THE 2/1 STEADY/HOPF MODE INTERACTION IN THE 2-LAYER BENARD-PROBLEM

The interaction of Hopf and steady modes with wavenumber ratio 2:1 is investigated for a critical situation. Two different immiscible liquids lie in layers between horizontal walls and are heated from below. A situation with a pair of complex conjugate eigenvalues at wavenumber rr and a real eigenvalue at wavenumber 2 alpha is at criticality. Weakly nonlinear amplitude equations are derived for the interaction of these oscillatory and steady modes. The two modes generate a two-parameter bifurcation. The coefficients involved in the equations are determined numerically, based on the physical parameters of the system at criticality. Three obvious equilibrium solutions of the amplitude equations are the steady solution, the traveling waves and the mixed standing waves, The eigenvalues governing the stability of these solutions are found explicitly. Numerical results and bifurcation diagrams are given for the critical situation. The steady solution and the traveling wave solution are unstable, There is a region of stability for the standing wave solution, A new equilibrium solution, the asymmetric mixed mode, is found to be stable in a parameter range. Bifurcations from the standing wave solution and the asymmetric mixed mode are described.