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An effective design method to produce stationary chemical reaction-diffusion patterns
1. | Centre De Recherche Paul Pascal, CNRS, Av. Schweitzer, 33600 Pessac, France |
2. | Institute of Chemistry, Laboratory of Nonlinear Chemical Dyanmics, Eötvös L. University, P.O. Box H-1518 Budapest 112, Hungary |
References:
[1] |
A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. Ser B, 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[2] |
J. D. Murray, "Mathematical Biology I-II,'' 3rd edition, Springer Verlag, Berlin, 2002.
doi: 10.1007/b98868. |
[3] |
H. Ikeda, M. Mimura and Y. Nishiura, Global bifurcation phenomena of travelling wave solutions for some bistable reaction-diffusion systems, Nonl. Anal. TMA, 13 (1989), 507.
doi: 10.1016/0362-546X(89)90061-8. |
[4] |
A. Hagberg and E. Meron, Complex patterns in reaction-diffusion systems: a tale of two front instabilities, Chaos, 4 (1994), 477.
doi: 10.1063/1.166047. |
[5] |
G. Nicolis and I. Prigogine, "Self Organization in Nonequilibrium Systems,'' Wiley, New York, 1977. |
[6] |
H. Meinhardt, "Models of Biological Pattern Formation,'' Academic Press, New York, 1982. |
[7] |
E. Ammelt, Y. A. Astrov and H. G. Purwins, Stripe Turing structures in a two-dimensional gas discharge system, Physical Review E , 55 (2001), 6731-6740.
doi: 10.1103/PhysRevE.55.6731. |
[8] |
L. A. Lugiato, C. Oldano and L. M. Narducci, Cooperative frequency locking and stationary spatial structures in lasers, Journal of the Optical Society of America B -Optical Physics, 5 (1988), 879-888.
doi: 10.1364/JOSAB.5.000879. |
[9] |
S. A. Levin, The problem of pattern and scale in ecology, Ecology, 73 (1992), 1943-1967.
doi: 10.2307/1941447. |
[10] |
F. Borgogno, P. D'Odorico and F. Laio, et al, Mathematical models of vegetation pattern formation in ecohydrology, Reviews of Geophysics, 47 (2009), RG1005.
doi: 10.1029/2007RG000256. |
[11] |
P. De Kepper, J. Boissonade and I. Szalai, From sustained oscillations to stationnary reaction-diffusion patterns, in "Chemomechanical Instabilities in Responsive Materials'' Eds. P. Borckmans, P. De Kepper, A. R. Khokhlov, S. Métens; Springer Series A, ISBN 978-90-481-2992-8(PB), (2009) 1-33.
doi: 10.1007/978-90-481-2993-5\_1. |
[12] |
V. Castets, E. Dulos, J. Boissonade and P. De Kepper, Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern, Phys. Rev. Lett., 64 (1990), 2953-2956.
doi: 10.1103/PhysRevLett.64.2953. |
[13] |
K. J. Lee, W. D. McCormick, Q. Ouyang and H. L. Swinney, Pattern formation by interacting chemical fronts, Science, 261 (1993), 192-194.
doi: 10.1126/science.261.5118.192. |
[14] |
E. Dulos, P. W. Davies, B. Rudovics and P. De Kepper, From quasi-2D to 3D Turing patterns in ramped systems, Physica D, 98 (1996), 53-66.
doi: 10.1016/0167-2789(96)00072-3. |
[15] |
P. De Kepper, J.-J. Perraud, B. Rudovics and E. Dulos, Experimental study of stationary Turing patterns and their interaction with traveling waves in a chemical system, Int. J. Bifurcation and Chaos, 6 (1994), 1077-1092.
doi: 10.1142/S0218127494000915. |
[16] |
K. J. Lee, W. D. McCormick, H. L. Swinney and J. E. Pearson, Experimental observation of self-replicating spots in a reaction-diffusion system, Nature, 369 (1994), 215-218.
doi: 10.1038/369215a0. |
[17] |
J. Horváth, I. Szalai and P. De Kepper, An experimental design method leading to chemical turing patterns, Science, 324 (2009), 772-775.
