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The coupling mechanism is one of most important approaches to generating multiple-scaled spatial-temporal patterns. In this paper, the mode interaction between two different Turing modes and the pattern forming mechanisms in the non-symmetric reaction diffusion system are numerically investigated by using a two-layered coupled model. This model is comprised of two different reaction diffusion models: the Brusselator model and the Lengyel-Epstein model. It is shown that the system gives rise to superlattice patterns if these two Turing modes satisfy the spatial resonance condition, otherwise the system yields simple patterns or superposition patterns. A suitable wave number ratio and the same symmetry are two necessary conditions for the spatial resonance of Turing modes. The eigenvalues of these two Turing modes can only vary in a certain range in order to make the two sub-system patterns have the same symmetry. Only when the long wave mode becomes the unstable mode, can it modulate the other Turing mode and result in the formation of spatiotemporal patterns with multiple scale. As the wave number ratio increases, the higher-order harmonics of the unstable mode appear, and the sub-system with short wave mode undergoes a transition from the black-eye pattern to the white-eye pattern, and finally to a temporally oscillatory hexagon pattern. It is demonstrated that the resonance between the Turing mode and its higher-order harmonics located in the wave instability region is the dominant mechanism of the formation of this oscillatory hexagon pattern. Moreover, it is found that the coupling strength not only determines the amplitudes of these patterns, but also affects their spatial structures. Two different types of white-eye patterns and a new super-hexagon pattern are obtained as the coupling strength increases. These results can conduce to understanding the complex spatial-temporal behaviors in the coupled reaction diffusion systems.
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Keywords:
- Turing mode /
- non-symmetric reaction diffusion system /
- super-lattice pattern /
- numerical simulation
[1] Ross T D, Lee H J, Qu Z J, Banks R A, Phillips R, Thomson M 2019 Nature 572 224
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
Li X Z, Bai Z G, Li Y 2019 Acta Phys. Sin. 68 068201
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 不同参数下耦合系统的色散关系 (a) Du1 = 12.6, Dv1 = 27.9, Du2 = 22, Dv2 = 420, α = 0.1; (b) Du1 = 5.3, Dv1 = 20, Du2 = 22, Dv2 = 500, α = 0.1
Figure 1. Dispersion relations of coupled systems under different parameters: (a) Du1 = 12.6, Dv1 = 27.9, Du2 = 22, Dv2 = 420, α = 0.1; (b) Du1 = 5.3, Dv1 = 20, Du2 = 22, Dv2 = 500, α = 0.1.
图 2 不同波数比下的超点阵斑图及其傅里叶频谱图 (a) 1∶2下的黑眼斑图,
${D_{u1}} = 13.5$ ,${D_{v1}} = 27.5$ ,${D_{u2}} = 22$ ,${D_{v2}} = 400$ ; (b) 1∶3下的白眼斑图,${D_{u1}} = 6$ ,${D_{v1}} = 12.3$ ,${D_{u2}} = 22$ ,${D_{v2}} = 400$ ; (c) 1∶4下的白眼斑图,${D_{u1}} = 3.4$ ,${D_{v1}} = 6.96$ ,${D_{u2}} = 21.9$ ,${D_{v2}} = 400$ .$\alpha = 0.1$ Figure 2. Superlattice pattern and fourier spectrum under different wave number ratios: (a) Black eye pattern at 1∶2,
${D_{u1}} = 13.5$ ,${D_{v1}} = 27.5$ ,${D_{u2}} = 22$ ,${D_{v2}} = 400$ ; (b) white eye pattern at 1∶3,${D_{u1}} = 6$ ,${D_{v1}} = 12.3$ ,${D_{u2}} = 22$ ,${D_{v2}} = 400$ ; (c) white eye pattern at 1∶4,${D_{u1}} = 3.4$ ,${D_{v1}} = 6.96$ ,${D_{u2}} = 21.9$ ,${D_{v2}} = 400$ .$\alpha = 0.1$ .
图 3 波数比为1∶5时的时间振荡超六边形斑图,
${D_{u1}} = 2.2$ ,${D_{v1}} = 4.5$ ,${D_{u2}} = 21.9$ ,${D_{v2}} = 400$ ,$\alpha = 0.1$ (a) 色散关系曲线; (b) 三个位置处u1的时间变化关系图; (c) 一个振荡周期内的斑图演化过程Figure 3. Oscillatory super-hexagon pattern with wave number ratio of 1∶5,
${D_{u1}} = 2.2$ ,${D_{v1}} = 4.5$ ,${D_{u2}} = 21.9$ ,${D_{v2}} = 400$ ,$\alpha = 0.1$ : (a) Dispersion curve; (b) time variation of u1 at three positions; (c) evolution of pattern in an oscillating period.
