||Nonlinear Dynamics at the Free University Berlin|
Prof. Dr. A. Azouani
BMS-Course Infinite-Dimensional Dynamical Systems
Recitation session: Dr. Stefan Liebscher
Schedule, Summer 2012
"Infinite-dimensional dynamical systems" is an attempt to look at partial differential equations (PDEs) from a dynamical systems point of view. Existence, uniqueness and regularity theory of solutions - so dominant over almost a century - are not the focus of this attempt. The questions at the core, instead, will be like the following:
OK, solutions exist. So what do they do? What do they look like? Where do they go? How do they fit together?
As a paradigm, throughout this lecture, we will study the simplest PDE of parabolic type: one equation, one space dimension. We will encounter an amazing emerging structure, and complexity, of the interplay between stationary equilibria, heteroclinic orbits, and periodicity, which keeps fascinating me since quite a while. The description of this structure will be combinatorial and geometric. Along the way we will learn, in a specific fixed setting, about old and new concepts like hyperbolicity, variational principles and their occasional absence, comparison principles and Sturm nodal properties, Morse-Smale systems and transversality, global PDE attractors, dimension reduction, blow-up, and the like.
With a book project in planning and many open problems linking dynamics, geometry, combinatorics, and global topology, there are many opportunities to contribute to a current and lasting project, on all thesis levels.
basic concepts of dynamical systems OR basic concepts of partial differential equation - and a fresh mind.
original papers by Angenent, Brunovsky, Hale, Härterich, Matano, Oliva, Rocha, Sandstede, Sola-Morales, Wolfrum, Fiedler and others, to be announced in the course.
Homework assignments, Summer 2012
|Last change: Oct. 30, 2012||