doi: 10.1126/science.1169973. |
[18] |
V. K. Vanag and I. R. Epstein, Pattern formation in a tunable medium: The Belousov-Zhabotinsky reaction in an aerosol ot microemulsion, Phys. Rev. Lett., 87 (2001), 228301.
doi: 10.1103/PhysRevLett.87.228301. |
[19] |
P. Érdi and J. Tóth, "Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models,'' Princeton University Press, Princeton, 1989. |
[20] |
J. Boissonade and P. De Kepper, Transitions from bistability to limit cycle oscillations. Theoretical analysis and experimental evidence in an open chemical system, J. Phys. Chem., 84 (1980), 501-506.
doi: 10.1021/j100442a009. |
[21] |
G. Dangelmayr and J. Guckenheimer, On a four parameter family of planar vector fields, Archive for Rational Mechanics and Analysis, 97 (1987), 321-352.
doi: 10.1007/BF00280410. |
[22] |
I. R. Epstein and J. A. Pojman, "An Introduction to Nonlinear Chemical Dynamics Oscillations, Waves, Patterns, and Chaos,'' Oxford University Press, New York, 1998. |
[23] |
P. Gray and S. K. Scott, "Chemical Oscillations and Instabilities,'' Clarendon Press, Oxford, 1990. |
[24] |
P. Blanchedeau and J. Boissonade, Resolving an experimental paradox in open spatial reactors: The role of spatial bistability, Phys. Rev. Lett., 81 (1998), 5007-5010.
doi: 10.1103/PhysRevLett.81.5007. |
[25] |
K. Benyaich, T. Erneux T, S. Metens, S, S. Villain and P. Borckmans, Spatio-temporal behaviors of a clock reaction in an open gel reactor, Chaos, 16 (2006), 037109.
doi: 10.1063/1.2219703 . |
[26] |
J. Boissonade, E. Dulos, F. Gauffre, M. N. Kuperman and P. De Kepper, Spatial bistability and waves in a reaction with acid autocatalysis, Faraday Discuss., 120 (2001), 353-361.
doi: 10.1039/b103240m. |
[27] |
P. Blanchedeau, J. Boissonade and P. De Kepper, Theoretical and experimental studies of spatial bistability in the chlorine-dioxide-iodide reaction, Physica D, 147 (2000), 283-299.
doi: 10.1016/S0167-2789(00)00169-X. |
[28] |
Z. Virányi, I. Szalai, J. Boissonade and P. De Kepper, Sustained Spatiotemporal Patterns in the Bromate-Sulfite Reaction, J. Phys. Chem. A, 111 (2007), 8090-8094.
doi: 10.1021/jp0723721. |
[29] |
I. Szalai and P. De Kepper, Spatial bistability, oscillations and excitability in the Landolt reaction, Phys. Chem. Chem. Phys., 8 (2006), 1105-1110.
doi: 10.1039/b515620c. |
[30] |
I. Szalai and P. De Kepper, Pattern formation in the ferrocyanide-iodate-sulfite reaction: The control of space scale separation, Chaos, 18 (2008), 026105.
doi: 10.1063/1.2912719. |
[31] |
J. Horváth, I. Szalai and P. De Kepper, Pattern formation in the Thiourea-Iodate-Sulfite system: spatial bistability, waves, and stationary patterns, Physica D, 239 (2010), 776-784.
doi: 10.1016/j.physd.2009.07.005. |
[32] |
I. Szalai and P. De Kepper, Patterns of the Ferrocyanide-Iodate-Sulfite reaction revisited: the role of immobilized carboxylic functions, J. Phys. Chem. A, 112 (2008), 783-786.
doi: 10.1021/jp711849m. |
[33] |
S. Ponce Dawson, M. V. D'Angelo and J. E. Pearson, Towards a global classification of excitable reaction-diffusion systems, Phys. Lett. A, 265 (2000), 346-352.