图 4 不同本征值
${h_2}$ 下的复杂斑图及其傅里叶频谱图 (a) 蜂窝状六边形斑图${h_2} = - 2.56$ ,${D_{u1}} = 8.5$ ,${D_{v1}} = 12.5$ ,${D_{u2}} = 22$ ,${D_{v2}} = 400$ ; (b) 白眼斑图${h_2} = - 1.3$ ,${D_{u1}} = 7$ ,${D_{v1}} = 12.3$ ,${D_{u2}} = 22$ ,${D_{v2}} = 400$ ; (c) 白眼斑图${h_2} = - 0.55$ ,${D_{u1}} = 6$ ,${D_{v1}} = 12.3$ ,${D_{u2}} = 22$ ,${D_{v2}} = 400$ ; (d) 超六边形斑图${h_2} = - 0.31$ ,${D_{u1}} = 5.9$ ,${D_{v1}} = 12.7$ ,${D_{u2}} = 22$ ,${D_{v2}} = 400$ ; (e) 条纹斑图${h_2} = 0.67$ ,${D_{u1}} = 5.5$ ,${D_{v1}} = 16$ ,${D_{u2}} = 22$ ,${D_{v2}} = 400$ .$\alpha = 0.1$ .Figure 4. Complex patterns and Fourier spectrum under different eigenvalues
${h_2}$ : (a) Honeycomb hexagon pattern${h_2} = - 2.56$ ,${D_{u1}} = 8.5$ ,${D_{v1}} = 12.5$ ,${D_{u2}} = 22$ ,${D_{v2}} = 400$ ; (b) white-eye pattern${h_2} = - 1.3$ ,${D_{u1}} = 7$ ,${D_{v1}} = 12.3$ ,${D_{u2}} = 22$ ,${D_{v2}} = 400$ ; (c) white-eye pattern${h_2} = - 0.55$ ,${D_{u1}} = 6$ ,${D_{v1}} = 12.3$ ,${D_{u2}} = 22$ ,${D_{v2}} = 400$ ; (d) super-hexagon pattern${h_2} = - 0.31$ ,${D_{u1}} = 5.9$ ,${D_{v1}} = 12.7$ ,${D_{u2}} = 22$ ,${D_{v2}} = 400$ ; (e) stripe pattern${h_2} = 0.67$ ,${D_{u1}} = 5.5$ ,${D_{v1}} = 16$ ,${D_{u2}} = 22$ ,${D_{v2}} = 400$ .$\alpha = 0.1$ .
图 5 不同本征值
$ {h_1}$ 下的复杂斑图及其傅里叶频谱图 (a) 白眼斑图,$ {h_1} = 0.2$ ,$ {D_{u1}} = 6.1$ ,$ {D_{v1}} = 12.7$ ,$ {D_{u2}} = 22.3$ ,$ {D_{v2}} = 403$ ; (b) 条纹点状斑图,$ {h_1} = 0.4$ ,$ {D_{u1}} = 6.1$ ,$ {D_{v1}} = 12.6$ ,$ {D_{u2}} = 20.3$ ,$ {D_{v2}} = 464$ ,$ \alpha = 0.1$ Figure 5. Complex patterns and fourier spectrum under different eigenvalues
${h_1}$ : (a) White-eye pattern,${h_1} = 0.2$ ,${D_{u1}} = 6.1$ ,${D_{v1}} = 12.7$ ,${D_{u2}} = 22.3$ ,${D_{v2}} = 403$ ; (b) stripe-spot pattern,${h_1} = 0.4$ ,${D_{u1}} = 6.1$ ,${D_{v1}} = 12.6$ ,${D_{u2}} = 20.3$ ,${D_{v2}} = 464$ ,$\alpha = 0.1$ .
