Dynamical Systems (Winter Semester 2013/14)
- Classes: Lecture (0105400), Problem class (0105500)
- Weekly hours: 4+2
With the usage of computer power to numerically approximate solutions to ordinary and partial differential equations, the interest shifted from analytically calculating the exact solution to analyzing their (time-)asymptotic behavior. This is the starting point of the theory of dynamical systems.
A dynamical system is given by a rule, describing how a current state evolves with time. Typical examples are ordinary and (time-dependent) partial differential equations as well as their discretization. Note that time is continuous in the first two examples and discrete in the last one.
This lecture gives an introduction into the theory of finite and infinite dimensional systems. The main objective of the lecture is then to analyze the asymptotic behavior of solutions.
Topics include:
- invariant sets and their stability properties (in the sense of Lyapunov)
- stable and unstable manifolds
- attractors
- behavior of these objects under the change of a parameter or discretization (structural stability)
We plan to continue the analysis of dynamical systems by a lecture about traveling waves and their stability in the next semester.
Important: Lecture on December 19th in K2 (Kronenplatz 32)
| Schedule | ||
|---|---|---|
| Lecture: | Tuesday 14:00-15:30 | Z 1 |
| Thursday 8:00-9:30 | 1C-03 | |
| Problem class: | Thursday 15:45-17:15 | Z 1 |
| Lecturers | ||
|---|---|---|
| Lecturer, Problem classes | JProf Dr. Jens Rottmann-Matthes | |
| Office hours: - | ||
| Room - Kollegiengebäude Mathematik (20.30) | ||
| Email: marion.ewald@kit.edu | ||
Exercises
References
The following list will be regularly updated.
Finite-Dimensional Dynamical Systems
- H. Amann: Gewöhnliche Differentialgleichungen, 2. Auflage, de Gruyter 1995
- B. Aulbach: Gewöhnliche Differenzialgleichungen, 2. Auflage, Spektrum Verlag 2004
- J. Guckenheimer, Ph. Holmes: Nonlinear Oscillations, Dynamical Systems, and Bifurcarions of Vectorfields, Springer 1983
Infinite-Dimensional Dynamical Systems
- J. Hale: Asymptotic Behavior of Dissipative Systems, American Mathematical Society 1988
- J.C. Robinson: Infinite-Dimensional Dynamical Systems, Cambridge University Press 2001
- G.R. Sell, Y. You: Dynamics of Evolutionary Equations, Springer Verlag 2002
Functionalanalysis
- H.W. Alt: Lineare Funktionalanalysis, 3. Auflage, Springer 1999
- T. Kato: Perturbation Theory for Linear Operators, 2. Auflage, Springer 1980