doi: 10.1016/S0375-9601(00)00008-6. |
[34] |
I. Lengyel and I. R. Epstein, A chemical approach to design Turing patterns in reaction-diffusion systems, Proc. Natl. Acad. Sci. USA, 89 (1992), 3977-3979.
doi: 10.1073/pnas.89.9.3977. |
[35] |
J. E. Pearson and W. Bruno, Pattern formation in an N+Q component reaction-diffusion system, Chaos, 2 (1992), 513-524.
doi: 10.1063/1.165893. |
[36] |
D. E. Strier and S. P. Dawson, Turing patterns inside cells, PLoS ONE, 2 (2007), 1053.
doi: 10.1371/journal.pone.0001053. |
[37] |
D. Horváth and Á. Tóth, Diffusion-driven front instabilities in the chlorite-tetrathionate reaction, J. Chem. Phys., 108 (1998), 1447.
doi: 10.1063/1.475355. |
[38] |
D. Horváth and Á. Tóth, Turing patterns in a single-step autocatalytic reaction, J. Chem. Soc. Farad. Trans., 93 (1997), 4301.
doi: 10.1039/a705895k. |
[39] |
I. Szalai and P. De Kepper, Turing patterns, spatial bistability, and front instabilities in a reaction-diffusion system, J. Phys. Chem. A, 108 (2004), 5315-5321.
doi: 10.1021/jp049168n. |
[40] |
I. Szalai N. Takács, J. Horváth and P. De Kepper, Sustained self-organizing pH patterns in hydrogen peroxide driven aqueous redox systems, (2011), submitted. |
[41] |
B. Rudovics, E. Barillot, P. W. Davies, E. Dulos, J. Boissonade and P. De Kepper, Experimental Studies and Quantitative Modeling of Turing Patterns in the (Chlorine Dioxide, Iodine, Malonic Acid) Reaction, J. Phys. Chem. A, 103 (1999), 1790-1800.
doi: 0.1021/jp983210v. |
[42] |
J. A. Vastano, J. E. Pearson, W. Horsthemke and H.L. Swinney, Turing patterns in an open reactor, J. Chem. Phys., 88 (1988), 6175.
doi: 10.1063/1.454456. |
show all references
References:
[1] |
A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. Ser B, 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[2] |
J. D. Murray, "Mathematical Biology I-II,'' 3rd edition, Springer Verlag, Berlin, 2002.
doi: 10.1007/b98868. |
[3] |
H. Ikeda, M. Mimura and Y. Nishiura, Global bifurcation phenomena of travelling wave solutions for some bistable reaction-diffusion systems, Nonl. Anal. TMA, 13 (1989), 507.
doi: 10.1016/0362-546X(89)90061-8. |
[4] |
A. Hagberg and E. Meron, Complex patterns in reaction-diffusion systems: a tale of two front instabilities, Chaos, 4 (1994), 477.
doi: 10.1063/1.166047. |
[5] |
G. Nicolis and I. Prigogine, "Self Organization in Nonequilibrium Systems,'' Wiley, New York, 1977. |
[6] |
H. Meinhardt, "Models of Biological Pattern Formation,'' Academic Press, New York, 1982. |
[7] |
E. Ammelt, Y. A. Astrov and H. G. Purwins, Stripe Turing structures in a two-dimensional gas discharge system, Physical Review E , 55 (2001), 6731-6740.
doi: 10.1103/PhysRevE.55.6731. |
[8] |
L. A. Lugiato, C. Oldano and L. M. Narducci, Cooperative frequency locking and stationary spatial structures in lasers, Journal of the Optical Society of America B -Optical Physics, 5 (1988), 879-888.
doi: 10.1364/JOSAB.5.000879. |
[9] |
S. A. Levin, The problem of pattern and scale in ecology, Ecology, 73 (1992), 1943-1967.
doi: 10.2307/1941447. |
[10] |
F. Borgogno, P. D'Odorico and F. Laio, et al, Mathematical models of vegetation pattern formation in ecohydrology, Reviews of Geophysics, 47 (2009), RG1005.
