Introduction to Dynamical Systems (Winter Semester 2023/24)
- Lecturer: Dr. Björn de Rijk, M.Sc. Joannis Alexopoulos
- Classes: Lecture (0100041), Problem class (0100042)
- Weekly hours: 3+1
| Schedule | |||
|---|---|---|---|
| Lecture: | Tuesday 8:00-9:30 (every 2nd week) | 20.30 SR 3.61 | Begin: 24.10.2023 |
| Wednesday 11:30-13:00 | 20.30 SR 3.69 | ||
| Problem class: | Tuesday 8:00-9:30 (every 2nd week) | 20.30 SR 3.61 | Begin: 31.10.2023 |
| Lecturers | ||
|---|---|---|
| Lecturer | Dr. Björn de Rijk | |
| Office hours: Office hours: by appointment | ||
| Room -1.019 Kollegiengebäude Mathematik (20.30) | ||
| Email: bjoern.rijk@kit.edu | Problem classes | M.Sc. Joannis Alexopoulos |
| Office hours: by appointment | ||
| Room -1.024 Kollegiengebäude Mathematik (20.30) | ||
| Email: joannis.alexopoulos@kit.edu | ||
Contents
A dynamical system consists of a state space and a dynamical rule describing the time evolution of points in the state space, i.e., what future states follow from the current state. In this course we focus on continuous, or differential, dynamical systems, where the dynamical rule is given by an ordinary (or partial) differential equation. Such systems form the basis of physical models that exhibit smooth change and naturally arise in many scientific disciplines such as physics, biology, chemistry and engineering. Rather than calculating explicit solutions (which are known in only very few examples), we develop analytical and geometrical techniques to study the qualitative properties of dynamical systems. In particular, we treat the following concepts:
- Flows
- Abstract dynamical systems
- Lyapunov functions
- Invariant sets
- Limit sets and attractors
- Hartman-Grobman theorem
- Local (un)stable manifold theorem
- Poincaré-Bendixson theorem
- Periodic orbits and Floquet theory
- Exponential dichotomies
- Melnikov functions
- Lin's method
- Hamiltonian dynamics
- Liénard systems
- Bifurcations
- Chaotic dynamics
- (Introduction to) Fenichel theory
- Center manifolds
- Dynamical systems associated with semilinear evolution equations
Examination
The module examination at the end of the semester takes place in the form of an oral exam of about 30 minutes.
References
- Differential Dynamical Systems by James D. Meiss (available as e-book within the KIT university network)
- Ordinary Differential Equations with Applications by Carmen Chicone
- Differential Equations and Dynamical Systems by Lawrence Perko
- Gewöhnliche Differentialgleichungen und dynamische Systeme by Mathias Wilke and Jan W. Prüss (in German, available as e-book within the KIT university network)