Control of delay-induced oscillation death by coupling phase in coupled oscillators

A coupling phase is deemed to be crucial in stabilizing behavior in nonlinear systems. In this paper, we study how the coupling phase influences the delay-induced oscillation death (OD) in coupled oscillators. The OD boundaries are identified analytically even in the presence of the coupling phase. We find that OD only occurs for a coupling phase belonging to a certain interval. The optimal coupling phase, under which the largest OD island forms, is characterized well by a power law scaling with respect to the frequency. The coupling phase turns out to be a key parameter that determines a delay-induced OD. Furthermore, the controlling role of the coupling phase generally is proved to hold fairly for networked delay-coupled oscillators.

Figure 1

The OD islands of the coupled system (1) in the parameter space [$\tau ,{\log }_{10}\left(K\right)$] for $\theta =-0.1\pi$ (the left one) and $\theta =0.1\pi$ (the right one). $\omega =10$ is fixed. The OD regions are enclosed by the two critical curves ${\tau }_1(K,\theta )$ and ${\tau }_2(K,\theta )$. The open circles represent the numerical results, which well confirm the theoretical prediction.

Figure 2

The OD islands of the coupled system (1) in the parameter space [$\tau ,{\log }_{10}\left(K\right)$] for $\theta =-0.4\pi ,-0.05\pi ,-0.02\pi ,0,0.05\pi$, and $0.1\pi$, respectively, $\omega =10$. All the OD islands, which have been proved by the direct numerical experiments, come from Eq. (3). The OD boundaries are indicated by the different colors and styles of lines.

Figure 3

(a)–(d) The OD island area $S\left(\theta \right)$ of coupled system (1) vs the coupling phase $\theta$ for $\omega =10,5,4$, and 2, respectively. The maximum point $S\left({\theta }_M\right)$ at the optimal phase ${\theta }_M$ is indicated by blue triangles, and the two critical points with ${\theta }_c^{-}$ and ${\theta }_c^{+}$ are indicated by red stars and green circles. OD only is possible for ${\theta }_c^{-}<\theta <{\theta }_c^{+}$.

Figure 4

(a) The two critical coupling phases ${\theta }_c^{-}$ (open squares) and ${\theta }_c^{+}$ (solid squares) and the optimal coupling phase ${\theta }_M$ (open circles) vs $w$ from the numerical calculation results of Fig. 3. The red (outward) line comes from Eq. (4), which well predicts ${\theta }_c^{-}$ and ${\theta }_c^{+}$. The minimum point is highlighted by a green triangle with $(w_c,{\theta }_c)=(2.27,-0.26\pi )$. The blue (inside) line is the fit by ${\theta }_M=\alpha w^{\beta }$ with $\alpha \approx -2.19$ and $\beta \approx -1.16$. The three horizontal dashed lines from top to bottom are $\theta =\frac{\pi }{2},0$, and $-\frac{\pi }{2}$, respectively. (b) The log-log plot of $-{\theta }_M$ vs $w$, which shows a perfect power law scaling. The blue line is a linear fit.

Figure 5

The OD islands of the coupled system Eq. (6) with a ring network for different coupled oscillators $N$. The OD islands remain unchanged for all the ring networks with even nodes $N$. For the odd values $N$, the OD regions are enclosed by four critical curves when $N\le 9$, and two curves otherwise, and as $N\to \infty$, the OD boundaries approach that for the even $N$. $\theta =-0.1\pi$ and $\omega =10$ are fixed. The four critical curves are indicated by the different colors and styles of lines.

Figure 6

(a)–(d) The OD island area $S\left(\theta \right)$ vs the coupling phase $\theta$ for a ring of delay-coupled Landau-Stuart oscillators with $N=3,5,7$, and 9, respectively, $\omega =10$. The structure is quite similar to that in Fig. 3.

Figure 7

(a) The OD islands of the coupled system Eq. (6) with the irregular network for different coupling phases $\theta$. The OD boundaries are indicated by the different colors and styles of lines. A schematic of the irregular network structure is depicted in the upper-right corner. (b) The OD island area $S\left(\theta \right)$ vs $\theta$ for the irregular network used in (a). Also, the structure is quite similar to that in Fig. 3. $\omega =10$ is fixed.