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The influence of Turing modes in two subsystems on pattern formation is investigated by the two-layer non-linearly coupled Brusselator model. It is found that the coupling coefficient and wave number ratio between two Turing modes take an important role in the pattern formation and pattern selection. The kind of pattern changes from simple pattern to complex one with the increase of wave number ratio. When nonlinear coupling coefficient is smaller than 0.1, the short wave mode in system 1 under the action of instability mode in system 2 can form not only simple pattern (such as simple hexagon and quadrilateral and stripe pattern), but also complex pattern due to the resonance coupling between the two Turing modes (such as honeycomb hexagon and super hexagon and complex black-eye pattern), and the transformation process of pattern from quadrilateral to superlattice pattern is observed for the first time under the specific parameters. When nonlinear coupling coefficient is more than 0.1, the simple patterns such as simple hexagon and stripe pattern are obtained only in system 1, because there is no resonance coupling between the two Turing modes in system 1.
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Keywords:
- Brusselator model /
- nonlinear coupling /
- Turing mode
[1] Schenk C P, Or-Guil M, Bode M, Purwins H G 1997 Phys. Rev. Lett. 78 3781
[2] Berenstein I, Dolnik M, Yang L, Zhabotinsky A M, Epstein I R 2004 Phys. Rev. E 70 046219
[3] Arbell H, Fineberg J 2002 Phys. Rev. E 65 036224
[4] Sharpe J P, Ramazza P L, Sungar N, Saunders K 2006 Phys. Rev. Lett. 96 094101
[5] Bois J S, Jlicher F, Grill S W 2011 Phys. Rev. Lett. 106 028103
[6] Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2002 Phys. Rev. Lett. 88 208303
[7] Berenstein I, Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2003 Phys. Rev. Lett. 91 058302
[8] Turing A M 1952 Phil. Trans. R. Soc. London B 237 37
[9] Dong L F, Fan W L, He Y F, Liu F C, Li S F, Gao R L, Wang L 2006 Phys. Rev. E 73 066206
[10] Duan X X, Ouyang J T, Zhao X F, He F 2009 Phys. Rev. E 80 016202
[11] Stollenwerk L, Laven J G, Purwins H G 2007 Phys. Rev. Lett. 98 255001
[12] Shirafuji T, Kitagawa T, Wakai T, Tachibana K 2003 Appl. Phys. Lett. 83 2309
[13] Shin J, Raja L L 2007 J. Phys. D: Appl. Phys. 40 3145
[14] Liu C Z, Brown N, Meenan B J 2006 Appl. Surf. Sci. 252 2297
[15] Dong L F, Liu W L, Wang H F, He Y F, Fan W L, Gao R L 2007 Phys. Rev. E 76 046210
[16] Nie Q Y, Ren C S, Wang D Z, Li S Z, Zhang J L 2007 Appl. Phys. Lett. 90 221504
[17] Cross M C, Hohenberg P C 1993 Rev. Mod. Phys. 65 851
[18] Barrio R A, Varea C, Aragón J L, Maini P K 1999 Bull. Math. Biol. 61 483
[19] Kytta K, Kaski K, Barrio R A 2007 Physica A 385 105
[20] Berenstein I, Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2005 J. Phys. Chem. A 109 5382
[21] Bai Z G, Dong L F, Li Y H, Fan W L 2011 Acta Phys. Sin. 60 118201 (in Chinese) [白占国, 董丽芳, 李永辉, 范伟丽 2011 60 118201]
[22] Míguez D G, Dolnik M, Epstein I R, Muñuzuri A P 2011 Phys. Rev. E 84 046210
[23] Rogers J L, Schatz M F, Brausch O, Pesch W 2000 Phys. Rev. Lett. 85 4281
