Pattern phase diagram for two-dimensional arrays of coupled limit-cycle oscillators

Roland Lauter, Christian Brendel, Steven J. M. Habraken, and Florian Marquardt

Phys. Rev. E 92, 012902 – Published 6 July 2015

Abstract

Arrays of coupled limit-cycle oscillators represent a paradigmatic example for studying synchronization and pattern formation. We find that the full dynamical equations for the phase dynamics of a limit-cycle oscillator array go beyond previously studied Kuramoto-type equations. We analyze the evolution of the phase field in a two-dimensional array and obtain a “phase diagram” for the resulting stationary and nonstationary patterns. Our results are of direct relevance in the context of currently emerging experiments on nano- and optomechanical oscillator arrays, as well as for any array of coupled limit-cycle oscillators that have undergone a Hopf bifurcation. The possible observation in optomechanical arrays is discussed briefly.

Received 12 February 2015
Revised 19 May 2015

DOI:https://doi.org/10.1103/PhysRevE.92.012902

Authors & Affiliations

Roland Lauter^1,2, Christian Brendel¹, Steven J. M. Habraken¹, and Florian Marquardt^1,2

¹Institut für Theoretische Physik II, Friedrich-Alexander-Universität Erlangen-Nürnberg, Staudtstr. 7, 91058 Erlangen, Germany
²Max Planck Institute for the Science of Light, Günther-Scharowsky-Straße 1/Bau 24, 91058 Erlangen, Germany

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Vol. 92, Iss. 1 — July 2015

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Figure 1
(a) A single Hopf oscillator undergoes dynamics on a limit cycle, with an amplitude set by external parameters (such as the drive providing the power) and a time-evolving phase $φ (t)$ . (b) Array of coupled Hopf oscillators. Often, the system can be well described with a single degree of freedom per lattice site, the phase $φ_{i}$ . Due to the coupling, the phases can lock and phase patterns can form.
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Figure 2
Spiral patterns and length scales in the Hopf-Kuramoto model. (a) Stationary spiral pattern emerging from random initial conditions for $S_{1} / C = 2.0, S_{2} / C = 0$ on a $N \times N$ square lattice ( $N = 128$ ) with periodic boundary conditions. (b) Vortex–anti-vortex pair (see inset) winding up to a stationary spiral–anti-spiral pair with a characteristic spiral arm width $λ$ . Parameters are like in (a). (c) Spatial correlations $Re 〈 exp [i (φ_{m} - φ_{n})] 〉$ as a function of the distance $| {\vec{r}}_{m} - {\vec{r}}_{n} |$ (rounded to the nearest integer). To obtain the data for (d), we extract the first correlation minimum position from parabolic fits and average over ten runs with different random initial conditions. (d) The location of the first correlation minimum [red (light grey)] and the spiral arm width $λ$ from (b) (black), as a function of the ratio $S_{1} / C$ , in units of the lattice constant. There can be hysteresis [blue (dark grey)].
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Figure 3
Pattern phase diagram of the Hopf-Kuramoto model, Eq. (2). Different colors indicate different patterns, which are discussed in the main text. Sharp transitions occur between stationary spirals and pulsating/mobile spirals (for small $S_{2} / C$ ), and in the appearance of “ $π$ defects”. Point markers indicate parameters explored by numerical simulation. Some typical phase patterns are shown in the insets (a) to (f).
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Figure 4
Spiral motion in the Hopf-Kuramoto model. (a) Instantaneous phase velocity field ${\dot{φ}}_{i}$ after a (long) integration time $T = 1000 / C .$ Mobile spiral centers are visible as local inhomogeneities in ${\dot{φ}}_{i}$ . Some stationary spirals (not visible) exist in the uniform regions. Parameters are $S_{1} / C = 1.6, S_{2} / C = 0.5, N = 64$ . The corresponding phase field is shown in Fig. 3. (b) Spiral positions at time $T$ . The lines are the spiral trajectories (from $T - 15 / C$ to $T$ ; the trajectories have been slightly smoothened for clarity). (c) Spiral dynamics of a single spiral over a longer period of time. It can be seen that the spiral remains fixed for some time before starting to move again (this is usually induced via a kick by a nearby mobile spiral). The spiral preferably moves in the cartesian directions set by the lattice.
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Figure 5
The eigenvalues $λ_{m}$ of the Hessian $\partial^{2} U / (\partial φ_{i} \partial φ_{j})$ for a homogeneous phase configuration with a single $π$ defect, as a function of the parameter $S_{1} / S_{2}$ for a lattice of size $20 \times 20$ . Note the eigenvalue $λ^{-}$ leading to the instability of the $π$ defect. The zero eigenvalue $λ^{T}$ belongs to the translational mode of the phase configuration.
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Figure 6
(a) Partial intensity ${〈| \sum_{j} e^{i \vec{q} \cdot {\vec{r}}_{j}} θ_{j} |^{2}〉}_{t}$ of the light reflected from the phase field given in Fig. 2. Parameter $θ^{\max} = 0.01$ . (b) Partial intensity ${〈| \sum_{j} e^{i \vec{q} \cdot {\vec{r}}_{j}} θ_{j} |^{2}〉}_{t}$ in dependance on the radial coordinate $q_{r} = \sqrt{q_{x}^{2} + q_{y}^{2}}$ , averaged over 11 different random initial phase configurations (black) and for a single phase configuration as in (a) (blue).
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