Pattern formation in stiff oscillatory media with nonlocal coupling: A numerical study of the hydrogen oxidation reaction on Pt electrodes in the presence of poisons

The impact of the strength of negative (desynchronizing) global coupling (NGC) on the spatiotemporal dynamics of an electrochemical relaxation oscillator is studied numerically with a prototypical model, the electro-oxidation of hydrogen in the presence of poisons. The results are compared with recent experiments. The NGC has a destabilizing effect on the homogeneous oscillations. Both, in theory and in experiments, the basic patterns found with increasing global coupling strength are modulated oscillations, target patterns (including an asymmetric variant), and modulated pulses, the average spatial inhomogeneity during an oscillation increasing with the intensity of the NGC. It is suggested that this scenario is typical for strong relaxation oscillations, and a comparison with an electrochemical oscillator exhibiting harmonic oscillations points to the fact that the critical coupling strength, upon which the complete synchronization is destroyed, is larger for relaxation oscillations than for harmonic oscillations. In addition, the numerical simulations predicted two- and three-phase cluster patterns at high coupling strength. Also in experiments cluster patterns were observed, however only in parameter regions of the local dynamics which were different from the one investigated in this study.

Figure 1

(a) One parameter continuation of the homogeneous steady states and homogeneous periodic solutions for the dimensionless model, Eqs. (1)–(4). Solid line and dashed lines, stable and unstable stationary steady state, respectively; dots, maximal amplitude of homogeneous periodic solutions (stable or unstable depending on $\rho$, s.b.). For parameters see Table , $\sigma ∕\beta \approx 0.296$. (b) Location of Hopf bifurcation points in the $(\sigma ∕\beta$-$U)$-parameter plane for the same parameter set.

Figure 2

(Color online) Spatiotemporal patterns in the $U$-$\rho$-parameter plane for $p_{\mathrm{Cu}}=p_{\mathrm{Cu}}^0\times {10}^{-4}$ (for other parameters see Table ). The homogeneous steady state is observed on the left and right borders of the figure (dashed lines) and homogeneous oscillations are found for $\rho ⩾-0.6$. Bistability is indicated by stacking the respective symbols. TP, target pattern; MO, modulated oscillations; CP, cluster pattern; A-TP, asymmetric target pattern.

Figure 3

(Color online) Pattern formation under varying global coupling $\rho$ and adjusting $\sigma$ at the same time to investigate the system behavior for fixed physical conductivity $\left[\sigma \right(1+\rho )=\mathrm{const}]$. The homogeneous steady state is globally stable outside the region marked with the dashed line (cf. Fig. 1). $p_{\mathrm{Cu}}=p_{\mathrm{Cu}}^0\times {10}^{-2}$, for other parameters see Table . MO, modulated oscillations; TP, target pattern.

Figure 4

(Color online) Approximate regions where various patterns induced by the negative global coupling are observed for $p_{\mathrm{Cu}}=p_{\mathrm{Cu}}^0$ (for other parameters see Table ). MO, modulated oscillations; TP, target pattern; A-TP, asymmetric target pattern.

Figure 5

(Color online) Rotating antiphase oscillation. Upper plate, ${\varphi }_{\mathrm{DL}}(x,t)$; lower plate, $i\left(t\right)$. Parameters: $U=5$, $\sigma =2.51069$, $p_{\mathrm{Cu}}=p_{\mathrm{Cu}}^0$, $\rho =-0.7$. The map used here and in the following is also displayed with minimum and maximum values of ${\varphi }_{\mathrm{DL}}$ (though ${\varphi }_{\mathrm{DL},min}$ and ${\varphi }_{\mathrm{DL},max}$ differ in the individual figures). For other parameters see Table .

Figure 6

(Color online) Typical pulse forms observed in the (a) oscillatory and (b) nonoscillatory regime. A second independent frequency is visible in the total current density. Upper plates, ${\varphi }_{\mathrm{DL}}(x,t)$; lower plates, $i\left(t\right)$. (a) Chaotically modulated pulse. Parameters: $U=6$, $p_{\mathrm{Cu}}=p_{\mathrm{Cu}}^0$, $\rho =-0.9$. (b) Periodically modulated pulse. Parameters: $U=2$, $\sigma =1.125534$, $p_{\mathrm{Cu}}=p_{\mathrm{Cu}}^0\times {10}^{-2}$, $\rho =-0.9$. For other parameters see Table .

Figure 7

(Color online) Modulated oscillations. (a) ${\varphi }_{\mathrm{DL}}(x,t)$; (b) Inhomogeneous part of the double layer potential, ${\varphi }_{\mathrm{DL}}-⟨{\varphi }_{\mathrm{DL}}⟩$; (c) Total current density as a function of time displaying the relaxationlike behavior of the underlying oscillations. Parameters: $U=14$, $p_{\mathrm{Cu}}=p_{\mathrm{Cu}}^0\times {10}^{-4}$, $\rho =-0.7$, for other parameters see Table .

Figure 8

(Color online) Two examples of type-1 target patterns (TPs). Upper plates, ${\varphi }_{\mathrm{DL}}(x,t)$; middle plates, ${\varphi }_{\mathrm{DL}}-⟨{\varphi }_{\mathrm{DL}}⟩$; lower plates, $i\left(t\right)$. (a) Typical TP observed for most $U$, $\rho$, and $p_{\mathrm{Cu}}$ values. Parameters: $U=12$, $p_{\mathrm{Cu}}=p_{\mathrm{Cu}}^0$, $\rho =-0.9$. (b) TP as observed for larger aspect ratios (smaller $\beta$). Parameters: $U=11$, $\beta =0.1$, $\sigma =0.03$, $p_{\mathrm{Cu}}=p_{\mathrm{Cu}}^0\times {10}^{-4}$, $\rho =-0.5$. For other parameters see Table .

Figure 9

(Color online) (a) Second type of target pattern found during the HOR. Parameters: $U=15$, $p_{\mathrm{Cu}}=p_{\mathrm{Cu}}^0\times {10}^{-2}$, $\rho =-0.9$. (b) Type-3 target pattern observed at the lower $U$ boundary. Parameters: $U=3$, $p_{\mathrm{Cu}}=p_{\mathrm{Cu}}^0\times {10}^{-4}$, $\rho =-0.7$. For other parameters see Table . Upper plates, ${\varphi }_{\mathrm{DL}}(x,t)$; middle plates, ${\varphi }_{\mathrm{DL}}-⟨{\varphi }_{\mathrm{DL}}⟩$; lower plates, $i\left(t\right)$.

Figure 10

(Color online) (a) Traveling two-phase cluster pattern. (b) Traveling three-phase cluster pattern observed for the same parameter values. Upper plates, ${\varphi }_{\mathrm{DL}}(x,t)$; middle plates, ${\varphi }_{\mathrm{DL}}-⟨{\varphi }_{\mathrm{DL}}⟩$; lower plates, $i\left(t\right)$. (c) ${\varphi }_{\mathrm{DL}}(x,t)$ at $t=21000$ and (d) at $t=15200$ for the cluster patterns presented in (a) and (b), respectively. Parameters: $U=5$, $p_{\mathrm{Cu}}=p_{\mathrm{Cu}}^0\times {10}^{-4}$, $\rho =-0.9$, for other parameters see Table .

Figure 11

Time series ${\varphi }_{\mathrm{DL}}(x_{1∕2},t)$ at two different points $x_{1∕2}$ in space representing the two-phase domains for the cluster pattern presented in Fig. 10a.

Figure 12

(Color online) Experimentally obtained bifurcation diagram. Experimental details can be found in [29]. RHE, reversible hydrogen electrode [31].