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用双层耦合的Lengel-Epstein模型, 研究了两个子系统的图灵模对斑图的影响,发现其波数比在斑图的形成和选择过程中起着重要作用.当波数比为1时,双层系统未能发生耦合,只能出现条纹和六边形斑图;当波数比处于1-√17 的范围时,两子系统发生耦合,图灵模之间发生共振相互作用,得到种类丰富的超点阵斑图,包括暗点、点-棒和复杂超六边、Ⅰ-型和Ⅱ-型白眼、类蜂窝和环状超六边等斑图;当波数比大于√17 , 系统选择的斑图类型不再变化,均为环状超六边斑图.数值模拟得到的条纹、六边形、超六边点阵、Ⅱ-型白眼斑图和类蜂窝斑图均已在介质阻挡放电系统实验中观察到. 另外,还得到了超点阵斑图的波数随两个扩散系数乘积DuDv的变化曲线,发现其随的DuDv增大而减小.The influence of Turing modes in two subsystems on pattern formation is investigated by using the two-layer coupled Lengel-Epstein model. It is found that the wave number ratio between two Turing modes play an important role in the pattern formation and pattern selection. When the wave number ratio is 1, no coupling behavior occurs between two subsystems and only stripe and hexagon patterns arise in system. If the wave number ratio lies in a range of 1-√17, a variety of superlattice patterns, such as dark-dot, bar-dot and complex super hexagons, I-type or II-type white-eye, honeycomb-like, and superhexagon of circle, are obtained due to the resonance interaction between the two Turing modes in the coupled systems. When the wave number ratio is greater than √17, the superhexagon of circle is always selected and unchanged. Some superlattice patterns above, including stripes, hexagons, super hexagon, Ⅱ-type white-eye, and honeycomb-like patterns, are observed experimentally in a dielectric barrier discharge (DBD) system. In addition, the curves for variation of hexagon pattern wave number with the increase of the product of two diffusion coefficients are obtained and it is found that the wave number becomes smaller with DuDv increasing.
[1] Barrio R A, Varea C, Aragon J L, Maini P K 1999 Math. Biol. 61 483
[2] Maini P K, Painter K J, Chau J 1999 Chem. Soc. Faraday Trans. 93 3601
[3] Epstein T, Fineberg J 2008 Phys. Rev. Lett. 100 134101
[4] Conway J M, Riecke H 2007 Phys. Rev. Lett. 99 218301
[5] Conway J M, Riecke H 2007 Phys. Rev. E 76 057202
[6] Hu H X, Li Q S, Ji L 2008 Phys. Chem. Chem. Phys. 10 438
[7] Dong L F, Xie W X, Zhao H T, Fan W L, He Y F, Xiao H 2009 Acta Phys. Sin. 58 4806 (in Chinese) [董丽芳、谢伟霞、赵海涛、范伟丽、贺亚峰、肖 红 2009 58 4806]
[8] Dong L F, Zhao H T, Xie W X, Wang H F, Liu W L, Fan W L, Xiao H 2008 Acta Phys. Sin. 57 5768 (in Chinese) [董丽芳、赵海涛、谢伟霞、王红芳、刘微粒、范伟丽、肖 红 2008 57 5768]
[9] Dong L F, Liu W L, Wang H F, He Y F, Fan W L, Gao R L 2007 Phys. Rev. E 76 046210
[10] He Y F, Dong L F, Liu W L, Wang H F, Zhao Z C, Fan W L 2007 Phys. Rev. E 76 017203
[11] 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
[12] Shao X J, Ma Y, Li Y X, Zhang G J 2010 Acta Phys. Sin. 59 8747 (in Chinese) [邵先军、马 跃、李娅西、张冠军 2010 59 8747]
[13] Xia G Q, Xue W H, Chen M L,Zhu Y, Zhu G Q 2011 Acta Phys. Sin. 60 015201 (in Chinese) [夏广庆、薛伟华、陈茂林、朱 雨、朱国强 2011 60 015201]
[14] Doelman A, Van Harten A 1995 Nonlinear Dynamics and Pattern Formation in the Natural Environment (Longman) p223
[15] Schenk C P, Schutz P, Bode M, Purwins H G 1998 Phys. Rev. E 57 6480
[16] Barrio R A, Varea C, Aragon J L, Maini P K 1999 Bull. Math. Biol. 61 483
[17] Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2002 Phys. Rev. Lett. 88 208303
[18] Zhou C X, Guo H Y, Ouyang Q 2002 Phys. Rev. E 65 036118
