The butterfly singularity in double-diffusive convection

The two-dimensional, two-component Benard problem in a finite box is analyzed. A Lyapunov-Schmidt reduction for the stationary solutions of the Boussinesq equations to an amplitude equation is performed up to fifth order and the existence and non-degeneracy conditions for a butterfly singularity in the sense of imperfect bifurcation theory are established. It is shown that the fifth order term is stabilizing. The butterfly singularity justifies the tricritical point of double-diffusive convection known from a non-rigorous five-mode expansion. For non-flux-boundary conditions at the side walls and homogeneous upper and lower boundaries Non-Boussinesq-effects, although they break the Z(2) x Z(2) symmetry inherent in the Boussinesq approximation, do not produce unfolding terms for the butterfly singularity. However the addition of inhomogeneous heat-ing, lateral heat flux or an inclined box can act as imperfections. All topologically different bifurcation diagrams for the universal unfolding of the butterfly singularity are determined. New hystereses phenomena are predicted.