This talk discusses two PDE dynamical systems: a Navier-Stokes problem, and a reaction-diffusion problem.
The first dynamical system is a Marangoni-flow problem. A cylindrical bar of silicon material is subjected to external heat, and is melting. Surface tension induces a flow in the melt. The problem has free surfaces. The outer surface of the cylinder is allowed to oscillate in time. Based on numerical analysis of C. Menke, we show a period-doubling scenario.
The second problem arises in image processing. The aims are twofold: Perturbations in the image must be smoothed out, and edges should be preserved. These aims can be pursued by a modal analysis of a reaction-diffusion problem. It turns out that stabilization of certain modes results in a problem with Turing bifurcation.