UBC Theses and Dissertations
Hopf bifurcations in magnetoconvection in the presence of sidewalls Zangeneh, Hamid
We study multiple Hopf bifurcations that occur in a model of a layer of a viscous, electrically conducting fluid that is heated from below in the presence of a. magnetic field. We assume that the fluid flow is two-dimensional, and consider the effects of sidewalls with stress-free boundary conditions. Our model partial differential equations together with the boundary conditions have two reflection symmetries. We use center manifold theory to reduce the partial differential equations to a two-parameter family of four-dimensional ordinary differential equations. We show that two different normal forms are appropriate-ate, depending on the sizes of certain magnetoconvection parameters for large aspect ratios. AVe denote the two normal forms by "Case I" and "Case II". In both cases we prove the primary Hopf bifurcation of standing wave (SW) solutions, and we prove the existence of secondary Hopf bifurcations of invariant tori from the SW solutions. We prove that the tori persist in 'wedges' in the parametric plane. In Case II we show that there are also secondary Bogdanov-Takens bifurcation points. Using this, we show there are additional secondary and tertiary bifurcations of periodic solutions and invariant tori, and also argue that generically, there exist transversal homoclinic and heteroclinc points, and consequently open regions of parameter space that correspond to chaos of chaotic regions, and show the existence of quasiperiodic saddle-node bifurcations of invariant tori. Also, we show that in this case the system is a small perturbation of a system with the symmetries of the square, as the aspect ratio approaches infinity.
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