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Workgroup Nonlinear Partial Differential Equations

Secretariat
Kollegiengebäude Mathematik (20.30)
Room 3.029

Address
Karlsruher Institut für Technologie
Institut für Analysis
Englerstraße 2
76131 Karlsruhe
Germany

Office hours:
Mon-Fri 10-12, Tue+Wed 14-16

Tel.: +49 721 608 42064

Fax.: +49 721 608 46530

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.


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