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Codimension 2 Bifurcation in Binary Convection with Square Symmetry

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

Abstract

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].

Keywords

Normal Form Rayleigh Number Bifurcation Point Invariant Subspace Center Manifold 
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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References

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Copyright information

© Plenum Press, New York 1990

Authors and Affiliations

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

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