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Chin. Phys. B, 2010, Vol. 19(10): 100509    DOI: 10.1088/1674-1056/19/10/100509
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Stationary patterns in a discrete bistable reaction–diffusion system: mode analysis

Zou Wei(邹为)a)b) and Zhan Meng(占萌) a)†ger
a Wuhan Institute of Physics and Mathematics, the Chinese Academy of Sciences, Wuhan 430071, China; b Graduate School of the Chinese Academy of Sciences, Beijing 100049, China
Abstract  This paper theoretically analyses and studies stationary patterns in diffusively coupled bistable elements. Since these stationary patterns consist of two types of stationary mode structure: kink and pulse, a mode analysis method is proposed to approximate the solutions of these localized basic modes and to analyse their stabilities. Using this method, it reconstructs the whole stationary patterns. The cellular mode structures (kink and pulse) in bistable media fundamentally differ from stationary patterns in monostable media showing spatial periodicity induced by a diffusive Turing bifurcation.
Keywords:  discrete reaction–diffusion system      stationary patterns      bistable      mode analysis  
Received:  18 January 2010      Revised:  09 April 2010      Accepted manuscript online: 
PACS:  02.30.Oz (Bifurcation theory)  
  05.45.-a (Nonlinear dynamics and chaos)  
Fund: Project partially supported by the Outstanding Oversea Scholar Foundation of the Chinese Academy of Sciences (Bairenjihua) and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

Cite this article: 

Zou Wei(邹为) and Zhan Meng(占萌) Stationary patterns in a discrete bistable reaction–diffusion system: mode analysis 2010 Chin. Phys. B 19 100509

[1] Cross M C and Hohenberg P C 1993 Rev. Mod. Phys. 65 851
[2] Kapral R and Showalter K 1995 Chemical Waves and Patterns (Dordrecht: Kluver Academic Publishers)
[3] Epstein I R and Pojman J A 1998 An Introduction to Nonlinear Chemical Dynamics: Oscillations, Waves, Patterns, and Chaos (New York: Oxford University Press)
[4] Kuramoto Y 1984 Chemical Oscillations, Waves, and Turbulence (Berlin: Springer)
[5] Field R J and Burger M 1985 Oscillations and Travelling Waves in Chemical Systems (New York: Wiley-Interscience)
[6] Keener J and Sneyd J 1998 Mathematical Physiology (New York: Springer-Verlag) Chapter 9
[7] Murray J D 2003 Mathematical Biology 3rd edn. (Berlin: Springer)
[8] Gao Z Y and Lu Q S 2007 Chin. Phys. 16 2479
[9] Wang B Y, Xing Z C and Xu W 2009 Acta Phys. Sin. 58 6590 (in Chinese)
[10] Turing A M 1952 Philos. Trans. Roy. Soc. London Ser. B bf 237 37
[11] Castets V, Dulos E, Boissonade J and De Kepper P 1990 Phys. Rev. Lett. 64 2953
[12] De Kepper P, Castets V, Dulos E and Boissonade J 1991 Physica D 49 161
[13] Langyel I and Epstein I R 1990 Science bf 251 650
[14] Ouyang Q and Swinney H L 1991 Nature 352 610
[15] Kladko K, Mitkov I and Bishop A R 2000 Phys. Rev. Lett. bf 84 4505
[16] Laplante J P and Erneux T 1992 J. Phys. Chem. 96 4931
[17] Erneux T and Nicolis G 1993 Physica D 67 237
[18] F'ath G 1998 Physica D 116 176
[19] Keener J P 1987 SIAM J. Appl. Math. 47 556
[20] Comte J C, Morfu S and Marqui'e P 2000 Phys. Rev. E bf 64 027102
[21] Mitkov I, Kladko K and Pearson J E 1998 Phys. Rev. Lett. bf 81 5453
[22] Preez-Munuzuri V, Perez-Villar V and Chua L O 1992 Int. J. Bifur. Chaos 2 403
[23] MacKay R S and Sepulchre J A 1995 Physica D bf 82 243
[24] Nekorkin V I and Makarov V A 1995 Phys. Rev. Lett. 74 4819
[25] Mu nuzhi A P and Chua L O 1997 Int. J. Bifur. Chaos bf 7 2807 % locally linearly coupled
[26] Nizhnik L P, Nizhnik I L and Hasler M 2002 Int. J. Bifur. Chaos 12 261
[27] Nagumo J, Arimoto S and Yoshisawa S 1962 Proc. IRE 50 2061
[28] Paz'o D and P'erez-Mu nuzuri V 2001 Phys. Rev. E bf 64 065203(R)
[29] Paz'o D and P'erez-Mu nuzuri V 2003 Chaos 13 812
[30] Parker T S and Chua L O 1989 Practical Numerical Algorithms for Chaotic Systems (New York: Springer-Verlag)
[31] Liu W Q, Wu Y, Zou W, Xiao J H and Zhan M 2007 Phys. Rev. E 76 036215 endfootnotesize
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