Codimension 2 Bifurcation in Binary Convection with Square Symmetry

  • Dieter Armbruster
Part of the NATO ASI Series book series (NSSB, volume 225)


Codimension two bifurcations with double zero eigenvalues (Takens-Bogdanov bifurcations) and square symmetries have recently been shown to exhibit chaotic behaviour [AGK]. Unlike chaotic solutions in other Codimension 2 situations [GH] the chaotic behaviour here is found in a “large” region in parameter space — a wedge with positive angle originating at the singularity. Hence Codimension 2 singularities with square symmetry should be particularly attractive from an experimental point of view. The intuitive reason for the appearance of these “large” chaotic regions is the existence of a nonintegrable Hamiltonian with two degrees of freedom in a scaling limit of the D 4 — symmetric Takens-Bogdanov normal form. This nonintegrable Hamiltonian system creates stochastic regions in phase space which, for a certain range of unfolding parameters are not quenched to the fixed points when one takes the dissipation into account. Section 2 gives a short overview on the D 4-symmetric Takens-Bogdanov normal form and its unfolding. We analyse the phase space for typical parameters and discuss chaotic and nonchaotic solutions. Section 3 deduces the physical meaning of these solutions in real space as opposed to phase space. In Section 4 we calculate the nonlinear terms in the normal form for the specific system of two sets of orthogonal convection rolls in a mixture of He 3/He 4 and analyse the resulting normal form. Section 5 deals with the implication of this theory for some experiments by Moses and Steinberg [MS] and Le Gal et al. [LeG].


Normal Form Rayleigh Number Bifurcation Point Invariant Subspace Center Manifold 


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  1. [AR]
    G.Ahlers, I.Rehberg: Convection in a binary mixture heated from below, Phys. Rev. Lett. 56 1373 (1986)MathSciNetADSCrossRefGoogle Scholar
  2. [AGK]
    D.Armbruster, J.Guckenheimer, Seunghwan Kim: Chaotic dynamics in systems with square symmetry, Physics Letters A 140 416 (1989)MathSciNetADSCrossRefGoogle Scholar
  3. [DK]
    G.Dangelmayr, E.Knobloch: The Takens-Bogdanov bifurcation with 0(2)-symmetry, Phil. Trans. R. Soc. London 322, 243 (1987)MathSciNetADSMATHCrossRefGoogle Scholar
  4. [GH]
    J.Guckenheimer, P.J.Holmes: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer New York (1986)Google Scholar
  5. [LeG]
    P.Le Gal, A.Pocheau, V.Croquette: Square versus roll pattern at convective threshold, Phys. Rev. Lett. 54, 2501 (1985)ADSCrossRefGoogle Scholar
  6. [Los]
    J.E.Los: Non-normally hyperbolic invariant curves for maps in R 3 and doubling bifurcation, Nonlinearity 2, 149 (1989)MathSciNetADSMATHCrossRefGoogle Scholar
  7. [MS]
    E.Moses, V.Steinberg: Competing patterns in a convective binary mixture, Phys. Rev. Lett. 57, 2018 (1986)ADSCrossRefGoogle Scholar
  8. [SG]
    F.Simonelli, J.P.Gollub: Surface wave mode interactions: Effects of symmetry and degeneracy, J. Fluid Mech. 199, 471 (1989)MathSciNetADSCrossRefGoogle Scholar
  9. [Wa]
    R.W.Waiden, P.Kolodner, A.Passner, C.M.Surko: Travelling waves and chaos in convection in binary fluid mixtures, Phys. Rev. Lett. 55, 496 (1985)ADSCrossRefGoogle Scholar

Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

  • Dieter Armbruster
    • 1
  1. 1.Institut für InformationsverarbeitungUniversität TübingenW. Germany

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