doi: 10.1029/2007RG000256. |
[11] |
P. De Kepper, J. Boissonade and I. Szalai, From sustained oscillations to stationnary reaction-diffusion patterns, in "Chemomechanical Instabilities in Responsive Materials'' Eds. P. Borckmans, P. De Kepper, A. R. Khokhlov, S. Métens; Springer Series A, ISBN 978-90-481-2992-8(PB), (2009) 1-33.
doi: 10.1007/978-90-481-2993-5\_1. |
[12] |
V. Castets, E. Dulos, J. Boissonade and P. De Kepper, Experimental evidence of a sustained standing Turing-type nonequilibrium chemical pattern, Phys. Rev. Lett., 64 (1990), 2953-2956.
doi: 10.1103/PhysRevLett.64.2953. |
[13] |
K. J. Lee, W. D. McCormick, Q. Ouyang and H. L. Swinney, Pattern formation by interacting chemical fronts, Science, 261 (1993), 192-194.
doi: 10.1126/science.261.5118.192. |
[14] |
E. Dulos, P. W. Davies, B. Rudovics and P. De Kepper, From quasi-2D to 3D Turing patterns in ramped systems, Physica D, 98 (1996), 53-66.
doi: 10.1016/0167-2789(96)00072-3. |
[15] |
P. De Kepper, J.-J. Perraud, B. Rudovics and E. Dulos, Experimental study of stationary Turing patterns and their interaction with traveling waves in a chemical system, Int. J. Bifurcation and Chaos, 6 (1994), 1077-1092.
doi: 10.1142/S0218127494000915. |
[16] |
K. J. Lee, W. D. McCormick, H. L. Swinney and J. E. Pearson, Experimental observation of self-replicating spots in a reaction-diffusion system, Nature, 369 (1994), 215-218.
doi: 10.1038/369215a0. |
[17] |
J. Horváth, I. Szalai and P. De Kepper, An experimental design method leading to chemical turing patterns, Science, 324 (2009), 772-775.
doi: 10.1126/science.1169973. |
[18] |
V. K. Vanag and I. R. Epstein, Pattern formation in a tunable medium: The Belousov-Zhabotinsky reaction in an aerosol ot microemulsion, Phys. Rev. Lett., 87 (2001), 228301.
doi: 10.1103/PhysRevLett.87.228301. |
[19] |
P. Érdi and J. Tóth, "Mathematical Models of Chemical Reactions: Theory and Applications of Deterministic and Stochastic Models,'' Princeton University Press, Princeton, 1989. |
[20] |
J. Boissonade and P. De Kepper, Transitions from bistability to limit cycle oscillations. Theoretical analysis and experimental evidence in an open chemical system, J. Phys. Chem., 84 (1980), 501-506.
doi: 10.1021/j100442a009. |
[21] |
G. Dangelmayr and J. Guckenheimer, On a four parameter family of planar vector fields, Archive for Rational Mechanics and Analysis, 97 (1987), 321-352.
doi: 10.1007/BF00280410. |
[22] |
I. R. Epstein and J. A. Pojman, "An Introduction to Nonlinear Chemical Dynamics Oscillations, Waves, Patterns, and Chaos,'' Oxford University Press, New York, 1998. |
[23] |
P. Gray and S. K. Scott, "Chemical Oscillations and Instabilities,'' Clarendon Press, Oxford, 1990. |
[24] |
P. Blanchedeau and J. Boissonade, Resolving an experimental paradox in open spatial reactors: The role of spatial bistability, Phys. Rev. Lett., 81 (1998), 5007-5010.
doi: 10.1103/PhysRevLett.81.5007. |
[25] |
K. Benyaich, T. Erneux T, S. Metens, S, S. Villain and P. Borckmans, Spatio-temporal behaviors of a clock reaction in an open gel reactor, Chaos, 16 (2006), 037109.
doi: 10.1063/1.2219703 . |
[26] |
J. Boissonade, E. Dulos, F. Gauffre, M. N. Kuperman and P. De Kepper, Spatial bistability and waves in a reaction with acid autocatalysis, Faraday Discuss., 120 (2001), 353-361.
