Weakly and strongly coupled Belousov-Zhabotinsky patterns

We investigate experimentally and numerically the synchronization of two-dimensional spiral wave patterns in the Belousov-Zhabotinsky reaction due to point-to-point coupling of two separate domains. Different synchronization modalities appear depending on the coupling strength and the initial patterns in each domain. The behavior as a function of the coupling strength falls into two qualitatively different regimes. The weakly coupled regime is characterized by inter-domain interactions that distorted but do not break wave fronts. Under weak coupling, spiral cores are pushed around by wave fronts in the other domain, resulting in an effective interaction between cores in opposite domains. In the case where each domain initially contains a single spiral, the cores form a bound pair and orbit each other at quantized distances. When the starting patterns consist of multiple randomly positioned spiral cores, the number of cores decreases with time until all that remains are a few cores that are synchronized with a partner in the other domain. The strongly coupled regime is characterized by interdomain interactions that break wave fronts. As a result, the wave patterns in both domains become identical.

Figure 1

The central region of a counterclockwise rotating spiral wave. The gray scale represents inhibitor concentration. The most salient features are the front of the wave (thick red line), the tail of the wave (blue thin line), the spiral tip (blue dot) where the front and tail meet, the core defined as the zone enclosed by the trajectory of the spiral tip (dashed circle), and the phase of the spiral $\psi$.

Figure 2

Schematic of the experimental setup. (a) Reactor and the pump arrangement. Arrows mark directions in which the chemical solutions are pumped. (b) Optical arrangement (see text for explanation).

Figure 3

(a) At weak forcing, wave fronts are distorted due to a local decrease of their propagation speed. The coupling strength here was $\mathcal{I}=0.4$. (b) At strong forcing, wave fronts break and give rise to new spiral cores. The coupling strength here was $\mathcal{I}=0.7$.

Figure 4

Typical example of a spiral tip trajectory in experiments in the weakly coupled regime. Solid circles in (a) mark the positions of the spiral in the $xy$ plane of domain one for different times (indicated by the symbol color - see color bar). Panels (b) and (c) show separately the $x$ and $y$ coordinates versus time for the spirals in both domains. Solid lines in (a), (b), and (c) are fits of Eq. (3) for the position of the spiral tip in the first domain.

Figure 5

Radii of orbits in domain 1 and 2 plotted against each other. Radii are normalized by a fitted value of the wavelength over four to highlight the $m$ dependence in Eq. (2); the fit yields a value of $\lambda =0.70$ mm and $\phi =1.5$ rad. The clustering of data around integer values of $R_i/\left(\frac{1}{4}\lambda \right)$ is indicative of quantization. While most orbits have equal radii [i.e., $R_1\left(m\right)=R_2\left(m\right)$; red dashed line], we observe some instances of orbits with $R_1\left(m\right)=R_2(m+1)$ (dash-dotted line). Inset: Un-normalized radii.

Figure 6

Evolution of initially random BZ patterns for an experiment in the weakly coupled regime ($\mathcal{I}=0.15$). Left: Gray-scale image of both domains. Right: Skeletonized image showing superimposed wave fronts extracted from the left [dark (blue)] and right [light (red)] domains. The small circular mark on the left side of the left domain is a small air bubble attached to the window of the reaction chamber; it did not affect the reaction pattern.

Figure 7

(a) Evolution of weakly coupled initially random BZ patterns in the simulation. Gray images of both domains are shown at the left side. The right column shows the extracted wave fronts of the first [dark (blue)] and the second domain [bright (red)] superimposed on each other. (b) Spiral core annihilation. Two counter-rotating spiral cores merge and subsequently disappear. In this way, spiral cores are destroyed, leading to an increasing synchronization of the patterns.

Figure 8

Number of spiral cores as a function of time for the simulation shown in Fig. 7 for domain 1 ($\circ$) and domain 2 ($\diamond$). The dashed and dotted lines mark fitted solutions $n_1\left(t\right)$ and $n_2\left(t\right)$ of Eq. (4). Inset: Same data plotted logarithmically; data for domain 2 ($\diamond$) shifted downwards for clarity. The vertical dashed line at $t=155$ marks a transition suggested by the change of the slope.

Figure 9

Time evolution of strongly coupled spiral patterns in the experiment ($\mathcal{I}=0.5$). The left and middle columns show the extracted wave fronts superimposed on the gray-scale pattern for various times. The right column shows the superposition of the wave fronts from cell 1 [dark (blue)] and cell 2 [bright (red)]. The region within the dashed square (green) is used in Fig. 10. The rotation period of the undisturbed spirals was 48 s.

Figure 10

Fragmentation of wave fronts and formation of new spirals during the evolution towards synchronization. Images cropped from Fig. 9.

Figure 11

Wavefront patterns synchronized by strong coupling in (a) experiments and [(c) and (d)] simulation. Bright areas correspond to high concentration of the oxidized catalyst [${{\mathrm{Ru}\left(\mathrm{bpy}\right)}_3}^{3+}$ in the experiment, $v$ in the simulation]. [(b) and (e)] Wavefronts extracted from (a), (c), and (d) illustrating the degree of synchronization.

Figure 12

Phase of domains 1 [(a) and (d)] and 2 [(b) and (e)], and their pointwise difference [(c) and (f)] for weak ($I_0=0.03$, top row) and strong coupling ($I_0=0.16$, bottom row).

Figure 13

Spatial standard deviation of the phase differences between domain 1 and domain 2, as a function of time for different coupling strength $I_0$ (color coded). The dashed vertical line marks an arbitrary chosen threshold $t_t=40$. Only data for $t>t_t$ were used to calculate statistical properties shown in Fig. 14.

Figure 14

(a) The difference between the average ${\sigma }_s$ at $I_0=0$ and at $I_0$ (blue circles). (b): The standard deviation (red squares, right $y$ axis) as a function of the coupling strength. These values were calculated based on the times larger than $t_t=40$ (black vertical line in Fig. 13). The solid line (blue) is a fit of $\propto \sqrt{I_0}$ to the data as expected for a pitchfork bifurcation.