| Summary: | The focus of this thesis is the influence of rotation or a magnetic field on the dynamics of pattern formation by thermal convection in a plane layer (the Rayleigh-Bénard problem). Chapter 1 of the thesis provides a short introduction to the theory of equivalent steady-state and Hopf bifurcations on doubly-periodic lattices in the plane. Known results and new observations for Hopf bifurcations on non-rotating and rotating square lattices are summarised. There is also a detailed presentation of new stability results for the Hopf bifurcation on a square superlattice. In Chapter 2 the effect on convection of an imposed vertical magnetic field is discussed. This leads to a detailed analysis of the simplest three-dimensional Hopf/steady-state mode interaction, where the ratio of the critical wavenumbers for oscillatory and steady convection is 1 : √2. Chapters 3, 4 and 5 contain results on pattern selection in rotating Rayleigh-Bénard convection, specifically for low Prandtl number fields. Calculations in the regime where the onset of convection is oscillatory are performed to determine possible forms of convection at onset. The transition (with increasing Prandtl number) from patterns involving oscillatory rolls to those involving steady rolls is influenced by a second 1 : √2 mode interaction. The resulting amplitude equations contain heteroclinic cycles and bursting behaviour in marked contrast to the magnetoconvection case. Finally, a new asymptotic regime of low Prandtl number and rapid rotation is explored. The relevance of this regime to various sets of experimental results is discussed.
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