P. Colinet et Jc. Legros, ON THE HOPF-BIFURCATION OCCURRING IN THE 2-LAYER RAYLEIGH-BENARD CONVECTIVE INSTABILITY, Physics of fluids, 6(8), 1994, pp. 2631-2639
The oscillating convective structures appearing at the threshold of th
e two-layer Rayleigh-Benard instability are analyzed in the nonlinear
regime. By deriving the amplitude equations for left- and right-travel
ing waves from the infinite Prandtl number Boussinesq equations, it is
shown that one of these waves should generally appear, rather than st
anding waves, in sufficiently large cells. Numerical results show that
these waves have a limited range of existence, because a hysteretic t
ransition to stationary convection occurs when the Rayleigh number is
increased (via approach of a heteroclinic orbit for standing waves, an
d steady-state bifurcation for traveling waves). From numerical eviden
ce and by comparison with similar behaviors encountered in the one-lay
er two-component problem, it is inferred that the overall behavior is
typical of a codimension-2 Takens-Bogdanov bifurcation.