Prerequisites
Prerequisite is material covered in most standard bachelor programs in mathematics, containing in particular a bachelor course on ordinary differential equations (including the existence and uniqueness theorem), analysis in multiple variables (including the implicit function theorem), linear algebra (including the Jordan normal form) and (point set) topology (including open, closed and compact sets). Some familiarity with the language of differential geometry (manifolds, tangent spaces), is useful but not required.
Aim of the course
The aim of this course is for students to learn the basic concepts, examples, results and techniques for studying smooth dynamical systems generated by ordinary differential equations or maps. Students learn in particular to apply techniques from analysis and topology to study properties of dynamical systems.
We provide a broad introduction to the subject of dynamical systems. In particular we develop theory for both discrete and continuous time dynamical systems, we cover both local and global techniques, and we discuss general results as well as their implications for concrete examples.
One aim of dynamical systems theory is to describe asymptotic properties of orbits for typical initial points and how this depends on varying parameters. The strength and beauty of the theory lies herein that techniques to do so work not only for special examples, but for large classes of dynamical systems. The focus of the course will be on learning techniques to analyse dynamical systems.
A global overview of topics:
- Examples of discrete and continuous dynamical systems, from circle homeomorphisms to Hamiltonian systems.
- Topological dynamics, invariant sets, limit sets, recurrence, topological conjugacy, topological entropy.
- Local behavior near fixed points and periodic orbits, center and (un-)stable manifolds.
- Bifurcations of critical points, periodic points and periodic orbits.
- Global dynamics, chaos, hyperbolic sets and attractors.
Lectures
Weekly meetings consist of 2 hours of plenary lectures and one hour of interactive exercise classes. During the exercise class students work on homework exercises and are expected to present solutions to these.
Rules about Homework/Exam
There will be two sets of hand-in exercises (counting 10% each), doing exercises at the blackboard during the tutorial (20% total) and one written final exam (60%). To pass the course, the average grade must be 5.5/10 or higher. In addition, a minimum score of 5.0/10 for the final exam is required.
Lecture notes/Literature
We will use (the ebook version of) the textbook "Dynamical Systems: Stability, Symbolic Dynamics, and Chaos” by Clark Robinson (ISBN 978-0849384950)
- Docent: Bob Rink