Interactions of Stationary Modes in Systems with Two and Three Spatial Degrees of Freedom
This paper deals with steady state mode-interactions in three-dimensional directional solidification and in two-dimensional thermohaline convection with variable aspect ratios. We show that a solidifying material can exhibit three coupled stationary cellular modes which occur at codimension-three bifurcation points and discuss the associated bifurcation diagrams. The thermohaline convection problem gives rise to a double tricritical point identifiable with a codimension-four bifurcation point. To obtain structurally stable configurations, non-Boussinesq effects have to be taken into account which give rise to temporally oscillating states. Mixed mode patterns and hysteretic behaviour between different unstable modes are modeled and simulated using cellular automata concepts.
KeywordsNormal Form Rayleigh Number Hopf Bifurcation Cellular Automaton Bifurcation Diagram
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