Multicluster and traveling chimera states in nonlocal phase-coupled oscillators

Chimera states consisting of domains of coherently and incoherently oscillating identical oscillators with nonlocal coupling are studied. These states usually coexist with the fully synchronized state and have a small basin of attraction. We propose a nonlocal phase-coupled model in which chimera states develop from random initial conditions. Several classes of chimera states have been found: (a) stationary multicluster states with evenly distributed coherent clusters, (b) stationary multicluster states with unevenly distributed clusters, and (c) a single cluster state traveling with a constant speed across the system. Traveling coherent states are also identified. A self-consistent continuum description of these states is provided and their stability properties analyzed through a combination of linear stability analysis and numerical simulation.

Figure 1

Stable splay states with (a) $G_1^{\left(1\right)}\left(x\right)\equiv cos\left(x\right)$ and (b) $G_2^{\left(1\right)}\left(x\right)\equiv cos\left(2x\right)$. In both cases $\beta =0.1$ and $N=512$. State (a) travels with speed $c=3.124$, while (b) travels with speed $c=1.563$, both towards the right.

Figure 2

Chimera states with (a) $G_1^{\left(1\right)}\left(x\right)\equiv cos\left(x\right)$ and (b) $G_2^{\left(1\right)}\left(x\right)\equiv cos\left(2x\right)$ obtained from random initial conditions. In both cases $\beta =0.1$ and $N=512$.

Figure 3

Chimera states with (a) $G_3^{\left(1\right)}\left(x\right)\equiv cos\left(3x\right)$ and (b) $G_4^{\left(1\right)}\left(x\right)\equiv cos\left(4x\right)$ obtained from random initial conditions. (a) $\beta =0.05$ and $N=512$; (b) $\beta =0.02$ and $N=1024$.

Figure 4

(a) The phase distribution $\theta$ in a chimera state with coupling $G_1^{\left(1\right)}\left(x\right)\equiv cos\left(x\right)$. (b) The local order parameters $R$ (red dashed line) and $\Theta$ (blue dotted line) computed from Eq. (7) and the definitions $z(x,t)\equiv e^{-i\Omega t}\tilde{z}\left(x\right)$ and $\tilde{Z}\equiv R\left(x\right)e^{i\Theta \left(x\right)}=K\tilde{z}$. The simulation was done with $\beta =0.1$ and $N=512$.

Figure 5

(a) The quantities $R_0$ and $\Omega$, and (b) the coherent fraction $e$, all as functions of $\beta$ for the two-cluster chimera state with $G_1^{\left(1\right)}\left(x\right)$ coupling [Fig. 4].

Figure 6

(a) Spectrum of the linearized operator in Eq. (20) for $G_1^{\left(1\right)}$ when $\beta \approx 0.83$. (b) Dependence of the point eigenvalue ${\lambda }_p$ on $\beta$.

Figure 7

The eigenvector $(v_1,v_2)$ of the unstable point eigenvalue with coupling $G_1^{\left(1\right)}\left(x\right)$ at threshold. Left panels: $|v_1\left(x\right)|$ and $|v_2\left(x\right)|$. Right panels: phase of $v_1\left(x\right)$ and $v_2\left(x\right)$. The phase jumps by $\pm \pi$ whenever the modulus vanishes.

Figure 8

The position $x_0$ of a coherent cluster as a function of $t$ in a chimera state obtained with the coupling $G_1^{\left(1\right)}\left(x\right)$ when $\beta =0.1$ and $N=256$.

Figure 9

The dependence of the standard deviation $\sigma$ of (a) ${\dot{x}}_0$, (b) $\Omega -\overline{\Omega }$, and (c) $a-\overline{a}$ on ${log}_{2}N$ when $\beta =0.1$.

Figure 10

(a) The dependence of the speed $c$ on $\beta$ for $n=3$ (solid line), and $n=4$ (dashed line) splay states with $G_3^{\left(2\right)}$ coupling. (b) The dependence of the frequency $\Omega$ on $\beta$ is the same in both cases.

