stable. This article is part of the theme issue ‘New trends in pattern formation and nonlinear dynamics of extended systems’.

Nonuniformly twisted states and traveling chimeras in a system of nonlocally coupled identical phase oscillators

“…They observed a chimera-like state, although they followed this motion for a relatively short time interval. This system is close to the KB setup, with an additional advective term in the coupling [24]. The basic observation is that a relatively ordered phase profile appears in such a system, which, however, can be well visualized for a large number of oscillators only.…”

“…The corresponding Poincaré map coincides with the Möbius transformation that takes the closed unit disk onto itself [26][27][28][29][30]. According to [24,30], the parameters of the canonical form of this transformation can be uniquely determined by employing two solutions of the supporting complex Riccati equation starting from the specific (specially selected) initial conditions. For fixed quantities Ω, v, h 0 , h 1 and h 2 , this approach allows one to obtain a periodic orbit for the phase variable ϕ(ξ) defined on a cylinder, which automatically meets the periodicity condition (10) and can be associated with the phase profile of the traveling wave we are looking for (see [24] for details).…”

“…As a result, for each set of the significant model parameters α, g 1 and g 2 , the problem of searching for a nonuniformly twisted state in the form of a continuous pattern with a permanent shape, moving at a constant velocity, reduces to the problem of detecting roots of equation (15), which can be solved numerically by a method based on the Newton-Raphson algorithm. A discussion of the details and subtleties (in particular, an iterative procedure for finding a good initial approximation close to the genuine roots of the nonlinear equations) of each stage of the developed procedure for constructing similar solutions in the model of oscillator lattices with advective-diffusive coupling can be found in the paper [24].…”

“…This traveling regime is non-stationary and weakly turbulent. Additionally, in [24], regular traveling wave solutions in a KB-type setting were constructed, but only some of them are stable.…”

Introduction to ‘New trends in pattern formation and nonlinear dynamics of extended systems’

“…Among them are chimera patterns combining order and disorder [ 30 ]. In [ 31 ], chimera patterns are studied in a one-dimensional array of phase oscillators coupled via a complex field, which is governed by an advection–diffusion equation with violated left–right symmetry. In contradistinction to the symmetric case, chimera patterns move.…”