Emergence of amplitude and oscillation death in identical coupled oscillators

We deduce rigorous conditions for the onset of amplitude death (AD) and oscillation death (OD) in a system of identical coupled paradigmatic Stuart-Landau oscillators. A nonscalar coupling and high frequency are beneficial for the onset of AD. In strong contrast, scalar diffusive coupling and low intrinsic frequency are in favor of the emergence of OD. Our finding contributes to clearly distinguish intrinsic geneses for AD and OD, and further substantially corroborates that AD and OD are indeed two dynamically distinct oscillation quenching phenomena due to distinctly different mechanisms.

Figure 1

Bifurcation diagrams of steady states of the coupled system (1) with instantaneous coupling [left column (a)–(d) with $\tau =0$] and time-delayed coupling [right column (e)–(h) with $\tau =0.16$] for $\alpha =0,0.0005,0.001$, and $0.002$, respectively. $w=10$. Thin dashed lines correspond to unstable steady states, whereas solid black (dark gray) and solid red (light gray) lines to stable HSS (AD) and stable IHSS (OD), respectively.

Figure 2

AD and OD regimes of the coupled system (1) in the parameter space of $\left(\tau ,K\right)$ for different coupling rates $\alpha$: (a) $\alpha =0$, (b) $\alpha =0.0005$, (c) $\alpha =0.001$, (d) $\alpha =0.002$. $w=10$. The black (dark gray) and red (light gray) regions represent the stable HSS (AD) and stable IHSS (OD), respectively. The dashed blue lines indicate the limit of $K$ for the emergence of IHSS.

Figure 3

AD and OD regimes of a system of coupled identical oscillators (1) with $\alpha =0.001$ on the $\left(\tau ,K\right)$ plane for different values of the frequency $w$: (a) $w=9$, (b) $w=8$, (c) $w=7$, and (d) $w=6$. The black (dark gray) and red (light gray) regions represent the stable HSS (AD) and stable IHSS (OD), respectively.

Figure 4

Minimum value $w_{\text{min}}$ (black triangles) of $w$ for AD and maximum value ${\alpha }_{\text{max}}$ (red squares) of $\alpha$ for OD. Both AD and OD occur in the same system of coupled identical oscillators when $\alpha$ and $w$ are located in the upper-left corner.

Figure 5

AD and OD of the dynamically coupled system (2) in the parameter space of $\left(w,K\right)$ for different coupling rates $\alpha$: (a) $\alpha =0$, (b) $\alpha =0.01$, (c) $\alpha =0.02$, (d) $\alpha =0.03$. The black (dark gray) and red (light gray) regions represent the stable HSS (AD) and stable IHSS (OD), respectively.