Figure 11

Chimera states with (a) $G_1^{\left(2\right)}\equiv cos\left(x\right)+cos\left(2x\right)$ and (b) $G_2^{\left(2\right)}\equiv cos\left(2x\right)+cos\left(3x\right)$ obtained from random initial conditions. In both cases $\beta =0.03$ and $N=512$.

Figure 12

Chimera states with (a) $G_3^{\left(2\right)}\equiv cos\left(3x\right)+cos\left(4x\right)$ and (b) $G_4^{\left(2\right)}\equiv cos\left(4x\right)+cos\left(5x\right)$ obtained from random initial conditions. In both cases $\beta =0.03$ and $N=512$.

Figure 13

The position $x_0$ of a coherent cluster in the three-cluster chimera state obtained with coupling $G_1^{\left(2\right)}\equiv cos\left(x\right)+cos\left(2x\right)$ as a function of $t$ for $t\ge 5000$ when $\beta =0.1$ and $N=512$, starting from random initial conditions at $t=0$.

Figure 14

(a) The phase distribution $\theta \left(x\right)$ in a three-cluster chimera state with coupling $G_1^{\left(2\right)}\equiv cos\left(x\right)+cos\left(2x\right)$ and $\beta =0.05$, $N=512$ [Fig. 11]. (b) The local order parameters $R$ (red dashed line) and $\Theta$ (blue dotted line).

Figure 15

(a) Spectrum of the linearized operator in Eq. (20) for $G_1^{\left(2\right)}$ coupling when $\beta \approx 0.83$. (b) Dependence of the point eigenvalues on $\beta$. Black line: real point eigenvalue. Red (or gray) solid line: real part of the complex point eigenvalues. Red (or gray) dashed line: imaginary part of the complex point eigenvalues.

Figure 16

Eigenvectors of (a) the real unstable mode and (b) the Hopf mode at threshold [Fig. 15]. In each plot, the left panels correspond to $|v_1\left(x\right)|$ and $|v_2\left(x\right)|$, while the right panels show the phase of $v_1\left(x\right)$ and $v_2\left(x\right)$. The phase jumps by $\pm \pi$ whenever the modulus vanishes.

Figure 17

(a) The phase distribution $\theta \left(x\right)$ in a four-cluster chimera state with coupling $G_1^{\left(2\right)}\equiv cos\left(x\right)+cos\left(2x\right)$ and $\beta =0.03$, $N=512$. (b) The local order parameters $R$ (red dashed line) and $\Theta$ (blue dotted line). (c, d) A related chimera state with order parameters $R$ and $-\Theta$.

Figure 18

(a) The phase distribution $\theta \left(x\right)$ in a four-cluster chimera state with coupling $G_2^{\left(2\right)}\equiv cos\left(2x\right)+cos\left(3x\right)$ and $\beta =0.03$, $N=512$. (b) The local order parameters $R$ (red dashed line) and $\Theta$ (blue dotted line).

Figure 19

(a) The solution of the self-consistency conditions (31, 32) as a function of $\beta$ ($b_r$: solid black line, $d_r$: solid red (or gray) line, $d_i$: dashed red (or gray) line, $\Omega$: dashed black line). (b) The predicted order parameter $R\left(x\right)$ at $\beta =0.24$ (solid line) in comparison with the line $R=\Omega$ (dashed line) indicating that four-cluster chimeras are present for $\beta \lesssim 0.24$; for $\beta \gtrsim 0.24$ only two-cluster chimeras are predicted.

Figure 20

(a) The computed phase distribution $\theta \left(x\right)$ at $\beta =0.24$ and (b) the corresponding order parameter $R\left(x\right)$ (red dashed line) and the associated phase $\Theta \left(x\right)$ (blue dotted line) for comparison with the prediction in Fig. 19.

Figure 21

(a) The phase distribution $\theta \left(x\right)$ at $\beta =0.6$ and (b) the corresponding order parameter $R\left(x\right)$ (red dashed line) and the associated phase $\Theta \left(x\right)$ (blue dotted line) for comparison with Fig. 20.

Figure 22

(a) The phase distribution $\theta \left(x\right)$ and (b) the corresponding order parameter $R\left(x\right)$ (red dashed line) and the associated phase $\Theta \left(x\right)$ (blue dotted line) for the stationary coherent state with $G_1^{\left(2\right)}$ coupling present at $\beta =0.96$.

