Chimera states as chaotic spatiotemporal patterns

Chimera states are a recently new discovered dynamical phenomenon that appears in arrays of nonlocally coupled oscillators and displays a spatial pattern of coherent and incoherent regions. We report here an additional feature of this dynamical regime: an irregular motion of the position of the coherent and incoherent regions, i.e., we reveal the nature of the chimera as a spatiotemporal pattern with a regular macroscopic pattern in space, and an irregular motion in time. This motion is a finite-size effect that is not observed in the thermodynamic limit. We show that on a large time scale, it can be described as a Brownian motion. We provide a detailed study of its dependence on the number of oscillators $N$ and the parameters of the system.

Figure 1

Phase snapshots of (a) symmetric and (b) nonsymmetric chimera states. $N=200$.

Figure 2

(Color online) Sensitivity to initial conditions: two chimera trajectories with initial conditions that differ by ${10}^{-3}$. (Color coded time-averaged phase velocities, dark region: coherent motion). $N=200$.

Figure 3

(Color online) (a) Snapshot of phase velocities (orange/gray dots); time averaged phase velocities (black dots) following the profile given by Kuramoto’s self-consistency equation. For the given phase velocities, the reference function $f\left(x,\xi \right)$ with $\xi =0$ (dashed green curve) minimizes the distance function $F\left(\xi \right)$ shown in (b).

Figure 4

(Color online) Time evolution $\xi \left(t\right)$ of a chimera’s position, $N=200$: (a) Brownian motion on large time scales and (b) deterministic irregular oscillations on a short time scale.

Figure 5

(Color online) Normalized autocorrelation function (5) for $\xi \left(t\right)$ evaluated from a time interval $T={10}^6$. $N=200$.

Figure 6

(a) Histograms of displacements $\Delta \xi$ corresponding to $\Delta t=200$ (circles) and $\Delta t=600$ (squares) with fitted normal distribution [Eq. (7)] (solid lines). (b) Variance $\sigma$ for coupling function $G_{step}\left(x,0.7\right)$ (squares) and $G_{\text{cos}}\left(x,0.99\right)$ (circles) shows linear dependence on time step $\Delta t$ [see Eq. (8)]. $N=100$.

Figure 7

The dependence of the diffusion coefficient $D$ on the number of oscillators $N$ follows a power law $D\left(N\right)\sim N^{-5/3}$.

Figure 8

(Color online) Dependence of the diffusion coefficient $D$ on the coupling range $r$ and the phase lag $\alpha$. The gray region at the bottom indicates the parameter region, where we found chimera trajectories for $N=100$. The green (light gray) line indicates the boundary of the existence region for the limit $N=\infty$.