Spatiotemporal patterns appear through self-organization in a variety of physical, chemical and biological systems.^[1–11] Some of them such as the patterns observed in the Belousov–Zhabotinsky reaction have been well studied by reaction-diffusion equations.^[1,2] In general, these reaction–diffusion equations depict that the molecules execute nearest neighbor jumps on sites. The mean square displacement of a molecule varies with time: 〈Δr²〉∼t^γ, here, γ = 1, the diffusion is normal. However, in some fractal media such as the porous glass and the gel, the molecules can execute not only long jumps but also long waiting between two jumps.^[12] This means that these molecules would undergo anomalous diffusion like superdiffusion or subdiffusion. For superdiffusion such as Lévy flight, γ > 1, the diffusion process is faster than normal diffusion, and the pattern formation is depicted by space fractional reaction-diffusion equations. For subdiffusion, γ < 1, the diffusion process is slower than normal diffusion, and the pattern formation is depicted by time fractional reaction–diffusion equations. It is necessary that anomalous diffusions should be considered on studying the pattern formation appearing in fractal media like porous glass.

Pattern formations occurring in anomalous diffusion cases have been studied recently. Golovin et al. found that it is not needed for superdiffusion that the inhibitor must diffuse faster than the activator for a turing pattern as in a normal diffusion case, and the Lévy flight could give rise to a transition from spiral to spiral chaos.^[13] Zanette et al. observed the acceleration of the front induced by superdiffusion in a bistable system.^[14] In subdiffusion cases, Langlands et al. obtained that the conditions for turing pattern formation are the same as for the normal reaction–diffusion systems when the diffusion index of the activator is equal to that of the inhibitor.^[15] Henry et al. also found that it is not needed for the appearance of the turing mode that the diffusion coefficient of the inhibitor must be larger than that of the activator.^[16] In this paper, we study the pattern formation in an Oregonator model with superdiffusion. Antispiral, stable turing patterns, and travelling patterns are observed by changing the diffusion indices of the activator. A travelling hexagonal pattern is observed for the first time. We also observe a transition between antispiral and spiral by changing the diffusion index of the inhibitor.

The starting point of our consideration is the dimensionless two-chemical reaction–diffusion Oregonator model with indices of different orders,

Here, u and v are dimensionless variables, which correspond to the concentrations of the activator and inhibitor, respectively. D_u and D_v denote their diffusion coefficients, respectively. ɛ, f, and q are rescaled kinetic parameters. The diffusion indices 1 < α,β < 2 depict the superdiffusion fractional and are defined the same as:

The time evolution of the system is given by the singularities of Eqs. (10) and (11), i.e., S = 0. Then, the instability analysis of the system could be predicted. For turing instability, it occurs when Re[s(k = 0)] < 0 and Re[s(k ≠ 0)] > 0. For Hopf instability, it occurs when Re[s(k = 0)] > 0 and Im[s(k = 0)] ≠ 0.

Figure 1 shows the dependance of s on the wavenumber k for different combinations of α and β, in which the solid (dashed) lines represent the real (imaginary) parts of s.

Fig. 1.

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	Fig. 1. Plots of s(k). Du = Dv = 0.001, f = 0.95, q = 0.002, ɛ = 0.75. Diffusion indices α and β are shown on the legends.

Numerical simulations are performed in two dimension space. In order to solve the spatial derivative in the fractional-in-space reaction diffusion equations, the Fourier spectral method is used with grid points 128 × 128. The temporal derivative of the equations is dealt with in a standard fourth order Runge–Kutta method. The time step is dt = 0.05 t.u. Different domain sizes are used, typically L × L = 20 × 20 s.u.² and L × L = 0.4 × 0.4 s.u.². In the following, we will discuss firstly two cases that when the activator and the inhibitor undergo superdiffusion, respectively, then we illustrate the full dynamics of the space fractional Oregonator model in terms of a phase diagram.

We firstly consider the case of β = 2, 1 < α < 2, that means only the activator is superdiffusion. Figure 1 shows dispersion relations of a system with normal (α = 2) and anomalous (α = 1.5) diffusions. When the activator is normal diffusion, only the Hopf mode appears on the dispersion curve. The Hopf instability leads to the formation of an antispiral as shown in Fig. 2(a). However, when the activator is superdiffusion, various patterns appear at different diffusion indices.

Fig. 2.