Figure 23

(a) The solution of the self-consistency conditions (34, 35) as a function of $\beta$ [$a_r$: solid black line, $c_r$: solid red (or gray) line, $c_i$: dashed red (or gray) line, $\Omega$: dashed black line]. (b) The predicted order parameter $R\left(x\right)$ at $\beta =0.77$ (solid line) in comparison with the line $R=\Omega$ (dashed line).

Figure 24

(a) The phase distribution $\theta \left(x\right)$ at (a) $\beta =0.77$ (symmetric distribution), (b) $\beta =0.76$ (asymmetric distribution) and (c) $\beta =0.66$ (asymmetric distribution), all for $N=512$. The state in (b) oscillates in time while drifting to the left; state (c) travels to the left at constant speed. Reflected solutions travel to the right.

Figure 25

(a) The position $x_0$ of the coherent state as a function of time at $\beta =0.762$. (b) The time-averaged speed $\overline{c}$ of the coherent state as a function of $\beta$ as $\beta$ decreases. Note the abrupt decrease in speed at $\beta \approx 0.7570$ associated with the disappearance of the oscillations. (c) The time-averaged speed $\overline{c}$ of the coherent state as a function of $\beta$ as $\beta$ increases. Oscillations reappear at $\beta \approx 0.7595$ as shown in (d). The position of the coherent state is measured mod $2\pi$. All calculations are for $N=512$.

Figure 26

Hidden line plots of the phase distribution $\theta$ as a function of time when (a) $\beta =0.762$ [oscillatory drift, Fig. 25] and (b) $\beta =0.755$ (constant drift), both for $N=512$.

Figure 27

Comparison of (a) the speed $c$ and (b) the frequency $\Omega$ obtained from the solution of the nonlinear eigenvalue problem (36) (solid lines) with measurements computed with $N=512$ oscillators (open circles), both as a function of $\beta$. The inset in (a) reveals the expected square root behavior near ${\beta }_c\approx 0.7644$. In contrast, the behavior of $\Omega$ is approximately linear everywhere.

Figure 28

Comparison of the phase and order parameter profiles from direct simulation (top panels) with those obtained from the nonlinear eigenvalue problem when $\beta =0.66$. The profiles are qualitatively similar modulo translation and overall phase rotation.

Figure 29

(a) A left-traveling one-cluster chimera state. (b) A right-traveling one-cluster chimera state. The simulation is done for $\beta =0.03$ with the coupling $G_2^{\left(2\right)}\equiv cos\left(2x\right)+cos\left(3x\right)$ and $N=512$.

Figure 30

(a) A left-traveling one-cluster chimera state. (b) A right-traveling one-cluster chimera state. The simulation is done for $\beta =0.03$ with the coupling $G_3^{\left(2\right)}\equiv cos\left(3x\right)+cos\left(4x\right)$ and $N=512$.

Figure 31

(a,b) Splay states with $n=3$ and $n=4$ for comparison with (c) the traveling chimera state with $G_3^{\left(2\right)}\equiv cos\left(3x\right)+cos\left(4x\right)$ coupling. (d) Instantaneous phase velocity of the state in (c). (e) Profile of the function $F$ used to track the position of the coherent structure.

Figure 32

(a) The position of the coherent cluster as a function of time when $N=512$, $\beta =0.03$. (b) The dependence of the time-averaged speed $c$ of the cluster on the oscillator number $N$ when $\beta =0.03$. (c) The dependence of the time-averaged speed $c$ of the cluster on the parameter $\beta$ when $N=512$.

Figure 33

(a) The local order parameter $R(x,t)$ in a space-time plot for the traveling chimera in Fig. 30. (b) Zoom of (a) showing additional detail.

Figure 34

(a) Snapshot of the phase distribution in a one-cluster traveling chimera state with $G_3^{\left(2\right)}\equiv cos\left(3x\right)+cos\left(4x\right)$ coupling. (b) Local order parameters $R$ (red dashed line) and $\Theta$ (blue dotted line). The simulation is done with $\beta =0.03$ and $N=512$.