Figure 1

(Color online) Phase of the oscillators along the ring for different time delays. (a) ${\tau }^{\prime }=0.3$, (b) 0.9, (c) 1.8, and (d) 3.5. $H\left(\theta \right)=\text{sin}\left(\theta \right)$. $\omega =\pi ∕2$, $N=1600$, $L=1$, ${\tau }^{\prime }\equiv L\tau$, $K=1.0$, and $\overline{n}=40$ $\left(p=0.025\right)$. (a) ${\tau }^{\prime }=0.3$, a near-synchronous state (winding number $m=0$, $R_0\approx 0.995$, $R_{m\ne 0}<0.01$), ${\Omega }_{\text{av}}\approx 1.137$, and ${\sigma }_{\Omega }<{10}^{-5}$. (b) ${\tau }^{\prime }=0.9$, a wave state $m=1$ ($R_1\approx 0.979$, $R_{m\ne 1}<0.02$), ${\Omega }_{\text{av}}\approx 1.635$, and ${\sigma }_{\Omega }<{10}^{-5}$. (c) ${\tau }^{\prime }=1.8$, a wave state $m=2$ ($R_2\approx 0.968$, $R_{m\ne 2}<0.02$), ${\Omega }_{\text{av}}\approx 1.696$, and ${\sigma }_{\Omega }$ oscillates irregularly in $[0.00253,0.00713]$. (d) ${\tau }^{\prime }=3.5$, a wave state $m=3$ ($R_3\approx 0.89$, $R_{m\ne 3}<0.03$), ${\Omega }_{\text{av}}$ oscillates irregularly in [1.295, 1.305], and ${\sigma }_{\Omega }$ oscillates irregularly in [0.057, 0.078].Reuse & Permissions

Figure 2

(Color online) Phase of the oscillators along the ring for different average number of neighbors (a) $\overline{n}=15$, (b) 20, (c) 40, and (d) 80. $H\left(\theta \right)=H_{\text{crook}}\left(\theta \right)$ [29]. $\omega =\pi ∕2$, $N=3200$, $L=1$, ${\tau }^{\prime }\equiv L\tau =4.64$, and $K=1.0$. (a) $\overline{n}=15$ $\left(p=0.0046875\right)$, $R_{-5}$ and $R_5$ oscillate irregularly in [0.005, 0.42] and [0.005, 0.395], respectively. The state changes irregularly between the high $R_{-5}$ state and the high $R_5$ state. $R_{-5}$ and $R_5$ move in an anticorrelated manner, $R_{m\ne \pm 5}<0.045$. ${\Omega }_{\text{av}}$ oscillates irregularly in [3.842, 3.908] and ${\sigma }_{\Omega }$ oscillates irregularly in [0.465, 0.602]. (b) $\overline{n}=20$ $\left(p=0.00625\right)$, a wave state with winding number $m=-5$. $R_{-5}$ oscillates irregularly in [0.682, 0.725] and $R_{m\ne -5}<0.025$. ${\Omega }_{\text{av}}$ oscillates irregularly in [3.785, 3.82] and ${\sigma }_{\Omega }$ oscillates irregularly in [0.237, 0.303]. (c) $\overline{n}=40$ $\left(p=0.0125\right)$, a wave state with $m=5$, $R_5\approx 0.94$, $R_{m\ne 5}<0.012$. ${\Omega }_{\text{av}}$ oscillates irregularly in [3.643, 3.648] and ${\sigma }_{\Omega }$ oscillates irregularly in [0.02, 0.09]. (d) $\overline{n}=80$ $\left(p=0.025\right)$, a wave state with $m=5$, $R_5\approx 0.993$, $R_{m\ne 5}<0.006$, ${\Omega }_{\text{av}}\approx 3.558$, and ${\sigma }_{\Omega }<{10}^{-5}$.Reuse & Permissions

Figure 3

(Color online) Typical trend of order parameter $R_m$ as a function of the average number $\overline{n}$ of neighbors. $H\left(\theta \right)=\text{sin}\left(\theta \right)$ and system parameters are the same as Fig. 1 except $N$, $\overline{n}$, and ${\tau }^{\prime }$. (a) $R_m$ for the states with ${\tau }^{\prime }=3.0$ and ${\tau }^{\prime }=3.5$. $N=1600$. We compute both $R_{m=3}$ and $R_{m=-3}$ for each state obtained from simulations, and denote the larger one. The symbols represent the time average of $R_m$ and the error bars denote the standard deviation of $R_m$ time series. The curves are guides for the eyes. (b) $R_m$ for the states with $N=1600$, 3200, and 6400 for ${\tau }^{\prime }=3.5$. Error bars with similar sizes for each data point as in (a) are omitted for the simplicity of the figure.Reuse & Permissions

Figure 4

(Color online) States and their stability as a function of scaled relative time delay ${\tau }^{\prime }=L\tau$ for the case with $H\left(\theta \right)=\text{sin}\left(\theta \right)$, $\omega =\pi ∕2$, and $K=1$. Thick parts of the curves in (a), (b), and (c) correspond to the stable states identified by linear stability analysis of Eq. (8). (a) Synchronization frequency $\Omega$. $m$ is the winding number of a state. Curves are obtained from Eq. (10). Symbols and error bars denote the time average of ${\Omega }_{\text{av}}\left(t\right)$ and ${\sigma }_{\Omega }\left(t\right)$ defined by Eq. (4), respectively. Simulations are carried out with $N=1600$ and $\overline{n}=40$ $\left(p=0.025\right)$. Dashed vertical lines denote ${\tau }^{\prime }$ values with which $\text{Re}{\left({\lambda }_{\alpha }\right)}_{\text{max}}$ of (b) has minimum values. (b) The maximum real part of eigenvalue ${\lambda }_{\alpha }$. $\text{Re}\left({\lambda }_{\alpha }\right)$ is given by Eq. (14). $\text{Re}{\left({\lambda }_{\alpha }\right)}_{\text{max}}<0$ is the condition for the linear stability. (c) $\text{Re}\left({\lambda }_{\alpha }\right)$ for perturbations with large wave vector $\alpha \gg 1$. $\text{Re}{\left({\lambda }_{\alpha }\right)}_{\alpha \gg 1}$ is given by Eq. (15) and $\text{Re}{\left({\lambda }_{\alpha }\right)}_{a\gg 1}<0$ is a necessary condition for the stability of corresponding states.Reuse & Permissions

Figure 5

(Color online) States and their stability as a function of scaled relative time delay ${\tau }^{\prime }=L\tau$ for the case with $H\left(\theta \right)=H_{\text{crook}}\left(\theta \right)$ [29], $\omega =\pi ∕2$, and $K=1$. (a) Synchronization frequency $\Omega$. Simulations are carried out with $N=1600$ and $\overline{n}=80$ $\left(p=0.05\right)$. (b) The maximum real part of the eigenvalue ${\lambda }_{\alpha }$. (c) $\text{Re}\left({\lambda }_{\alpha }\right)$ for perturbations with large wave vector $\alpha \gg 1$. For other details, refer to the caption of Fig. 4.Reuse & Permissions