Chimera states in systems of nonlocal nonidentical phase-coupled oscillators

Chimera states consisting of domains of coherently and incoherently oscillating nonlocally coupled phase oscillators in systems with spatial inhomogeneity are studied. The inhomogeneity is introduced through the dependence of the oscillator frequency on its location. Two types of spatial inhomogeneity, localized and spatially periodic, are considered and their effects on the existence and properties of multicluster and traveling chimera states are explored. The inhomogeneity is found to break up splay states, to pin the chimera states to specific locations, and to trap traveling chimeras. Many of these states can be studied by constructing an evolution equation for a complex order parameter. Solutions of this equation are in good agreement with the results of numerical simulations.

Figure 1

The phase distribution $\theta \left(x\right)$ for splay states observed with $G\left(x\right)=cos\left(x\right),\omega \left(x\right)={\omega }_{0}exp(-2|x\left|\right)$, and $\beta =0.05$. (a) ${\omega }_0=0.006$. (b) ${\omega }_0=0.02$. (c) ${\omega }_0=0.05$. (d) ${\omega }_0=0.1$. The states travel to the right ($\Omega >0$). The inset in (a) shows an enlargement of the region near $x=0$.

Figure 2

(a) A snapshot of the phase distribution $\theta (x,t)$ for $G\left(x\right)=cos\left(x\right)$ and $\omega \left(x\right)=0.02exp(-2|x\left|\right)$. (b) The corresponding $R\left(x\right)$. (c) The corresponding $\Theta \left(x\right)$. (d) The oscillator frequency ${\overline{\theta }}_t$ averaged over the time interval $0<t<1000$ (solid line). In the coherent region ${\overline{\theta }}_t$ coincides with the global oscillation frequency $-\Omega$ (open circles). The inset in (a) shows an enlargement of the region near $x=0$. All simulations are done with $\beta =0.05$ and $N=512$.

Figure 3

(a) The overall frequency $\Omega$ and (b) the fraction $e$ of the domain occupied by the coherent oscillators as a function of the parameter ${\omega }_0$. (c) The two regions of incoherence $x_l\le x\le x_r$ that open up at $\omega \approx 0.0063$ and $\omega \approx 0.065$ corresponding to the transitions visible in panels (a) and (b). The calculation is for $\kappa =2,\beta =0.05$ and $N=512$.

Figure 4

Comparison of $\Omega +\omega \left(x\right)$ and $R\left(x\right)$ at the critical values ${\omega }_0$ for the appearance of new regions of incoherence around (a) $x=0$ for ${\omega }_0=0.0063$ and (b) $x=1.83$ for ${\omega }_0=0.065$. The calculation is for $\kappa =2,\beta =0.05$ and $N=512$.

Figure 5

(a) A snapshot of the phase distribution $\theta (x,t)$ in a 2-cluster chimera state for $G\left(x\right)=cos\left(x\right)$ and $\omega \left(x\right)=0.1exp(-2|x\left|\right)$. (b) The corresponding order parameter $R\left(x\right)$. (c) The corresponding order parameter $\Theta \left(x\right)$. Note that the oscillators in the two clusters oscillate with the same frequency but $\pi$ out of phase. The calculation is done with $\kappa =2,\beta =0.05$, and $N=512$.

Figure 6

The dependence of (a) $\Omega$ and the order parameter amplitude $b$ on ${\omega }_0$. (b) The fraction $e$ of coherent oscillators as a function of ${\omega }_0$. The calculation is done with $\kappa =2,\beta =0.05$, and $N=512$.

Figure 7

The position $x_0\left(t\right)$ of the coherent cluster as a function of time when ${\omega }_0=0.1,\beta =0.05,N=512$, and (a) $\kappa =10$, (b) $\kappa =6$, and (c) $\kappa =2$.

Figure 8

The standard deviation of the position $x_0\left(t\right)$ as a function of the parameter $\kappa$ for ${\omega }_0=0.1,\beta =0.05$, and $N=512$.

