Transition to chaos in Rayleigh-Bénard convection

The Boussinesq equations for the Rayleigh-Bénard problem have been solved by analytical and numerical methods. Two different sequences of Hopf bifurcations, leading from, stationary two-dimensional rolls to non-periodic motion, have been identified. For one of the sequences the first bifurcation results in transverse oscillations of the rolls. The next bifurcation gives quasiperiodic flow, and the sequence ends in chaotic motion after the third instability. The second route is characterized by waves in the periodic regime, travelling along the rolls. Thereafter two quasiperiodic regimes follow with two and three frequencies, respectively. Both types of sequences have been detected in the experiments reported by Gollub and Benson (1980). The regime of travelling waves is also analysed by a perturbation method.