Summer 2010

BMS-Course Infinite-Dimensional Dynamical Systems

Prof. Dr. Bernold Fiedler

Recitation session: Dr. Stefan Liebscher

Schedule, Summer 2012

Lecture:

Tuesday+Thursday, 10.15-12.00, seminar room 130, Arnimallee 3 (rear building)

Recitation session:

Friday, 8.30-10.00, seminar room 130, Arnimallee 3 (rear building)

Written examination:

Thursday, July 5, 10.15-11.45, seminar room 130, Arnimallee 3 (rear building)

Wednesday, October 31, 16.15-17.45, seminar room 140, Arnimallee 7 (rear building)

Topics

"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.

Prerequisites

basic concepts of dynamical systems OR basic concepts of partial differential equation - and a fresh mind.

References

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

1. assignment, due Apr 17, 2012, (PDF)
2. assignment, due Apr 24, 2012, (PDF)
3. assignment, due May 08, 2012, (PDF)
4. assignment, due May 15, 2012, (PDF)
5. assignment, due May 22, 2012, (PDF)
6. assignment, due May 29, 2012, (PDF)
7. assignment, due Jun 05, 2012, (PDF)
8. assignment, due Jun 12, 2012, (PDF), corrected version: (PDF)
9. assignment, due Jun 19, 2012, (PDF)
10. assignment, due Jun 26, 2012, (PDF)