图 6 不同耦合强度下的超六边形斑图 (a) 白眼斑图,
$\alpha = 0.01$ ,${D_{u1}} = 5.8$ ,${D_{v1}} = 11.4$ ,${D_{u2}} = 22$ ,${D_{v2}} = 367$ ; (b) 白眼斑图,$\alpha = 0.1$ ,${D_{u1}} = 6.3$ ,${D_{v1}} = 12.9$ ,${D_{u2}} = 21.8$ ,${D_{v2}} = 395$ ; (c) 新型超六边形斑图,$\alpha = 0.2$ ,${D_{u1}} = 6.1$ ,${D_{v1}} = 13$ ,${D_{u2}} = 22$ ,${D_{v2}} = 432$ ; (d) 新白眼斑图,$\alpha = 0.3$ ,${D_{u1}} = 6.3$ ,${D_{v1}} = 13.98$ ,${D_{u2}} = 22$ ,${D_{v2}} = 460$ Figure 6. Super-hexagon patterns with different coupling strength: (a) White-eye pattern,
$\alpha = 0.01$ ,${D_{u1}} = 5.8$ ,${D_{v1}} = 11.4$ ,${D_{u2}} = 22$ ,${D_{v2}} = 367$ ; (b) white-eye pattern,$\alpha = 0.1$ ,${D_{u1}} = 6.3$ ,${D_{v1}} = 12.9$ ,${D_{u2}} = 21.8$ ,${D_{v2}} = 395$ ; (c) new super-hexagon pattern,$\alpha = 0.2$ ,${D_{u1}} = 6.1$ ,${D_{v1}} = 13$ ,${D_{u2}} = 22$ ,${D_{v2}} = 432$ ; (d) new white-eye pattern,$\alpha = 0.3$ ,${D_{u1}} = 6.3$ ,${D_{v1}} = 13.98$ ,${D_{u2}} = 22$ ,${D_{v2}} = 460$ . -
[1] Ross T D, Lee H J, Qu Z J, Banks R A, Phillips R, Thomson M 2019 Nature 572 224
Google Scholar
[2] Wang Y, Zhang R P, Wang Z, Han Z J 2019 Chin. Phys. B 28 050503
Google Scholar
[3] Gaskins D K, Pruc E E, Epstein I R, Dolnik M 2016 Phys. Rev. Lett. 117 056001
Google Scholar
[4] Hannabuss J, Lera-Ramirez M, Cade N I, Fourniol F J, Nedelec F, Surrey T 2019 Curr. Biol. 29 2120
Google Scholar
[5] 黄志精, 李倩昀, 白婧, 唐国宁 2019 68 110503
Google Scholar
Huang Z J, Li Q Y, Bai J, Tang G N 2019 Acta Phys. Sin. 68 110503
Google Scholar
[6] Qian Y, Gao H Y, Yao C G, Cui X H, Ma J 2018 Chin. Phys. B 27 108902
Google Scholar
[7] Turing A M 1952 Philos. Trans. R. Soc. London, Ser. B 237 37
Google Scholar
[8] Guiu-Souto J, Carballido-Landeira J, Munuzuri A P 2012 Phys. Rev. E 85 056205
Google Scholar
[9] Epstein I R, Berenstein I B, Dolnik M, Vanag V K, Yang L F, Zhabotinsky A M 2008 Phil. Trans. R. Soc. A 366 397
Google Scholar
[10] 张荣培, 王震, 王语, 韩子健 2018 67 050503
Google Scholar
Zhang R P, Wang Z, Wang Y, Han Z J 2018 Acta Phys. Sin. 67 050503
Google Scholar
[11] Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2002 Phys. Rev. Lett. 88 208303
Google Scholar
[12] Yang L F, Epstein I R 2004 Phys. Rev. E 69 026211
Google Scholar
[13] Liu F C, He Y F, Pan Y Y 2010 Commun. Theor. Phys. 53 971
Google Scholar
[14] Catlla A J, McNamara A, Topaz C M 2012 Phys. Rev. E 85 026215
Google Scholar
[15] 白占国, 刘富成, 董丽芳 2015 64 210505
Google Scholar
Bai Z G, Liu F C, Dong L F 2015 Acta Phys. Sin. 64 210505
Google Scholar
[16] 白占国, 董丽芳, 李永辉, 范伟丽 2011 60 118201
Google Scholar
Bai Z G, Dong L F, Li Y H, Fan W L 2011 Acta Phys. Sin. 60 118201
Google Scholar
[17] 李新政, 白占国, 李燕 2019 68 068201
Google Scholar
Li X Z, Bai Z G, Li Y 2019 Acta Phys. Sin. 68 068201
Google Scholar
[18] 李新政, 白占国, 李燕, 赵昆, 贺亚峰 2013 62 220503
Google Scholar
Li X Z, Bai Z G, Li Y, Zhao K, He Y F 2013 Acta Phys. Sin. 62 220503
Google Scholar
[19] Gambino G, Lombardo M C, Sammartino M, Sciacca V 2013 Phys. Rev. E 88 042925
Google Scholar
[20] Biancalani T, Fanelli D, Di Patti F 2010 Phys. Rev. E 81 046215
Google Scholar
[21] Berenstein I, Munuzuri A P, Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2008 Phys. Rev. E 78 025101
Google Scholar
[22] Li J, Wang H L, Ouyang Q 2014 Chaos 24 023115
Google Scholar
[23] Berenstein I, Yang L, Dolnik M, Zhabotinsky A M, Epstein I R 2003 Phys. Rev. Lett. 91 058302
Google Scholar
[24] Feng F, Yan J, Liu F C, He Y F 2016 Chin. Phys. B 25 104702
Google Scholar
[25] Li C X, Dong L F, Feng J Y, Huang Y P 2019 Phys. Plasmas 26 023505
Google Scholar
[26] Sun H Y, Dong L F, Fan W L, Mi Y L, Liu B B, Huang J Y, Li C X, Pan Y Y 2018 Phys. Plasmas 25 123511
Google Scholar
[27] Han R, Dong L F, Huang J Y, Sun H Y, Liu B B, Mi Y L 2019 Chin. Phys. B 28 075204
Google Scholar
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