Abstract
Spatiotemporal structures arising in two identical cells, which are governed by higher autocatalator kinetics and coupled via diffusive interchange of autocatalyst, are discussed. The stability of the unique homogeneous steady state is obtained by the linearized theory. A necessary condition for bifurcations in spatially non-uniform solutions in uncoupled and coupled systems is given. Further information about Turing pattern solutions near bifurcation points is obtained by weakly nonlinear theory. Finally, the stability of equilibrium points of the amplitude equation is discussed by weakly nonlinear theory, with the bifurcation branches of the weakly coupled system.
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References
Wollkind David J, Stephenson Laura E. Chemical Turing pattern formation analyses:comparison of theory with experiment[J]. SIAM J Appl Math, 2000, 61(2):387–431.
Hill R, Merkin J H. Patten formation in a coupled cubic autocatalator system[J]. IMA J Appl Math, 1994, 53(3):265–322.
Hill R, Merkin J H, Needham D J. Stable pattern and standing wave formation in a simple isothermal cubic autocatalytic reaction scheme[J]. Journal of Engineering Mathematics, 1995, 29(3):413–436.
Hill R, Merkin J H. The effects of coupling on pattern formation a simple autocatalytic system[J]. IMA J Appl Math, 1995, 54(3):257–281.
Metcalf M J, Merkin J H. Reaction diffusion waves in coupled isothermal autocatalytic chemical systems[J]. IMA Journal of Applied Mathematics, 1993, 51(3):269–298.
Collier S M, Merkin J H, Scott S K. Multistability, oscillations and travelling waves in a product-feedback autocatalator model. II The initiation and propagation of travelling waves[J]. Phil Trans R Soc Lond A, 1994, 349(1691):389–415.
Merkin J H, Needham D J, Scott S K. Coupled reaction-diffusion waves in an isothermal autocatalytic chemical system[J]. IMA J Appl Math, 1993, 50(1):43–76.
Hovvath Dezso, Toth Agota. Turing patterns in a single-step autocatalytic reaction[J]. J Chem Soc Faraday Trans, 1997, 93(24):4301–4303.
Merkin J H, Sevcikova H, Snita D. The effect of an electric field on the local stoichiometry of front waves in an ionic chemical system[J]. IMA J Appl Math, 2000, 64(2):157–188.
Merkin J H, Sevcikova H. The effects of a complexing agent on travelling waves in autocatalytic systems with applied electric fields[J]. IMA J Appl Math, 2005, 70(4):527–549.
Metcalf M J, Merkin J H, Scott S K. Oscillating wave fronts in isothermal chemical systems with arbitrary powers of autocatalysis[J]. Proc R Soc Lond A, 1994, 447(1929):155–174.
Hubbard M E, Leach J A, Wei J C. Pattern formation in a 2D simple chemical system with general orders of autocatalysis and decay[J]. IMA J Appl Math, 2005, 70(6):723–747.
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Communicated by LI Ji-bin
Project supported by the National Natural Science Foundation of China (No. 60574075) Corresponding author LIU San-yang, Professor, Doctor
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Zhang, L., Liu, Sy. Bifurcation and pattern formation in a coupled higher autocatalator reaction diffusion system. Appl Math Mech 28, 1235–1248 (2007). https://doi.org/10.1007/s10483-007-0912-1
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DOI: https://doi.org/10.1007/s10483-007-0912-1