Quenching and revival of oscillations induced by coupling through adaptive variables

An adaptive coupling based on a low-pass filter (LPF) is proposed to manipulate dynamic activity of diffusively coupled dynamical systems. A theoretical analysis shows that tracking either the external or internal signal in the coupling via a LPF gives rise to distinctly different ways of regulating the rhythmicity of the coupled systems. When the external signals of the coupling are attenuated by a LPF, the macroscopic oscillations of the coupled system are quenched due to the emergence of amplitude or oscillation death. If the internal signals of the coupling are further filtered by a LPF, amplitude and oscillation deaths are effectively revoked to restore dynamic behaviors. The applicability of this approach is demonstrated in laboratory experiments of coupled oscillatory electrochemical reactions by inducing coupling through LPFs. Our study provides additional insight into (ar)rhythmogenesis in diffusively coupled systems.

Figure 1

Quenching and revival of oscillations in two coupled Stuart-Landau oscillators (1). (a) The AD regions in the parameter space of $(\mathrm{\Delta },K)$ for $\alpha =0$ (bounded by the dashed lines) and $\alpha =0.165$ (bounded by the solid lines). $w_1=5-\mathrm{\Delta }/2$, $w_2=5+\mathrm{\Delta }/2$, and $\beta =0$. (b) The AD interval vs $\alpha$ for $w_1=w_2=5$ and $\beta =0$. AD emerges for $\alpha >{\alpha }_c=0.16$ implying that stable limit-cycle oscillations are quenched in identical oscillators. (c) The AD interval vs $\beta$ for $w_1=w_2=5$ and $\alpha =10$. AD is completely revoked as $\beta >{\beta }_c=0.5$. (d) The plot of time series of real($Z_1$). Limit-cycle oscillation is lost for $\alpha =10$ when $\beta =0$, which is regained as $\beta$ switched from 0 to 1 at $t=15$. $K=3$ and $w_1=w_2=5$.

Figure 2

Quenching oscillations in the coupled system (4) by implementing the LPF associated with only the external term of the coupling. $\beta =0$ and $w=5$ are fixed. (a)–(d) The bifurcation diagrams of the steady-state solutions for $\alpha =0$, 0.15, 0.27, and 0.35, respectively. Solid black (dark gray) and solid red (light gray) lines denote the stable HSS (AD) and the stable IHSS (OD). (b) The stable interval of AD (black region) and OD (red region) vs $\alpha$ for $\beta =0$. The external LPF facilitates the onset of OD at small values of coupling strength, and further even induces AD as $\alpha >{\alpha }_c=0.257$.

Figure 3

Reviving oscillations from AD and OD in the coupled system (4) with $\alpha =0.5$ by implementing LPF in the internal term of the coupling. $w=5$ is fixed. (a)–(d) The bifurcation diagrams of the steady-state solutions for $\beta =0$, 0.2, 0.4, and 0.54, respectively. (b) The stable interval of AD (black region) and OD (red region) vs $\beta$ for $\alpha =0.5$. The internal LPF destabilizes AD from the lower coupling strength, which disappears at ${\beta }_{c,1}=0.3$. For $\beta >{\beta }_{c,1}$, OD begins to be destabilized and is completely dismissed as $\beta >{\beta }_{c,2}=0.55$.

Figure 4

Quenching and revival of oscillations in two coupled chaotic Rössler oscillators (7). (a) Bifurcation diagram obtained by plotting the local maxima of $X=\frac{x_1+x_2}{2}$ as a function of $\alpha$ for $\beta =0$ and $K=3$. The coupled system (7) undergoes a reverse period-doubling bifurcation from chaos to one cycle, and then experiences AD for $\alpha >{\alpha }_c=0.24$. (b) Bifurcation diagram as a function of $\beta$ for $\alpha =0.4$ and $K=3$. AD is destabilized at $\beta ={\beta }_c=0.444$. For $\beta >{\beta }_c=0.444$, the coupled system (7) experiences a period-doubling bifurcation from period-1 oscillation to chaos. (c) The plot of time series of $x_1$. Chaotic oscillation is regained from AD as $\beta$ is switched from 0 to 0.5 at $t=200$. $K=3$ and $\alpha =0.4$ are fixed. (d) $K$ vs $\alpha$ for the onset of AD for $\beta =0$, 0.3, 0.5, and 0.7.

Figure 5

Inducing AD and reviving the electrochemical oscillations with coupling through LPFs. (a) Experimental setup. $R_{\text{ind}}$: Individual resistances attached to the Ni wires. $V_k\left(t\right)$ and $i_k\left(t\right)$ are the applied circuit potentials and the measured currents for electrodes $k=1,2$, respectively. (b) Experimental implementation of feedback scheme. The measured currents $\left[i_k\left(t\right)\right]$ are filtered with coefficients $\alpha$ and $\beta$ in Eqs. (2) and (3) to obtain the corresponding $u_k\left(t\right)$ and $v_k\left(t\right)$ variables. The potentiostat (GILL IK64) applies feedback with set potential $V_0$ and gain $K$. (c) Synchronized oscillations, AD, and regained oscillations with coupling ($K=-0.3\mathrm{V}/\mathrm{mA}$) without LPF ($t<20\mathrm{s},\alpha =0,\beta =0$), LPF in external signal ($20\mathrm{s}\le t<110\mathrm{s},\alpha =10\mathrm{s},\beta =0$), and LPFs in both external and internal signals ($t\ge 110\mathrm{s},\alpha =10\mathrm{s},\beta =0.5\mathrm{s}$), respectively. $V_0=1100\mathrm{mV}$, $R_{\mathrm{ind}}=5$ kOhm.