Robustness of free and pinned spiral waves against breakup by electrical forcing in excitable chemical media

We present an investigation on the breakup of free and pinned spiral waves under an applied electrical current in the Belousov-Zhabotinsky reaction. Spiral fronts propagating towards the negative electrode are decelerated. A breakup of the spiral waves occurs when some segments of the fronts are stopped by a sufficiently strong electrical current. In the absence of obstacles (i.e., free spiral waves), the critical value of the electrical current for the wave breakup increases with the excitability of the medium. For spiral waves pinned to circular obstacles, the critical electrical current increases with the obstacle diameter. Analysis of spiral dynamics shows that the enhancement of the robustness against the breakup of both free and pinned spiral waves is originated by the increment of wave speed when either the excitability is strengthened or the obstacle size is enlarged. The experimental findings are reproduced by numerical simulations using the Oregonator model. In addition, the simulations reveal that the robustness against the forced breakup increases with the activator level in both cases of free and pinned spiral waves.

Figure 1

Breakup of free spiral waves by an applied electrical current in the BZ reaction with $\left[{\mathrm{H}}_2\mathrm{S}{\mathrm{O}}_4\right]=160$ (top row) and 200 $\mathrm{m}M$ (bottom row). (a,b) spiral waves in the absence of an electrical current, while in (c,d) a wave breakup is induced by the current with $J_{\mathrm{break}}=70$ and $105\mathrm{mA}\mathrm{c}{\mathrm{m}}^{-2}$, respectively. The positive and negative electrodes are placed on the left- and the right-hand sides, respectively.

Figure 2

Influence of excitability on the propagation and the breakup of free spiral waves in the BZ reaction: (a) wavelength λ, (b) period $T$, (c) speed $s$, and (d) critical value of the current density for breakup $J_{\mathrm{break}}$.

Figure 3

Breakup of a pinned spiral wave by an applied electrical current in the BZ reaction. The obstacle diameter is 2.46 mm and positive and negative electrodes are placed on the left- and the right-hand sides, respectively. (a) The initial structure of the spiral wave is isotropic. (b) The innermost part of the wave front is stopped by the electrical current with a density of $75\mathrm{mA}\mathrm{c}{\mathrm{m}}^{-2}$. Images (c,d) illustrate two free spiral tips evolving near the obstacle in addition to one pinned tip at 7 min and 30 min, respectively, after the applied current is switched off.

Figure 4

Influence of obstacle diameter on the propagation and the breakup of pinned spiral waves in the BZ reaction: (a) wavelength λ, (b) period $T$, (c) speed $s$, and (d) critical value of the current density for breakup $J_{\mathrm{break}}$. Open and filled circles depict free and pinned spiral waves, respectively.

Figure 5

Breakup of free and pinned spiral waves in the Oregonator model. (a,b) Free spiral waves at different excitability ${\varepsilon }^{-1}=10$ and 100, respectively. (c) A spiral wave pinned to circular obstacle (diameter 5.0 s.u.) and ${\varepsilon }^{-1}=100$. (d–f) Breakups of the spiral waves in (a–c) due to the critical electric field $E_{\mathrm{break}}=0.300$, 0.850, and 1.710, respectively. The field $E$ is pointing to the right of the images.

Figure 6

Influence of excitability on (a) wavelength λ, (b) period $T$, (c) speed $s$, and (d) critical density of the field $E_{\mathrm{break}}$ of free spiral waves in the Oregonator model.

Figure 7

Influence of obstacle diameter on the propagation and breakup of pinned spiral waves in the Oregonator model: (a) wavelength λ, (b) period $T$, (c) speed $s$, and (d) critical value of electric field for breakup $E_{\mathrm{break}}$. Open and filled circles depict free and pinned spiral waves, respectively.

Figure 8

Profiles of activator $u$ and inhibitor $v$ along a horizontal line (x) perpendicular to the fronts of free spiral waves at excitability (a) ${\varepsilon }^{-1}=10$ and (b) ${\varepsilon }^{-1}=100$. (c) A spiral wave pinned to a circular obstacle (diameter 10.0 s.u.) for ${\varepsilon }^{-1}=100$. $u_{\max }, w, v_{\max }$, and $v_{\min }$ represent the maximum of $u$, the wave width, the maximum of $v$, and the minimum of $v$, respectively.

Figure 9

(a–d) Dependence of $u_{\max }, w, v_{\max }$, and $v_{\min }$ on the excitability ${\varepsilon }^{-1}$, and (a′–d′) the obstacle diameter $d$, respectively. Open and filled circles depict free and pinned spiral waves, respectively.

Figure 10

Critical values of electrical forcing for breakup vs speed of wave fronts (a) in the BZ reaction and (b) in the Oregonator model and (c) $u_{\max }$ in the Oregonator model. Open and filled circles represent free and pinned spiral waves, respectively.