Figure 9

The histogram of the residues $\varepsilon \equiv x_0(t+\Delta t)-A{x}_0\left(t\right)-B$ is well approximated by a normal distribution when $\Delta t=100$.

Figure 10

A near-splay state for (a) $\omega \left(x\right)=0.1cos\left(2x\right)$. Chimera splay states for (b) $\omega \left(x\right)=0.2cos\left(2x\right)$, (c) $\omega \left(x\right)=0.3cos\left(2x\right)$, and (d) $\omega \left(x\right)=0.4cos\left(2x\right)$. In all cases $\beta =0.05$ and $N=512$.

Figure 11

Chimera splay states for $G\left(x\right)=cos\left(x\right)$ and (a) $\omega \left(x\right)=0.2cos\left(3x\right)$ (3-cluster state), (b) $\omega \left(x\right)=0.2cos\left(4x\right)$ (4-cluster state), and (c) $\omega \left(x\right)=0.2cos\left(5x\right)$ (5-cluster state). In all cases $\beta =0.05$ and $N=512$.

Figure 12

The dependence of (a) $\Omega$ and (b) the fraction $e$ of the domain occupied by the coherent oscillators on ${\omega }_0$, with $\omega \left(x\right)={\omega }_{0}cos\left(x\right)$ and $\beta =0.05$.

Figure 13

The dependence of (a) $\Omega$ and (b) the fraction $e$ of the domain occupied by the coherent oscillators on ${\omega }_0$, with $\omega \left(x\right)={\omega }_{0}cos\left(2x\right)$ and $\beta =0.05$.

Figure 14

Comparison of $\Omega +\omega \left(x\right)$ and $R\left(x\right)$ at the critical value ${\omega }_0\approx 0.16$. Parameters: $l=2$ and $\beta =0.05$.

Figure 15

Comparison of $\Omega +\omega \left(x\right)$ and $R\left(x\right)$ at the critical values ${\omega }_0$ for the appearance of a new region of incoherence around (a) $x=0$ for ${\omega }_0=0.0075$ and (b) $x=2.8$ for ${\omega }_0=0.13$. Parameters: $l=1$ and $\beta =0.05$.

Figure 16

Dependence of (a) $\Omega$ and (b) the coherent fraction $e$ on ${\omega }_0$, with $\omega \left(x\right)={\omega }_{0}cos\left(3x\right)$ and $\beta =0.05$.

Figure 17

(a) A snapshot of the phase distribution $\theta (x,t)$ in a 1-cluster chimera state for $G\left(x\right)=cos\left(x\right)$ and $\omega \left(x\right)=0.1cos\left(x\right)$. (b) The corresponding order parameter $R\left(x\right)$. (c) The corresponding order parameter $\Theta \left(x\right)$. Panel (a) shows the presence of a nearly coherent region near $x=0$ with oscillators that oscillate $\pi$ out of phase with the coherent cluster. The calculation is done with $\beta =0.05$ and $N=512$.

Figure 18

(a) The quantities $a$ and $\Omega$, and (b) the coherent fraction $e$, all as functions of ${\omega }_0$ for the chimera state shown in Fig. 17. (c) $\Omega +\omega \left(x\right)$ and $R\left(x\right)$ as functions of $x$ when ${\omega }_0=0.0463$.

Figure 19

2-cluster chimera states for $G\left(x\right)=cos\left(x\right)$ and (a) $\omega \left(x\right)=0.1cos\left(2x\right)$ and (b) $\omega \left(x\right)=0.1cos\left(3x\right)$. In both cases $\beta =0.05$ and $N=512$.

Figure 20

The dependence of (a) $\Omega$ (solid line) and $b$ (dashed line), and (b) the coherent fraction $e$ on ${\omega }_0$ when $l=3$ and $\beta =0.05$.

Figure 21

The three possible locations of the 2-cluster chimera state when $\omega \left(x\right)=0.1cos\left(3x\right)$. The simulation is done with $\beta =0.05$ and $N=512$.

Figure 22

(a) A snapshot of the phase pattern in a traveling coherent state when $\omega$ is constant. (b) The position $x_0$ of this state as a function of time. The simulation is done for $G\left(x\right)=cos\left(x\right)+cos\left(2x\right)$ with $\beta =0.75$ and $N=512$.

