Revoking amplitude and oscillation deaths by low-pass filter in coupled oscillators

When in an ensemble of oscillatory units the interaction occurs through a diffusion-like manner, the intrinsic oscillations can be quenched through two structurally different scenarios: amplitude death (AD) and oscillation death (OD). Unveiling the underlying principles of stable rhythmic activity against AD and OD is a challenging issue of substantial practical significance. Here, by developing a low-pass filter (LPF) to track the output signals of the local system in the coupling, we show that it can revoke both AD and OD, and even the AD to OD transition, thereby giving rise to oscillations in coupled nonlinear oscillators under diverse death scenarios. The effectiveness of the local LPF is proven to be valid in an arbitrary network of coupled oscillators with distributed propagation delays. The constructive role of the local LPF in revoking deaths provides a potential dynamic mechanism of sustaining a reliable rhythmicity in real-world systems.

Figure 1

Revoking AD (stable HSS) in the coupled system (1) with $\tau =0,w_1=10-\mathrm{\Delta }/2$, and $w_2=10+\mathrm{\Delta }/2$. (a) The AD interval vs $\alpha$ with $\mathrm{\Delta }=10$. (b) AD regions in the parameter space of $(K,\mathrm{\Delta })$ for $\alpha =0$, 0.05 (red region), 0.1 (green region), and 0.15 (blue region).

Figure 2

Revoking AD (stable HSS) in the coupled system (1) with $\tau >0$ and $w_1=w_2=w=10$. (a) AD islands in the parameter space of $(\tau ,K)$ for $\alpha =0$, 0.01, 0.02, and 0.03. (b) The AD island ratio $R=S\left(\alpha \right)/S(\alpha =0)$ vs $\alpha$.

Figure 3

Revoking AD (stable HSS) in networks of coupled Stuart-Landau oscillators with the propagation delays uniformly distributed over $\tau \pm 0.02,w_j=w=30$, and ${\rho }_N=-1$. (a)–(d) The stability regions of AD (stable HSS) for $\alpha =0$, 0.007, 0.008, and 0.01, respectively.

Figure 4

Revoking AD (stable HSS) and OD (stable IHSS) in the coupled system (7) with $w_1=2$ and $w_2=8$. (a)–(d) The bifurcation diagrams of steady-state solutions for $\alpha =0$, 0.03, 0.051, and 0.07, respectively. Solid black (dark gray) and solid red (light gray) lines mark the stable HSS (AD) and the stable IHSS (OD). Thin dashed lines denote the unstable steady states. (e) The stable interval of coupling for AD (black region) and OD (red region) vs $\alpha$.

Figure 5

Stability diagrams of AD and OD of the coupled system (7) in the parameter space of $(K,\delta )$ for $\alpha =0$ (a), 0.02 (b), 0.065 (c), and 0.082 (d). The black (dark gray) and red (light gray) regions denote the stable HSS (AD) and IHSS (OD), respectively. The dashed blue line represents the critical coupling strength $K_c$ for the birth of IHSS. $w_1=2$ and $w_2=\delta w_1$.