Stationary and Oscillatory Localized Patterns, and Subcritical Bifurcations

Stationary and oscillatory localized patterns (oscillons) are found in the Belousov-Zhabotinsky reaction dispersed in Aerosol OT water-in-oil microemulsion. The experimental findings are analyzed in terms of subcritical Hopf instability, subcritical Turing instability, and their combination.

Figure 1

Turing patterns (a),(b), localized Turing patterns (c), and oscillons (d)–(f) in the $\mathrm{Ru}(\mathrm{bpy}{)}_3^{2+}$- (a),(c),(e),(f) and ferroin- (b),(d) catalyzed BZ-AOT systems. Period of the oscillon is 47 s for (e),(f) and about 250 s for (d). ${\phi }_d=0.36$ (a),(d), 0.48 (b), 0.41 (c),(e),(f), $\omega \equiv [{\mathrm{H}}_2\mathrm{O}]/;[\mathrm{AOT}]=(\mathrm{a}\mathrm{)}\mathrm{,}\mathrm{(}\mathrm{c}\mathrm{)}\mathrm{,}\mathrm{(}\mathrm{e}\mathrm{)}\mathrm{,}\mathrm{(}\mathrm{d})$ 15, (b),(d) 18.3, $[{\mathrm{H}}_2{\mathrm{SO}}_4{]}_w/\mathrm{M}=(\mathrm{a})$ 0.32, (c),(e),(f) 0.25, (b),(d) 0.1, $[{\mathrm{NaBrO}}_3{]}_w/\mathrm{M}=(\mathrm{a})$ 0.215, (c),(e),(f) 0.2, (b),(d) 0.1 $[\mathrm{MA}{]}_w=0.25\mathrm{M}$, $[\mathrm{Ru}(\mathrm{bpy}{)}_3^{2+}{]}_w=[\mathrm{ferroin}{]}_w=4.2\mathrm{mM}$ (subscript $w$ refers to concentrations in the aqueous phase). Frame size, $\mathrm{m}{\mathrm{m}}^{\mathrm{2}}=(\mathrm{a})–(\mathrm{c})5.1\times 3.75$, (d) $7.6\times 5.6$, (e),(f) $2.13\times 1.87$. Arrows in (a) mark directions of phase waves.

Figure 2

Oscillon in 1D BZ-AOT model (4, 5, 6, 7) (c),(d) and oscillon and localized stationary Turing spot in 2D Samogyi-Stucki model (1, 2, 3) (a),(b),(f). (c) activator ($x$) (curve 1) and catalyst ($z$) (curves 2, 3), curve 1 corresponds to curve 2, which is separated in time from curve 3 by $T/2$, $T=3.46$, (a) cross section of oscillon (curves 1, 2) and stationary spot (curve 3); (f) is a contour map of the oscillon in 2D, $\mathrm{area}=20\times 20$. (b),(d) Dispersion curves for models (1, 2, 3) and (4, 5, 6, 7), respectively: curves 3 and 2 are imaginary part [divided by 6 in (b) and by 10 in (d)] and real part, respectively, of the complex eigenvalue; curve 1 is the real eigenvalue. Parameters for model (1, 2, 3): $k_1=2.1$, $k_2=1$, $K=3.4$, $k_4=1.8$, $k_5=0.47$, $k_6=0.05$, $k_7=0.6$, $k_8=0.14$, $D_x=D_z=0.01$, and $D_y=0.035$. Initial perturbations: $x_0=x_{\mathrm{SS}}+1.2\cos [0.7\pi (x{x}^2+y{y}^2{)}^{1/2}]\exp [-0.1(x{x}^2+y{y}^2)]\theta (c^2-x{x}^2-y{y}^2)$ with center at $xx=yy=0$, $y_0=y_{\mathrm{SS}}$, $z_0=z_{\mathrm{SS}}$, where $xx$ and $yy$ are spatial coordinates, subscript SS means steady state, and $\theta (a)=1$ if $a>0$ and $\theta (a)=0$ if $a\le 0$. Oscillon and stationary spot are generated when $c=2$ and $c=4$, respectively. Parameters for model (4, 5, 6, 7): $f_0=0.52$ (0.5), $q=0.0004$, $K_1=50$, $\alpha =0.1$, $\beta =0.72$, ${\epsilon }_1=0.05$, ${\epsilon }_2=0.001$, ${\epsilon }_3=2$, $D_x=0.01$, $D_z=0.3$, $D_u=1$, and $D_s=0.028$; (e) $k_5-D_y$ parameter plane for dynamical behavior of model (1, 2, 3). Turing patterns occupying the entire area (△), localized Turing patterns (▲), oscillons (●), steady state (+). Vertical lines 1 and 2 mark Hopf and subcritical Hopf bifurcations, respectively; curve 3 is Turing bifurcation.

Figure 3

Dependence of the behavior of model (1, 2, 3) on initial perturbations in 1D. (a) Toothlike initial perturbation $x_0=x_{\mathrm{SS}}+1.2\theta [(l_p/2{)}^2-x{x}^2]$ with center at $xx=0$. Ordinate, which distinguishes among states, is arbitrary. Abbreviations: SS, steady state (assigned ordinate value 0); 1T, single stationary Turing peak (value 1); 1O, single oscillon (value 2); (b) temporal behavior of $\mathrm{T}+\mathrm{O}+\mathrm{T}$ structure ($l_p/\Lambda =0.34$, $(g+l_p)/\Lambda =2.7$, $\Lambda =2.21$), small oscillations in curve 1 (stationary Turing peak) are induced by the oscillatory middle peak. (c) Two identical toothlike initial perturbations separated by a gap of length $g$. Parameters: $k_1=2.1$, $k_2=1$, $K=3.4$, $k_4=1.8$, $k_5=0.44$, $k_6=0.05$, $k_7=0.6$, $k_8=0.14$, $D_x=D_z=0.01$, and $D_y=0.06$. Symbols: +, SS; □, oscillon with two synchronously oscillating peaks; ○, stationary Turing pattern with two peaks; ■, oscillon with three synchronously oscillating peaks; ▲, pattern with three peaks, the middle one oscillating and the outer ones stationary ($\mathrm{T}+\mathrm{O}+\mathrm{T}$); ×, single oscillon; solid line, single stationary peak; ◇, oscillon with two peaks oscillating antiphase; $*$, two independent Turing or oscillatory peaks.