doi: 10.1039/b103240m. |
[27] |
P. Blanchedeau, J. Boissonade and P. De Kepper, Theoretical and experimental studies of spatial bistability in the chlorine-dioxide-iodide reaction, Physica D, 147 (2000), 283-299.
doi: 10.1016/S0167-2789(00)00169-X. |
[28] |
Z. Virányi, I. Szalai, J. Boissonade and P. De Kepper, Sustained Spatiotemporal Patterns in the Bromate-Sulfite Reaction, J. Phys. Chem. A, 111 (2007), 8090-8094.
doi: 10.1021/jp0723721. |
[29] |
I. Szalai and P. De Kepper, Spatial bistability, oscillations and excitability in the Landolt reaction, Phys. Chem. Chem. Phys., 8 (2006), 1105-1110.
doi: 10.1039/b515620c. |
[30] |
I. Szalai and P. De Kepper, Pattern formation in the ferrocyanide-iodate-sulfite reaction: The control of space scale separation, Chaos, 18 (2008), 026105.
doi: 10.1063/1.2912719. |
[31] |
J. Horváth, I. Szalai and P. De Kepper, Pattern formation in the Thiourea-Iodate-Sulfite system: spatial bistability, waves, and stationary patterns, Physica D, 239 (2010), 776-784.
doi: 10.1016/j.physd.2009.07.005. |
[32] |
I. Szalai and P. De Kepper, Patterns of the Ferrocyanide-Iodate-Sulfite reaction revisited: the role of immobilized carboxylic functions, J. Phys. Chem. A, 112 (2008), 783-786.
doi: 10.1021/jp711849m. |
[33] |
S. Ponce Dawson, M. V. D'Angelo and J. E. Pearson, Towards a global classification of excitable reaction-diffusion systems, Phys. Lett. A, 265 (2000), 346-352.
doi: 10.1016/S0375-9601(00)00008-6. |
[34] |
I. Lengyel and I. R. Epstein, A chemical approach to design Turing patterns in reaction-diffusion systems, Proc. Natl. Acad. Sci. USA, 89 (1992), 3977-3979.
doi: 10.1073/pnas.89.9.3977. |
[35] |
J. E. Pearson and W. Bruno, Pattern formation in an N+Q component reaction-diffusion system, Chaos, 2 (1992), 513-524.
doi: 10.1063/1.165893. |
[36] |
D. E. Strier and S. P. Dawson, Turing patterns inside cells, PLoS ONE, 2 (2007), 1053.
doi: 10.1371/journal.pone.0001053. |
[37] |
D. Horváth and Á. Tóth, Diffusion-driven front instabilities in the chlorite-tetrathionate reaction, J. Chem. Phys., 108 (1998), 1447.
doi: 10.1063/1.475355. |
[38] |
D. Horváth and Á. Tóth, Turing patterns in a single-step autocatalytic reaction, J. Chem. Soc. Farad. Trans., 93 (1997), 4301.
doi: 10.1039/a705895k. |
[39] |
I. Szalai and P. De Kepper, Turing patterns, spatial bistability, and front instabilities in a reaction-diffusion system, J. Phys. Chem. A, 108 (2004), 5315-5321.
doi: 10.1021/jp049168n. |
[40] |
I. Szalai N. Takács, J. Horváth and P. De Kepper, Sustained self-organizing pH patterns in hydrogen peroxide driven aqueous redox systems, (2011), submitted. |
[41] |
B. Rudovics, E. Barillot, P. W. Davies, E. Dulos, J. Boissonade and P. De Kepper, Experimental Studies and Quantitative Modeling of Turing Patterns in the (Chlorine Dioxide, Iodine, Malonic Acid) Reaction, J. Phys. Chem. A, 103 (1999), 1790-1800.
doi: 0.1021/jp983210v. |
[42] |
J. A. Vastano, J. E. Pearson, W. Horsthemke and H.L. Swinney, Turing patterns in an open reactor, J. Chem. Phys., 88 (1988), 6175.
doi: 10.1063/1.454456. |
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