February 28th, 2019
Room: W1 0-015
Boris Vertman (Oldenburg)
Discrete determinants and regularized integrals
We discuss the question of asymptotics for determinants of discrete Laplacians under refinement of the discretization mesh. This is an open question and we address its solution in the special case of tori. This is joint work with Matthias Lesch.
Marc Keßeböhmer (Bremen)
Characteristic exponents for recurrence and transience
We study dynamical systems with transient and recurrent behavior. With the help of the thermodynamical formalism we try to quantify its transient behavior in stochastic and geometric terms.
University of Bremen, January 24th, 2019
Hannes Uecker (Oldenburg)
Introduction to pde2path (and some newer features)pde2path is a Matlab package for numerical continuation and bifurcation analysis of partial differential equations. Its main design goals are flexibility (to be able to treat large classes of problems), easy usage, and hackability (easy customization). It is based on the finite element method and arclength continuation. Currently, it can handle branches of steady solutions and time periodic orbits for various classes of systems of PDEs, and the associated bifurcations. We first briefly review the basic setup using the Allen-Cahn equation as a simple example, and then explain some more advanced features such as bifurcations of higher multiplicity, including some connections to geometry by considering problems of pattern formation on curved surfaces.
Jens Rademacher (Bremen)
Dynamics of fronts in 1D Allen-Cahn equations with large scale couplingSharp interfaces are perhaps the most fundamental pattern. In this talk, recent progress regarding one-dimensional interface models in the presence of strong scale separation is discussed. Here the interplay of dynamical systems theory and parabolic PDE can be made surprisingly explicit. This combines geometric singular perturbation theory, the Evans-function for stability analysis, center manifold reduction, normal form and singularity theory. This allows to detect and unfold degenerate Takens-Bodganov points for the interface dynamics, which features various periodic, homoclinic and heteroclinic solutions. The results are illustrated with numerical computations.