Winter 2012/13

Seminar on differential equations: invariant manifolds and local bifurcations

PD Dr. Pavel Gurevich, Dr Sergey Tikhomirov, Eyal Ron

Schedule, Winter 2012/13

Seminar:

Friday 14.15-16.00, Arnimallee 3 R 130

Description

The seminar allows the students to deepen their knowledge of ordinary differential equations in the direction of dynamical systems. It deals with local qualitative behavior of solutions from the geometrical point of view. In particular, the talks given by students will cover the following fundamental topics: stable and unstable manifolds, center manifolds, and local bifurcations. A profound understanding of these notions not only allows one to master the modern language of ordinary differential equations, but also significantly simplifies the comprehension of dynamical properties of partial differential equations (to be taught in the courses Differential Equations II and III).

The language of the seminar is supposed to be English
(with the help of German if needed).

Topics

1. Stable Manifold theorem.
2. Proof of Grobman-Hartman Theorem.
3. Definitions and Examples of the main notions.
4. Center Manifold. Theorem on Existence. Remark on Non-uniqueness.
5. Reduction to Center Manifold.
6. Simple criterion for bifurcation types.
7. Hopf Bifurcation
8. Slow-Fast Systems