Skip to main content
Log in

The search for Turing structures

  • Articles
  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

Albeit their prediction by Turing, chemical periodic stationary patterns under far from equilibrium conditions, which are considered as prototypes of dissipative structures, have not yet been observed in a clear-cut way. We here assess the situation and discuss the perturbative influence of convection on their formation and selection mechanisms.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. H. Benard,Rev. Gen. Sci. Pure Appl. 11:1261 (1900).

    Google Scholar 

  2. G. I. Taylor,Phil. Trans. R. Soc. Lond. A 223:289 (1923).

    Google Scholar 

  3. T. Tanaka, S. Sun, Y. Hirokawa, S. Katayama, J. Kucera, Y. Hirose, and T. Amiya,Nature 325:796 (1987); J. S. Preston, H. M. van Driel, and J. E. Sipe,Phys. Rev. Lett. 58:289 (1987).

    Google Scholar 

  4. A. T. Winfree,J. Chem. Ed. 61:661 (1984).

    Google Scholar 

  5. P. Glansdorff and I. Prigogine,Thermodynamic Theory of Structures, Stability and Fluctuations (Wiley, New York, 1971).

    Google Scholar 

  6. R. J. Field and M. Burger (eds.),Oscillations and Traveling Waves in Chemical Systems (Wiley, New York, 1984).

    Google Scholar 

  7. R. Imbihl, M. P. Cox, G. Ertl, H. Müller, and W. Brenig,J. Chem. Phys. 83:1578 (1985); L. F. Razon and R. A. Schmitz,Catal. Rev. Sci. Eng. 28:89 (1986).

    Google Scholar 

  8. A. M. Turing,Phil. Trans. R. Soc. Lond. B 237:37 (1952).

    Google Scholar 

  9. G. Nicolis and I. Prigogine,Self Organization in Nonequilibrium Systems (Wiley, New York, 1977); D. Walgraef, G. Dewel, and P. Borckmans,Adv. Chem. Phys. 49:311 (1982).

    Google Scholar 

  10. P. K. Becker and R. Field,J. Phys. Chem. 89:118 (1985).

    Google Scholar 

  11. A. Boiteux and B. Hess,Ber. Bunsenges. Phys. Chem. 84:392 (1980).

    Google Scholar 

  12. A. M. Zhabothinsky and A. N. Zaikin,J. Theor. Biol. 40:45 (1973).

    Google Scholar 

  13. K. Showalter,J. Chem. Phys. 73:3735 (1980).

    Google Scholar 

  14. M. Orban,J. Am. Chem. Soc. 102:4311 (1980).

    Google Scholar 

  15. M. Marek, inModelling of Patterns in Space and Time, W. Jäger and J. D. Murray, eds. (Springer, Berlin, 1984), p. 214; K. Bar-Eli and W. Geiseler,J. Phys. Chem. 85:3461 (1981).

    Google Scholar 

  16. P. Möckel,Naturwissenschaften 64:224 (1977).

    Google Scholar 

  17. M. Kagan, A. Levi, and D. Avnir,Naturwissenschaften 69:548 (1982);70:144 (1983).

    Google Scholar 

  18. M. Gimenez and J.-C. Micheau,Naturwissenschaften 70:90 (1983).

    Google Scholar 

  19. D. Avnir and M. Kagan,Nature 307:717 (1984).

    Google Scholar 

  20. J.-C. Micheau, M. Gimenez, P. Borckmans, and G. Dewel,Nature 305:43 (1983).

    Google Scholar 

  21. S. Müller and Th. Plesser, inModelling of Patterns in Space and Time, W. Jäger and J. D. Murray, eds. (Springer, Berlin, 1984); S. Müller, Th. Plesser, and B. Hess, inTemporal Order, Synergetics No. 29, L. Rensing and N. Jaeger, eds. (Springer, Berlin, 1985).