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Fig. 2. Scenarios of pattern formations with decreasing the diffusion indices of the activator. (a): antispiral, α = 2; (b): antispiral, α = 1.8; (c): stable negative hexagonal pattern, α = 1.73; (d): stable stripe, α = 1.7; (e): travelling hexagonal pattern, α = 1.5; (f): travelling wave, α = 1.3. (g): dependance of the amplitude of antispiral (a)–(b) on the time; (h): dependance of the amplitude of patterns (c)–(f) on the time. The oscillation period in panel (g) is 16.72 (16.58) t.u. for α = 1.8 (α = 2). The amplitudes of panels (g) and (h) are measured from the center site of patterns (a)–(f), respectively. Domain sizes are set L × L = 20 × 20 s.u.2 in panels (a) and (b), L × L = 0.4 × 0.4 s.u.2 in panels (c)–(f), respectively.

Figure 2 shows the scenarios of pattern formation with decreasing the diffusion indices of the activator. When α > 1.75, antispirals occupy the system domain of L × L = 20 × 20 s.u.² as shown in Figs. 2(a) and 2(b). With decreasing α, the wavelength of the antispiral becomes shorter. Figure 2(g) illustrates that the oscillation period of the antispiral decreases a little with increasing α. When α = 1.73, the turing mode appears and becomes positive beside the Hopf mode. Here, the wave number k of the turing mode is very large, which means the wavelength of the turing pattern is very small. Therefore, we use a small system with a domain size of L × L = 0.4 × 0.4 s.u.² in the numerical simulation. Figure 2(c) shows the stable negative hexagonal pattern. When α = 1.7, a stable stripe pattern appears as shown in Fig. 2(d). The dependance of the amplitude on the time in Fig. 2(h) indicates that the negative hexagonal and the stripe patterns are stable.

Travelling patterns occur when α < 1.6. Figure 2(e) shows the travelling hexagonal pattern which is first observed in a reaction diffusion system. It is very different from the stable hexagonal patterns observed widely in chemical experiments and simulations. It has been proved that the stable hexagonal pattern origins from pure turing instability.

Figure 3 shows the travelling hexagonal pattern and its spatial spectrum in which the three primitive vectors k_i (i = 1,2,3) satisfy k₃ = −k₂ − k₁. The subharmonic vector k_f satisfies k_f = k₂ − k₁. From Fig. 3 we get |k_i| = 42 and |k_f| = 71. In order to explain the origin of the travelling hexagonal pattern we analyze the stability of the system. From Fig. 1 we find that the Hopf mode grows positively, and the system has a limit cycle solution and follows bulk oscillation. A Floquet multiplier is often used to analyze the stability of the limit cycle solution. Figure 4 shows the absolute value of Floquet multipliers as a function of wave number k. When the absolute value of the Floquet multiplier is less than 1 the limit cycle is stable. When the maximum of the absolute values of Floquet multipliers is larger than 1, the limit cycle solution would lose its stability. It can be seen from Fig. 4 that the the maximum of the absolute values of Floquet multipliers crosses the horizontal line of 1 firstly at the most unstable mode k_c = 48, which means that the limit cycle loses its stability at the most unstable mode k_c = 48. This value is close to the value of primitive vector k_i of the travelling hexagonal pattern and also close to the maximum at k_T = 49 of the turing band in Fig. 1. Therefore, one can view the travelling hexagonal pattern as a result of resonance between k_c and k_T. The travelling hexagonal pattern moves in the direction of subharmonic vector k_f. On decreasing the diffusion indices α, a travelling wave appears as shown in Fig. 2(f).

Fig. 3.

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	Fig. 3. Travelling hexagonal pattern (a) and its spatial spectrum (b). ki (i = 1,2,3) present the primitive vectors of the travelling hexagonal pattern. The arrow in panel (a) and the subharmonic vector kf in panel (b) indicate the direction of the travelling hexagonal pattern.

Fig. 4.

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	Fig. 4. Dependance of the absolute value of Floquet multiplier on wave number. kc = 48 indicates the most unstable mode at where the limit cycle solution loses stability. α = 1.5.