[24] Ni W M, Tang M X 2005 Trans. Amer. Math. Soc. 357 3953
[25] Mikhailova A S, Showalter K 2006 Physics Reports 425 79
[26] Yuan X J, Shao X, Liao H M, Ouyang Q 2009 Chin. Phys. Lett. 26 024702
[27] Liu H Y, Yang C Y, Tang G N 2013 Acta Phys. Sin. 62 010505 (in Chinese) [刘海英, 杨翠云, 唐国宁 2013 62 010505]
[28] Wang W M, Liu H Y, Cai Y L, Li Z Q 2011 Chin. Phys. B 20 074702
[29] Dong L F, Li S F, Liu F, Liu F C, Liu S H, Fan W L 2006 Acta Phys. Sin. 55 362 (in Chinese) [董丽芳, 李树锋, 刘峰, 刘富成, 刘书华, 范伟丽 2006 55 362]
[30] Dong L F, Yang Y J, Fan W L, Yue H, Wang S, Xiao H 2010 Acta Phys. Sin. 59 1917 (in Chinese) [董丽芳, 杨玉杰, 范伟丽, 岳晗, 王帅, 肖红 2010 59 1917]
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[1] Schenk C P, Or-Guil M, Bode M, Purwins H G 1997 Phys. Rev. Lett. 78 3781
[2] Berenstein I, Dolnik M, Yang L, Zhabotinsky A M, Epstein I R 2004 Phys. Rev. E 70 046219
[3] Arbell H, Fineberg J 2002 Phys. Rev. E 65 036224
[4] Sharpe J P, Ramazza P L, Sungar N, Saunders K 2006 Phys. Rev. Lett. 96 094101
[5] Bois J S, Jlicher F, Grill S W 2011 Phys. Rev. Lett. 106 028103
[6] Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2002 Phys. Rev. Lett. 88 208303
[7] Berenstein I, Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2003 Phys. Rev. Lett. 91 058302
[8] Turing A M 1952 Phil. Trans. R. Soc. London B 237 37
[9] Dong L F, Fan W L, He Y F, Liu F C, Li S F, Gao R L, Wang L 2006 Phys. Rev. E 73 066206
[10] Duan X X, Ouyang J T, Zhao X F, He F 2009 Phys. Rev. E 80 016202
[11] Stollenwerk L, Laven J G, Purwins H G 2007 Phys. Rev. Lett. 98 255001
[12] Shirafuji T, Kitagawa T, Wakai T, Tachibana K 2003 Appl. Phys. Lett. 83 2309
[13] Shin J, Raja L L 2007 J. Phys. D: Appl. Phys. 40 3145
[14] Liu C Z, Brown N, Meenan B J 2006 Appl. Surf. Sci. 252 2297
[15] Dong L F, Liu W L, Wang H F, He Y F, Fan W L, Gao R L 2007 Phys. Rev. E 76 046210
[16] Nie Q Y, Ren C S, Wang D Z, Li S Z, Zhang J L 2007 Appl. Phys. Lett. 90 221504
[17] Cross M C, Hohenberg P C 1993 Rev. Mod. Phys. 65 851
[18] Barrio R A, Varea C, Aragón J L, Maini P K 1999 Bull. Math. Biol. 61 483
[19] Kytta K, Kaski K, Barrio R A 2007 Physica A 385 105
[20] Berenstein I, Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2005 J. Phys. Chem. A 109 5382
[21] Bai Z G, Dong L F, Li Y H, Fan W L 2011 Acta Phys. Sin. 60 118201 (in Chinese) [白占国, 董丽芳, 李永辉, 范伟丽 2011 60 118201]
[22] Míguez D G, Dolnik M, Epstein I R, Muñuzuri A P 2011 Phys. Rev. E 84 046210
[23] Rogers J L, Schatz M F, Brausch O, Pesch W 2000 Phys. Rev. Lett. 85 4281
[24] Ni W M, Tang M X 2005 Trans. Amer. Math. Soc. 357 3953
[25] Mikhailova A S, Showalter K 2006 Physics Reports 425 79
[26] Yuan X J, Shao X, Liao H M, Ouyang Q 2009 Chin. Phys. Lett. 26 024702
[27] Liu H Y, Yang C Y, Tang G N 2013 Acta Phys. Sin. 62 010505 (in Chinese) [刘海英, 杨翠云, 唐国宁 2013 62 010505]
[28] Wang W M, Liu H Y, Cai Y L, Li Z Q 2011 Chin. Phys. B 20 074702
[29] Dong L F, Li S F, Liu F, Liu F C, Liu S H, Fan W L 2006 Acta Phys. Sin. 55 362 (in Chinese) [董丽芳, 李树锋, 刘峰, 刘富成, 刘书华, 范伟丽 2006 55 362]
[30] Dong L F, Yang Y J, Fan W L, Yue H, Wang S, Xiao H 2010 Acta Phys. Sin. 59 1917 (in Chinese) [董丽芳, 杨玉杰, 范伟丽, 岳晗, 王帅, 肖红 2010 59 1917]
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