    Google Scholar 

  22. I. R. Epstein, M. Morgan, C. Steel, and O. Valdes-Aguilar,J. Phys. Chem. 87:3955 (1983).

    Google Scholar 

  23. M. Gimenez, J.-C. Micheau, D. Lavabre, and J.-P. Laplante,J. Phys. Chem. 89:1 (1985).

    Google Scholar 

  24. V. Croquette, P. Le Gal, A. Pocheau, and R. Guglielmetti,Europhys. Lett. 1:393 (1986).

    Google Scholar 

  25. J. Bdzil and H. Frisch,Phys. Fluids 14:476, 1077 (1971).

    Google Scholar 

  26. J. K. Platten and J.-C. Legros,Convection in Liquids (Springer, Berlin, 1984), Part D.

    Google Scholar 

  27. G. Dewel, D. Walgraef, and P. Borckmans,Proc. Natl. Acad. Sci. USA 80:6429 (1983).

    Google Scholar 

  28. T. J. McDougall,J. Fluid Mech. 126:379 (1983).

    Google Scholar 

  29. M. Kagan and D. Avnir, inInterfacial Phenomena: Proceedings NATO ASI and EPS Summerschool and Conference on Physicochemical Hydrodynamics, Spain 1986, M. G. Velarde and B. Nichols, eds. (Plenum Press, New York, in press).

    Google Scholar 

  30. V. Steinberg and E. Moses, inPatterns, Defects and Microstructures in Nonequilibrium Systems: NATO ASI (Series E: Applied Sciences No. 121), D. Walgraef, ed. (Martinus Nijhoff, Dordrecht, 1987); C.M. Surko, P. Kolodner, A. Passner, and R. W. Waiden,Physica D 23:220 (1986).

    Google Scholar 

  31. E. Moses, J. Fineberg, and V. Steinberg,Phys. Rev. A 35:2757 (1987); R. Heinrichs, G. Ahlers, and D. S. Cannell,Phys. Rev. A 35:2761 (1987).

    Google Scholar 

  32. G. I. Taylor,Proc. R. Soc. Lond. A 219:186 (1953);225:473 (1954).

    Google Scholar 

  33. F. Sagues and W. Horsthemke,Phys. Rev. A 34:4136 (1986).

    Google Scholar 

  34. J. P. Gollub and T. H. Solomon, inChaos and Related Nonlinear Phenomena: Where Do We Go from Here?, I. Procaccia, ed. (Plenum Press, New York, in press).

  35. G. Dewel, P. Borckmans, and D. Walgraef, inChemical Instabilities: NATO ASI (Series C: Mathematical and Physical Sciences No. 120), G. Nicolis and F. Baras, eds. (Reidel, Dordrecht, 1984).

    Google Scholar 

  36. M. Marek and E. Svobodova,Biophys. Chem. 3:263 (1975).

    Google Scholar 

  37. E. A. Spiegel and S. Zalesky,Phys. Lett. 106A:335 (1984).

    Google Scholar 

  38. S. Kai, S. Müller, and J. Ross,J. Chem. Phys. 76:1392 (1982).

    Google Scholar 

  39. G. Martin,Phys. Rev. B 21:2122 (1980).

    Google Scholar 

  40. K. Krishan,Nature 287:420 (1980).

    Google Scholar 

  41. D. J. Scalapino and B. A. Huberman,Phys. Rev. Lett. 39:1365 (1977).

    Google Scholar 

  42. K. F. Berggren and B. A. Huberman,Phys. Rev. B 18:3369 (1978).

    Google Scholar 

  43. D. Walgraef and E. C. Aifantis,J. Appl. Phys. 58:688 (1985).

    Google Scholar 

  44. L. A. Lugiato and R. Lefever,Phys. Rev. Lett. 58:2209 (1987).

    Google Scholar 

  45. L. W. Anacker and R. Kopelman,Phys. Rev. Lett. 58:289 (1987).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Borckmans, P., Dewel, G., Walgraef, D. et al. The search for Turing structures. J Stat Phys 48, 1031–1044 (1987). https://doi.org/10.1007/BF01009531

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01009531

Key words

Navigation