Amplitude death in nonlinear oscillators with mixed time-delayed coupling

Amplitude death (AD) is an emergent phenomenon whereby two or more autonomously oscillating systems completely lose their oscillations due to coupling. In this work, we study AD in nonlinear oscillators with mixed time-delayed coupling, which is a combination of instantaneous and time-delayed couplings. We find that the mixed time-delayed coupling favors the onset of AD for a larger set of parameters than in the limiting cases of purely instantaneous or completely time-delayed coupling. Coupled identical oscillators experience AD under instantaneous coupling mixed with a small proportion of time-delayed coupling. Our work gives a deeper understanding of delay-induced AD in coupled nonlinear oscillators.

Figure 1

The stability regions of AD in two coupled nonidentical oscillators described by Eq. (1) for (a) the purely instantaneous coupling with $\alpha =0$, (b) the fully time-delayed coupling with $\alpha =1.0$, and the mixed time-delayed coupling with (c) $\alpha =0.99$ and (d) $\alpha =0.98$, respectively. $w_1=10+\Delta /2$ and $w_2=10-\Delta /2$. $\tau =0.2$ is fixed.

Figure 2

The AD islands of the two coupled identical oscillators of Eq. (1) on the $(\tau ,K)$ space for different values of the mixing parameter $\alpha$. $\alpha =1$, $0.99$, $0.98$, $0.9$, $0.55$, and $0.45$ for (a)–(f), respectively. The AD islands are surrounded by the critical curves ${\tau }_a$, ${\tau }_b$, ${\tau }_c$, and ${\tau }_d$ in Eq. (7), which are indicated by the different colors and styles of lines. The black star represents the minimal point of intersection of ${\tau }_a$ and ${\tau }_b$: $({\tau }_a={\tau }_b=\pi /w,K=1/2\alpha )$; and the red triangle is the maximal point of intersection of ${\tau }_c$ and ${\tau }_d$: $[{\tau }_c={\tau }_d=\pi /w,K=1/2(1-\alpha )]$. The open circles are the numerical results, which are in good agreement with the theoretical predictions. $w_1=w_2=w=10$ is fixed.

Figure 3

The AD island as a function of the intrinsic frequency $w$ for (a) $\alpha =0.9$ and (b) $\alpha =0.45$. The size of the AD island monotonically decreases as $w$ decreases. The AD island disappears if $w$ is below a certain critical value $w_{\min }$.

Figure 4

The critical frequency $w_{\min }\left(\alpha \right)$ vs the mixing parameter $\alpha$. $w_{\min }\left(\alpha \right)$ decreases as the mixing parameter $\alpha$ decreases from 1 to $0.5$, and retains at a constant value $\pi /2$ for $0<\alpha \le 0.5$.

Figure 5

The study of the coupled systems Eq. (11) for a ring topology with $N=11$. $w=10$ is fixed. The maximal point of intersection of ${\tau }_c$ and ${\tau }_d$ is given by $\{\pi /w,1/\left[1+(2\alpha -1){\rho }_N\right]\}$ marked by the red triangles. ${\tau }_a$, ${\tau }_b$, ${\tau }_c$, and ${\tau }_d$ are given in Eq. (14), which are indicated by differen colors and styles.

Figure 6

The stability regions of AD in the two coupled nonidentical chaotic Rössler oscillators of Eq. (15) for (a) the fully instantaneous coupling with $\alpha =0$, (b) the fully time-delayed coupling with $\alpha =1.0$, and the mixed coupling with (c) $\alpha =0.995$ and (d) $\alpha =0.99$, respectively. $w_1=1+\Delta /2$ and $w_2=1-\Delta /2$. $\tau =2.18$ is fixed.

Figure 7

The AD regions of the two coupled identical chaotic Rössler oscillators of Eq. (15) in the parameter space of $(\tau ,K)$ for (a) $\alpha =1$, (b) $\alpha =0.9$, (c) $\alpha =0.7$, (d) $\alpha =0.5$, (e) $\alpha =0.3$, and (f) $\alpha =0.1$, respectively. $w_1=w_2=1$ is fixed.