Time-Dependent and Chaotic Behaviour in Systems with O(3)-Symmetry

  • R. Friedrich
  • H. Haken
Conference paper
Part of the Springer Series in Synergetics book series (SSSYN, volume 37)


It is by now well known that chaotic temporal behaviour may be generated by completely deterministic evolution laws. The main theoretical interest in chaotic phenomena has been devoted to the examination of the stochastic properties of low dimensional dynamical systems, although experimental confirmations of certain theoretical predictions have been mainly pursued in spatially extended systems, e.g. in lasers, hydrodynamic flows, and chemical reactions. In such systems low dimensional temporal chaotic behaviour evidently has to be strongly interrelated with coherent spatial patterns. The success of synergetics in the explanation of spontaneous pattern formation in selforganizing systems allows a straightforward treatment of time-dependent and chaotic behaviour in spatially extended systems of finite size which is caused by the nonlinear interaction of several order parameters, each connected with a coherent spatial pattern. In these cases a complete examination of the interrelations between temporal disorder and spatial coherence becomes possible. Spatial patterns are naturally connected to the symmetries of a system. Therefore, it can be expected that quite different systems, which, however, possess the same spatial symmetries, may show comparable behaviour. To demonstrate this we shall consider instabilities in systems with 0(3)-symmetry. For the special case of mode interaction between two groups of modes belonging to two different irreducible representations of the 0(3)-group with ℓ=1 and ℓ=2 we shall derive the order parameter equations.


Rayleigh Number Convection Cell Unstable Mode Critical Rayleigh Number Phase Point 
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  1. [1]
    H. Haken, Synergetics. An Introduction, third edition (Springer-Verlag, Berlin, Heidelberg, New York 1983)zbMATHGoogle Scholar
  2. [2]
    H. Haken, Advanced Synergetics (Springer-Verlag, Berlin, Heidelberg, New York 1983)zbMATHGoogle Scholar
  3. [3]
    A. Wunderlin und H. Haken, Z. Physik. B 44, 135 (1981)MathSciNetADSGoogle Scholar
  4. [4]
    M. Hammermesh, Group Theory and Its Application to Physical Problems (Addison- Wesley, Reading, Mass., 1962)Google Scholar
  5. [5]
    D.H. Sattinger, J. Math. Phys. 19, 1720 (1978)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. [6]
    D.H. Sattinger, Group Theoretic Methods in Bifurcation Theory Lecture Notes in Math., 762 (Springer-Verlag, Berlin, Heidelberg, New York 1979)zbMATHGoogle Scholar
  7. [7]
    R. Friedrich und H. Haken, in Complex Systems-Operational Approaches, Proceedings of the International Symposium on Synergetics 1985, Schloß Elmau, edited by H. Haken (Springer-Verlag, Berlin, Heidelberg, New York 1985)Google Scholar
  8. [8]
    R. Friedrich und H. Haken, Phys. Rev. A 34, 2100 (1986)ADSGoogle Scholar
  9. [9]
    F.H. Busse, J. Fluid Mech. 72, 65 (1975)ADSCrossRefGoogle Scholar
  10. [10]
    F.H. Busse and N. Riahi, J. Fluid Mech. 123, 283 (1981)MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    M. Golubitsky and D. Schaeffer, Commun. Pure a. Applied Math. 35, 81 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  12. [12]
    P. Chossat: SIAM J. Appl. Math. 37, 624 (1979)MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. [13]
    F.H. Busse, Rep. Progr. Phys. 41. 1929 (1978)ADSCrossRefGoogle Scholar
  14. [14]
    S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Clarendon, Oxford 1961)zbMATHGoogle Scholar
  15. [15]
    L. P. Silnikov, Sov. Math. Dokl. 6, 163 (1965)Google Scholar
  16. [15a]
    L. P. Silnikov Sov. Math. Dokl. 8, 54 (1967)MathSciNetGoogle Scholar
  17. [15b]
    L. P. Silnikov Math. USSR SB 10, 91 (1970)CrossRefGoogle Scholar
  18. [16]
    J. Guckenheimer und P. Holmes, Nonlinear Oscillations, Dynamical Systems and Bifurcation of Vector Fields (Springer-Verlag, Berlin, Heidelberg, New York 1983)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1987

Authors and Affiliations

  • R. Friedrich
    • 1
  • H. Haken
    • 1
  1. 1.Institut für theoretische Physik und SynergetikUniversität StuttgartStuttgart 80Fed. Rep. of Germany

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