### Einführung in die dynamischen Systeme (Wintersemester 2023/24)

• Dozent*in:
• Veranstaltungen: Vorlesung (0100041), Übung (0100042)
• Semesterwochenstunden: 3+1
Termine
Vorlesung: Dienstag 8:00-9:30 (14-tägig) 20.30 SR 3.61 Beginn: 24.10.2023
Mittwoch 11:30-13:00 20.30 SR 3.69
Übung: Dienstag 8:00-9:30 (14-tägig) 20.30 SR 3.61 Beginn: 31.10.2023
Lehrende
Dozent Dr. Björn de Rijk
Sprechstunde: Sprechstunde nach Vereinbarung
Zimmer -1.019 Kollegiengebäude Mathematik (20.30)
Email: bjoern.rijk@kit.edu
Übungsleiter M.Sc. Joannis Alexopoulos
Sprechstunde: Nach Vereinbarung
Zimmer -1.024 Kollegiengebäude Mathematik (20.30)
Email: joannis.alexopoulos@kit.edu

## Inhalt

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

# Prüfung

The module examination at the end of the semester takes place in the form of an oral exam of about 30 minutes.