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The four hexagonal grid state patterns and a variety of non-grid states are obtained by changing the values of intensity ratio between two Turing modes in the two-layer coupled Lengel-Epstein model system. Results of numerical investigation show that those grid states in reaction diffusion are interleaving structures of three sets of different sublattices, which result from the interaction of both the wave number ratio and intensity ratio between Turing modes in the two subsystems; and the specific expressions of three-wave resonance in physical space are governed by the mode intensity ratio. Furthermore, the value of intensity ratio between the two Turing modes in the grid state patterns is greater than that of non-grid state structures, and the type of pattern selected by the system changes from complex to simple pattern with the increase of mode intensity ratio. Finally, it is found that these four hexagonal grid states correspond to different number pair (a, b) having different stability, and the grid state with the number pair (3, 2) is the most stable structure.
[1] Cross M C, Hohenberg P C 1993 Rev. Mod. Phys. 65 851
[2] Kytta K, Kaski K, Barrio R A 2007 Physica A 385 105
[3] Nie Q Y, Ren C S, Wang D Z, Li S Z, Zhang J L 2007 Appl. Phys. Lett. 90 221504
[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] Rogers J L, Pesch W, Brausch O, Schatz M F 2005 Phys. Rev. E 71 066214
[7] 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 物理学报 55 362]
[8] Besson T, Edwards W S, Tuckerman L S 1996 Phys. Rev. E 54 507
[9] Pesch M, Ackemann T, Lange W 2003 Phys. Rev. E 68 016209
[10] Westhoff E G, Herrero R, Ackemann T, Lange W 2003 Phys. Rev. E 67 025203
[11] Epstein T, Fineberg J 2006 Phys. Rev. E 73 055302
[12] Arbell H, Fineberg J 2002 Phys Rev E 65 036224
[13] Pampaloni E, Residori S, Soria S, Arecchi F T 1997 Phys. Rev. Lett. 78 1042
[14] Dong L F, Li S F, Fan W L, Pan Y Y 2009 Phys. Plasmas 16 122308
[15] Judd S L, Silber M 2000 Physica D 136 45
[16] Míguez D G, Dolnik M, Epstein I R, Muñuzuri A P 2011 Phys. Rev. E 84 046210
[17] Rogers J L, Schatz M F, Brausch O, Pesch W 2000 Phys. Rev. Lett. 85 4281
[18] 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 物理学报 62 010505]
[19] Wang W M, Liu H Y, Cai Y L, Li Z Q 2011 Chin. Phys. B 20 074702
[20] Mikhailova A S, Showalter K 2006 Physics Reports 425 79
[21] Yuan X J, Shao X, Liao H M, Ouyang Q 2009 Chin. Phys. Lett.26 024702
[22] Shang W L, Wang D Z 2007 Chin. Phys. Lett. 24 1992
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[1] Cross M C, Hohenberg P C 1993 Rev. Mod. Phys. 65 851
[2] Kytta K, Kaski K, Barrio R A 2007 Physica A 385 105
[3] Nie Q Y, Ren C S, Wang D Z, Li S Z, Zhang J L 2007 Appl. Phys. Lett. 90 221504
[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] Rogers J L, Pesch W, Brausch O, Schatz M F 2005 Phys. Rev. E 71 066214
[7] 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 物理学报 55 362]
[8] Besson T, Edwards W S, Tuckerman L S 1996 Phys. Rev. E 54 507
[9] Pesch M, Ackemann T, Lange W 2003 Phys. Rev. E 68 016209
[10] Westhoff E G, Herrero R, Ackemann T, Lange W 2003 Phys. Rev. E 67 025203
[11] Epstein T, Fineberg J 2006 Phys. Rev. E 73 055302
[12] Arbell H, Fineberg J 2002 Phys Rev E 65 036224
[13] Pampaloni E, Residori S, Soria S, Arecchi F T 1997 Phys. Rev. Lett. 78 1042
[14] Dong L F, Li S F, Fan W L, Pan Y Y 2009 Phys. Plasmas 16 122308
[15] Judd S L, Silber M 2000 Physica D 136 45
[16] Míguez D G, Dolnik M, Epstein I R, Muñuzuri A P 2011 Phys. Rev. E 84 046210
[17] Rogers J L, Schatz M F, Brausch O, Pesch W 2000 Phys. Rev. Lett. 85 4281
[18] 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 物理学报 62 010505]
[19] Wang W M, Liu H Y, Cai Y L, Li Z Q 2011 Chin. Phys. B 20 074702
[20] Mikhailova A S, Showalter K 2006 Physics Reports 425 79
[21] Yuan X J, Shao X, Liao H M, Ouyang Q 2009 Chin. Phys. Lett.26 024702
[22] Shang W L, Wang D Z 2007 Chin. Phys. Lett. 24 1992
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