We consider here the case of α = 2, 1 < β < 2 which means only the inhibitor is superdiffusion. Figure 5 shows the scenarios of pattern formation with decreasing the diffusion indices of the inhibitor. It can be seen that the antispiral transits to a spiral pattern with decreasing the diffusion index β. Figure 5(a) shows an antispiral with a large wavelength in the domain size of L × L = 10 × 10 s.u.². The arrows indicate the propagation direction of the antispiral. The frequency of bulk oscillation ω₀ = 0.4174 t.u.⁻¹ which is larger than that of antispiral ω_as = 0.3796 t.u.⁻¹. Figure 5(c) shows a spiral with the arms propagating outward as indicated by the arrows. However, the spiral induced by the superdiffusion of the inhibitor is very different from that in the normal diffusion case. The cores of spirals obtained here are localized and oscillate around turing spots, which is similar with the dual-mode spiral vortices observed by Meron near a Hopf-turing codimension-two point.^[3] The frequency of spiral ω_s = 0.3817 t.u.⁻¹, which is still smaller than that of bulk oscillation. Between the antispiral and the spiral, we observe oscillatory patterns with nonuniform distributions in space such as Fig. 5(b). This means that one can eliminate a spiral by changing the diffusion index of the inhibitor.

Fig. 5.

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	Fig. 5. Pattern formation at different diffusion indices of inhibitor. (a): antispiral, β = 1.9; (b): oscillatory pattern, β = 1.7; (c): spiral, β = 1.3. L × L = 10 × 10 s.u.2, α = 2. The arrows indicate the propagation direction of waves.

Both activator and inhibitor could undergo superdiffusion in real chemical reactions. In order to illustrate the full dynamics of pattern formation of the space fractional Oreganator model we give a phase diagram in α − β space as shown in Fig. 6. The phase diagram can be divided into five regions by the lines L₁–L₄. Regions (I: spiral) and (III: antiSpiral) represent spiral wave and antispiral wave, respectively. Oscillatory patterns with nonuniform distributions in space such as Fig. 5(b) occur in region (II: oscillation). Stable pattern formations such as the negative hexagonal pattern and stripe pattern are observed only in a narrow region (IV: turing). Travelling patterns such as the travelling hexagonal pattern and the travelling wave appear in region (V: travelling). Below the line L₁, the turing mode with a large wavenumber grows as indicated in Fig. 1, therefore, a small domain size (such as 0.4 × 0.4 s.u.² we used) is needed. Large domain size (i.e., small spatial resolution) would lead to uniform oscillation. Above line L₁, we still use the domain size 20 × 20 s.u.² in order to obtain antispiral and spiral patterns with large wavelength.

Fig. 6.

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	Fig. 6. Phase diagram of pattern formation at different diffusion indices. The domain size below (above) line L1 is 0.4 × 0.4 s.u.2 (20 × 20 s.u.2).

In this paper we have studied the pattern formations in the Oregonator model with superdiffusion. The space fractional reaction–diffusion equations are analyzed by applying Fourier and Laplace transform. If we consider the superdiffusion of the inhibitor only, i.e., α = 2, we observed the transition from antispiral to spiral with decreasing the diffusion index β of the inhibitor. However, if we consider the superdiffusion of the activator only, i.e., β = 2, stable negative hexagonal pattern, stable stripe, travelling hexagonal pattern, and travelling wave are obtained with decreasing the diffusion index α of the activator. The wave number k_i of the travelling hexagonal pattern is close to the wave number at where the Floquet multiplier loses stability.

From the expression of S in Eqs. (10) and (11), one can find that the imaginary part would appear as long as the diffusion index α,β ≠ 2. In terms of this, even for the turing pattern, it would exhibit temporal characters such as travelling. However, we have obtained stable stripe and negative hexagonal patterns in numerical simulations as shown in Figs. 2(d) and 2(c). On the other hand, the spatial characteristics such as the wavelength of patterns can still be described from Fig. 1. Therefore, linear stability analysis used often in a normal diffusion case has became inadequate for depicting fully the stability analysis of the system in the anomalous diffusion case. In addition, in the case of normal diffusion, spiral and antisprial are characterized by the relations between the frequency of bulk oscillation ω₀ and that of spiral ω_s (antispiral ω_as). For spiral (antispiral), the frequency of bulk oscillation is smaller (larger) than that of spiral (antispiral).^[17] However, in the case of superdiffusion these relations between frequencies do not hold anymore. In our simulations the frequencies of bulk oscillation are always larger than those of both antispiral and spiral. Thus, a new method is required in order to distinguish the spiral and antispiral and clarify their origins, which would be our future work.

Superdiffusion has been observed in the chemical reaction with the porous glass and the gel. The molecules in these fractal media could diffuse faster. Molecules with different size in chemical reaction diffuse at different speeds, therefore, they should be depicted by a different diffusion index in the reaction–diffusion equations with superdiffusion. Here, we studied the pattern formation induced by the superdiffusion of the activator and/or inhibitor. Our results are helpful to exploring the dynamics of pattern formation occurring in fractal media.