Codimension 2 Bifurcation in Binary Convection with Square Symmetry

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].