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Spatial Patterns

  • Hermann Haken
Part of the Springer Series in Synergetics book series (SSSYN, volume 20)

Abstract

In the introduction we got acquainted with a number of systems in which spatial patterns evolve in a self-organized fashion. Such patterns may arise in continuous media such as fluids, or in cell assemblies in biological tissues. In this chapter we want to show how the methods introduced in the preceding chapters allow us to cope with the formation of such patterns. We note that such patterns need not be time independent but may be connected with oscillations or still more complicated time-dependent motions. Throughout this chapter we shall consider continuous media or problems in which a discrete medium, e.g., a cell assembly, can be well approximated by a continuum model.

Keywords

Nusselt Number Reaction Diffusion Equation Stochastic Partial Differential Equation Finite Geometry Preceding Chapter 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Bibliography

  1. H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. (Springer, Berlin, Heidelberg, New York 1983)Google Scholar
  2. O. A. Ladyzhenskaya: The Mathematical Theory of Viscous Incompressible Flow (Gordon and Breach, New York 1963)zbMATHGoogle Scholar
  3. D. D. Joseph: Stability of Fluid Motions I and II, Springer Tracts Natural Philos., Vols. 27, 28 (Springer, Berlin, Heidelberg, New York 1976)Google Scholar
  4. P. C. Fife: In Dynamics of Synergetic Systems, Springer Ser. Synergetics, Vol. 6, ed. by H. Haken (Springer, Berlin, Heidelberg, New York 1980) p. 97, with further referencesGoogle Scholar
  5. J. S. Turner: Adv. Chem. Phys. 29, 63 (1975)CrossRefGoogle Scholar
  6. J. W. Turner: Trans. NY Acad. Sci. 36, 800 (1974), Bull. Cl. Sci. Acad. Belg. 61, 293 (1975)Google Scholar
  7. Y. Schiffmann: Phys. Rep. 64, 87 (1980)ADSCrossRefMathSciNetGoogle Scholar
  8. H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. (Springer, Berlin, Heidelberg, New York 1983)Google Scholar
  9. H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. (Springer, Berlin, Heidelberg, New York 1983)Google Scholar
  10. H. Haken: Z. Phys. B21, 105 (1975)ADSGoogle Scholar
  11. H. Haken: Z. Phys. B22, 69 (1975); B23, 388 (1975)ADSGoogle Scholar
  12. For a different approach (for a more restricted class of problems) based on scaling see Y. Kuramoto, T. Tsusuki: Prog. Theor. Phys. 52, 1399 (1974)ADSCrossRefGoogle Scholar
  13. A. Wunderlin, H. Haken: Z. Phys. B21, 393 (1975)ADSGoogle Scholar
  14. H. Haken: Unpublished materialGoogle Scholar
  15. Equation (9.5.15) with A ≅ 0 was derived differently by J. Swift, P. C. Hohenberg: Phys. Rev. A15, 319 (1977)ADSGoogle Scholar
  16. H. Haken: Synergetics, Springer Ser. Synergetics, Vol. 1, 3rd. ed. (Springer, Berlin, Heidelberg, New York 1983)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • Hermann Haken
    • 1
  1. 1.Institut für Theoretische PhysikUniversität StuttgartStuttgart 80Fed. Rep. of Germany

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