Chimera States in Continuous Media: Existence and Distinctness

Zachary G. Nicolaou, Hermann Riecke, and Adilson E. Motter

Phys. Rev. Lett. 119, 244101 – Published 13 December 2017

Abstract

The defining property of chimera states is the coexistence of coherent and incoherent domains in systems that are structurally and spatially homogeneous. The recent realization that such states might be common in oscillator networks raises the question of whether an analogous phenomenon can occur in continuous media. Here, we show that chimera states can exist in continuous systems even when the coupling is strictly local, as in many fluid and pattern forming media. Using the complex Ginzburg-Landau equation as a model system, we characterize chimera states consisting of a coherent domain of a frozen spiral structure and an incoherent domain of amplitude turbulence. We show that in this case, in contrast with discrete network systems, fluctuations in the local coupling field play a crucial role in limiting the coherent regions. We suggest these findings shed light on new possible forms of coexisting order and disorder in fluid systems.

Received 24 February 2017

DOI:https://doi.org/10.1103/PhysRevLett.119.244101

Physics Subject Headings (PhySH)

Chimera states Spatiotemporal chaos Synchronization

Coupled oscillators Nonlinear waves

Nonlinear Dynamics

Authors & Affiliations

Zachary G. Nicolaou¹, Hermann Riecke^2,3, and Adilson E. Motter^1,3

¹Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
²Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA
³Northwestern Institute on Complex Systems, Northwestern University, Evanston, Illinois 60208, USA

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Issue

Vol. 119, Iss. 24 — 15 December 2017

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Images

Figure 1
Coexistence of coherent spirals and amplitude turbulence in the CGL system (1): (a) with $c_{1} = 1.5$ and $c_{3} = 0.77$ and random initial conditions after a time $t = 10^{4}$ ; (b) with $c_{1} = 2.0$ and $c_{3} = 0.85$ and a spiral initial condition after a time $t = 10^{4}$ . The phase $θ \equiv \arg (A)$ is depicted in the upper left and the amplitude $r \equiv | A |$ in the lower right.
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Figure 2
Time-averaged angular frequency $ω_{av} \equiv ⟨ d θ / d t ⟩$ for the frozen vortex chimera in Fig. 1, where the arrow indicates the diameter $2 ρ$ of the coherent region. The mean frequency in the coherent region is $Ω = 0.15$ , while the frequency is significantly higher ( $> 0.40$ ) in the incoherent region. The distinction between the regions does not depend sensitively on the averaging time interval (here taken to be $10^{4}$ ). For an animation of the time evolution of this chimera state, see Supplemental Material [48].
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Figure 3
(a) Diagram of dynamical phases of the CGL system, where the frozen vortex chimeras (FVC), vortex glass (VG), and amplitude turbulence (AT) occupy the intermediate, small, and large $c_{3}$ regions, respectively. The boundary between VG and FVC was determined by calculating $η$ from five realizations of spiral nucleation from an amplitude turbulence initial state; the black lines show a modest change in this boundary as the threshold $η$ changes from $1 0^{2}$ (dashed line) to $1 0^{3}$ (solid line). The boundary between FVC and AT was determined by adiabatically varying $c_{1}$ and $c_{3}$ to move a spiral initial condition past the point of spiral destabilization, where $η$ is predicted to diverge. (b) Log-log plot of $η$ for $c_{1} = 2.0$ as a function of $0.86 - c_{3}$ , corresponding to the white line in (a). The error bars show the estimated standard deviation, and the line marks the linear fit.
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Figure 4
Time-averaged local coupling field $R_{av}$ for the conditions in Fig. 2 as a function of: (a) the two spatial coordinates; (b) the time-averaged amplitude $r_{av}$ for points in the coherent region (yellow dots), halo region (red dots), and remaining incoherent region (blue dots); and (c) the $x$ coordinate along the line $y = 0$ (magenta dots). The inset in (b) magnifies the box, which includes most points in the incoherent region. In (c) we also show the fluctuations $σ_{R}$ of $R$ (green dots). The dotted lines in (a) and (c) delimitate the coherent domain in Fig. 2. The dashed line in (b) marks the solution in Eq. (5) and in (c) marks the critical value $R = R_{c}$ above which this (real) solution exists.
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