Dynamische Systeme

What it is about

You learn the theory of dynamical systems for ordinary differential equations and mappings in finite-dimensional phase spaces. The module provides tools for analyzing stable and unstable solutions, bifurcations, chaos and ergodic properties; at the end you can analyze geometric and topological properties of trajectories and attractors and transfer the methods to differentiable manifolds and infinite-dimensional spaces.

What you will be able to do

• Stability analysis and application of Lyapunov functions
• Investigation of Poincaré mappings and stable/unstable manifolds
• Application of Hartman–Grobman and structural stability
• Recognition and classification of local and global bifurcations
• Use of normal forms and center manifolds
• Analysis of chaos: Smale horseshoe, symbolic dynamics and Lyapunov exponents
• Study of special examples: unimodal mappings, circle mappings, Sharkovsky theorem
• Foundations of invariant measures, ergodicity, Poincaré recurrence and entropy

What the module consists of

• VorlesungDelivery of theoretical foundations and demonstrative examples
• ÜbungDeepening through exercises, problem sheets and supervised self-study

Teaching method

• VorlesungPresentation of content with examples and discussion to stimulate independent, in-depth study
• ÜbungskursWorking on exercises to deepen understanding, with problem sheets and solutions for self-check
• SelbststudiumIndependent preparation of the topics and literature work