On the Hopf bifurcation occurring in the two-layer Rayleigh–Bénard convective instability

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Abstract
The oscillating convective structures appearing at the threshold of the two-layer Rayleigh–Bénard instability are analyzed in the nonlinear regime. By deriving the amplitude equations for left- and right-traveling waves from the infinite Prandtl number Boussinesq equations, it is shown that one of these waves should generally appear, rather than standing waves, in sufficiently large cells. Numerical results show that these waves have a limited range of existence, because a hysteretic transition to stationary convection occurs when the Rayleigh number is increased (via approach of a heteroclinic orbit for standing waves, and steady-state bifurcation for traveling waves). From numerical evidence and by comparison with similar behaviors encountered in the one-layer two-component problem, it is inferred that the overall behavior is typical of a codimension-2 Takens–Bogdanov bifurcation.