Interactions of Stationary Modes in Systems with Two and Three Spatial Degrees of Freedom

  • M. Neveling
  • D. Lang
  • P. Haug
  • W. Güttinger
  • G. Dangelmayr
Part of the Springer Series in Synergetics book series (SSSYN, volume 37)


This paper deals with steady state mode-interactions in three-dimensional directional solidification and in two-dimensional thermohaline convection with variable aspect ratios. We show that a solidifying material can exhibit three coupled stationary cellular modes which occur at codimension-three bifurcation points and discuss the associated bifurcation diagrams. The thermohaline convection problem gives rise to a double tricritical point identifiable with a codimension-four bifurcation point. To obtain structurally stable configurations, non-Boussinesq effects have to be taken into account which give rise to temporally oscillating states. Mixed mode patterns and hysteretic behaviour between different unstable modes are modeled and simulated using cellular automata concepts.


Normal Form Rayleigh Number Hopf Bifurcation Cellular Automaton Bifurcation Diagram 
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  1. 1.
    D. Armbruster & G. Dangelmayr, Math. Proc. Cambr. Phil. Soc. 101 (1987), 167MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    D. Armbruster & M. Neveling, J. Nonequ. Thermodyn. (1987, in press)Google Scholar
  3. 3.
    G. Dangelmayr, in “Structural Stability in Physics”, W. Güttinger & H. Eikemeier (eds.), Springer 1979, pp. 84–103CrossRefGoogle Scholar
  4. 4.
    G. Dangelmayr & D. Armbruster, Contemporary Mathematics 56 (1986), 53MathSciNetCrossRefGoogle Scholar
  5. 5.
    R. Friedrich & H. Haken, this volumeGoogle Scholar
  6. 6.
    U. Frisch, B. Hasslacher & Y. Pomeau, Phys. Rev. Lett. 56 (1986), 1505ADSCrossRefGoogle Scholar
  7. 7.
    C. Geiger, W. Güttinger & P. Haug, in “Complex Systems — Operational Approaches”, H. Haken (ed.), Springer 1985, pp. 279–299CrossRefGoogle Scholar
  8. 8.
    M. Golubitsky & D. Schaeffer, “Singularities and Groups in Bifurcation Theory, vol. 1”, Springer 1984Google Scholar
  9. 9.
    M. Golubitsky, J. Marsden & D. Schaeffer, in “Partial Differential Equations and Dynamical Systems”, W. Fitzgibbon (ed.), Pitman Press 1984, pp. 181–210Google Scholar
  10. 10.
    J. Guckenheimer & P. Holmes, “Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields”, Springer 1983Google Scholar
  11. 11.
    P. Haug, Phys. Rev. A (1987, in press)Google Scholar
  12. 12.
    M. Kerzsberg, Phys. Rev. B 27 (1983), 6796ADSCrossRefGoogle Scholar
  13. 13.
    J.K. Platten & G. Chavepeyer, Adv. Chem. Phys. 32 (1977), 581Google Scholar
  14. 14.
    E. Knobloch & J. Guckenheimer, Phys. Rev. A 27 (1983), 408MathSciNetADSGoogle Scholar
  15. 15.
    J.S. Langer, Rev. Mod. Phys. 52 (1980), 1ADSCrossRefGoogle Scholar
  16. 16.
    D. Schaeffer & M. Golubitsky, Comm. Math. Phys. 69 (1979), 209MathSciNetADSzbMATHCrossRefGoogle Scholar
  17. 17.
    L.H. Ungar & R.A. Brown, Phys. Rev. B 29 (1984), 1367ADSCrossRefGoogle Scholar
  18. 18.
    S. Wolfram, “Theory and Applications of Cellular Automata”, World Scientific, Singapore, 1986zbMATHGoogle Scholar
  19. 19.
    M.J. Bennett, R.A. Brown & L.H. Ungar, this volumeGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • M. Neveling
  • D. Lang
    • 1
  • P. Haug
    • 1
  • W. Güttinger
    • 1
  • G. Dangelmayr
    • 1
  1. 1.Institute for Information SciencesUniversity of TübingenTübingenFed. Rep. of Germany

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