[19] Berenstein I, Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2005 J. Phys. Chem. A 109 5382
[20] Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2006 Chaos 16 037114
[21] Bachir M, Metens S, Borckmans P, Dewel G 2001 Europhys. Lett. 54 612
[22] Page K M, Maini P K, Monk N A M 2005 Physica D 202 95
[23] Lengyel I, Epstein I R 1991 Science 251 650
[24] Ouyang Q 2000 Pattern Formation in Reaction-Diffusion Systems (Shanghai: Shanghai Scientific & Technological Education Publishing House) p12 (in Chinese) [欧阳颀 2000 反应扩散系统中的斑图动力学(上海:上海科技教育出版社)第12页]
[25] Dong L F, Xiao H, Fan W L Zhao H T, Yue H 2010 IEEE Trans. on Plas. Sci. 38 2486
[26] Dong L F, Qi Y Y, Zhao Z C, Li Y H 2008 Plas. Sourc. Sci. Technol. 17 015015
[27] Francis F C 1974 Introduction to Plasma Physics(California: Plenum Press)p90
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[1] Barrio R A, Varea C, Aragon J L, Maini P K 1999 Math. Biol. 61 483
[2] Maini P K, Painter K J, Chau J 1999 Chem. Soc. Faraday Trans. 93 3601
[3] Epstein T, Fineberg J 2008 Phys. Rev. Lett. 100 134101
[4] Conway J M, Riecke H 2007 Phys. Rev. Lett. 99 218301
[5] Conway J M, Riecke H 2007 Phys. Rev. E 76 057202
[6] Hu H X, Li Q S, Ji L 2008 Phys. Chem. Chem. Phys. 10 438
[7] Dong L F, Xie W X, Zhao H T, Fan W L, He Y F, Xiao H 2009 Acta Phys. Sin. 58 4806 (in Chinese) [董丽芳、谢伟霞、赵海涛、范伟丽、贺亚峰、肖 红 2009 58 4806]
[8] Dong L F, Zhao H T, Xie W X, Wang H F, Liu W L, Fan W L, Xiao H 2008 Acta Phys. Sin. 57 5768 (in Chinese) [董丽芳、赵海涛、谢伟霞、王红芳、刘微粒、范伟丽、肖 红 2008 57 5768]
[9] Dong L F, Liu W L, Wang H F, He Y F, Fan W L, Gao R L 2007 Phys. Rev. E 76 046210
[10] He Y F, Dong L F, Liu W L, Wang H F, Zhao Z C, Fan W L 2007 Phys. Rev. E 76 017203
[11] 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
[12] Shao X J, Ma Y, Li Y X, Zhang G J 2010 Acta Phys. Sin. 59 8747 (in Chinese) [邵先军、马 跃、李娅西、张冠军 2010 59 8747]
[13] Xia G Q, Xue W H, Chen M L,Zhu Y, Zhu G Q 2011 Acta Phys. Sin. 60 015201 (in Chinese) [夏广庆、薛伟华、陈茂林、朱 雨、朱国强 2011 60 015201]
[14] Doelman A, Van Harten A 1995 Nonlinear Dynamics and Pattern Formation in the Natural Environment (Longman) p223
[15] Schenk C P, Schutz P, Bode M, Purwins H G 1998 Phys. Rev. E 57 6480
[16] Barrio R A, Varea C, Aragon J L, Maini P K 1999 Bull. Math. Biol. 61 483
[17] Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2002 Phys. Rev. Lett. 88 208303
[18] Zhou C X, Guo H Y, Ouyang Q 2002 Phys. Rev. E 65 036118
[19] Berenstein I, Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2005 J. Phys. Chem. A 109 5382
[20] Yang L F, Dolnik M, Zhabotinsky A M, Epstein I R 2006 Chaos 16 037114
[21] Bachir M, Metens S, Borckmans P, Dewel G 2001 Europhys. Lett. 54 612
[22] Page K M, Maini P K, Monk N A M 2005 Physica D 202 95
[23] Lengyel I, Epstein I R 1991 Science 251 650
[24] Ouyang Q 2000 Pattern Formation in Reaction-Diffusion Systems (Shanghai: Shanghai Scientific & Technological Education Publishing House) p12 (in Chinese) [欧阳颀 2000 反应扩散系统中的斑图动力学(上海:上海科技教育出版社)第12页]
[25] Dong L F, Xiao H, Fan W L Zhao H T, Yue H 2010 IEEE Trans. on Plas. Sci. 38 2486
[26] Dong L F, Qi Y Y, Zhao Z C, Li Y H 2008 Plas. Sourc. Sci. Technol. 17 015015
[27] Francis F C 1974 Introduction to Plasma Physics(California: Plenum Press)p90
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