Figure 23

Hidden line plot of the phase distribution $\theta (x,t)$ when (a) ${\omega }_0=0$. (b) ${\omega }_0=0.04$. In both cases, $\kappa =2,\beta =0.75$, and $N=512$.

Figure 24

Hidden line plot of the phase distribution $\theta (x,t)$ when (a) ${\omega }_0=0.08$. (b) ${\omega }_0=0.12$. In both cases, $\kappa =2,\beta =0.75$, and $N=512$.

Figure 25

The position $x_0$ of the coherent state in Figs. 23 and 24 as a function of time for (a) ${\omega }_0=0$, (b) ${\omega }_0=0.04$, (c) ${\omega }_0=0.08$, and (d) ${\omega }_0=0.12$. In all cases $\kappa =2,\beta =0.75$, and $N=512$.

Figure 26

The profiles of $\Omega +\omega \left(x\right)$ and $R=|acos(x)+ccos(2x\left)\right|$ for ${\omega }_0=0.11,\kappa =2$, and $\beta =0.75$.

Figure 27

Hidden line plot of the phase distribution $\theta (x,t)$ when ${\omega }_0=0.12$ and $\beta =0.76$. (a) Unpinned state obtained from a traveling coherent state with ${\omega }_0=0$ and $\beta =0.76$ on gradually increasing ${\omega }_0$ to 0.12. (b) Pinned state obtained from a traveling coherent with ${\omega }_0=0$ and $\beta =0.75$ on gradually increasing ${\omega }_0$ to 0.12, and then increasing $\beta$ to 0.76. In both cases $\kappa =2$ and $N=512$.

Figure 28

(a) A snapshot of the phase distribution $\theta (x,t)$ of a coherent solution when ${\omega }_0=0.028$. (b) The position $x_0$ of the coherent solution as a function of time. In both cases $l=1,\beta =0.05$, and $N=512$.

Figure 29

(a) The mean angular velocity $\Omega$ and (b) the mean drift speed $v$, both as functions of ${\omega }_0$ when $l=1,\beta =0.7$.

Figure 30

(a) A snapshot of the phase distribution $\theta (x,t)$ for a traveling chimera state in a spatially homogeneous system. (b) The position $x_0$ of the coherent cluster as a function of time. The simulation is done for $G\left(x\right)\equiv cos\left(3x\right)+cos\left(4x\right)$ with $\beta =0.03$ and $N=512$.

Figure 31

Dependence of the pinning threshold ${\omega }_0$ on $\kappa$ when $\beta =0.03$.

Figure 32

The position $x_0$ of the pinned coherent cluster in a traveling chimera state as a function of time when (a) ${\omega }_0=0.04$, (c) ${\omega }_0=0.08$, (e) ${\omega }_0=0.12$. The average rotation frequency ${\overline{\theta }}_t$ for (b) ${\omega }_0=0.04$, (d) ${\omega }_0=0.08$, (f) ${\omega }_0=0.12$. In all cases $\beta =0.03$ and $\kappa =10,N=512$.

Figure 33

The position $x_0$ of the pinned coherent cluster in a traveling chimera state as a function of time when (a) ${\omega }_0=0.005$, (c) ${\omega }_0=0.01$. The average rotation frequency ${\overline{\theta }}_t$ for (b) ${\omega }_0=0.005$, (d) ${\omega }_0=0.01$. In all cases $\beta =0.03,l=1$, and $N=512$.

Figure 34

The position $x_0$ of the coherent cluster in a traveling chimera state as a function of time when (a) ${\omega }_0=0.001$, (b) ${\omega }_0=0.002$. In all cases $\beta =0.03,l=1$, and $N=512$.

Figure 35

(a), (c) The two possible phase distributions $\theta (x,t)$ of pinned traveling chimera states when ${\omega }_0=0.01,\beta =0.03$, and $l=2$. (b), (d) The corresponding average rotation frequencies ${\overline{\theta }}_t$. In both cases $N=512$.