Dynamical Systems - M1 - 8EC

Prerequisites
Prerequisite is material covered in most standard bachelor programs in mathematics, containing in particular a bachelor course on ordinary differential equations, analysis in multiple variables (including the implicit function theorem), linear algebra (including Jordan normal form) and (point set) topology. Some familiarity with concepts from differential geometry, such as manifolds and tangent spaces/bundles, is useful but not required.

A bachelor course on differential equations treats how a differential equation gives rise to a flow, i.e. a dynamical system, and starts a study of its qualitative properties. An example of a textbook that develops this theory is R.C. Robinson, An Introduction to Dynamical Systems.

Notions and techniques from topology are used throughout the course and require knowledge of topology as taught in a bachelor program. For instance, see part I (General Topology) from J.R. Munkres, Topology: we expect students to be proficient in using basic notions and techniques as in Chapters 1-4 from this book.

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 in finite dimensions.

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 always be on learning techniques to analyze dynamical systems.

Global overview of topics:

  • Topological dynamics and notions to describe invariant sets, limit sets, recurrence, topological conjugacy, attractors, chaos.
  • Flows, local behavior near fixed points and periodic orbits, center and stable manifolds.
  • Bifurcations of critical points, periodic points and periodic orbits.
  • Attractors and repellers and lattices of attractors and repellers.
  • Lyapunov functions.
  • Morse representations and Morse relations (Morse theory and Conley theory).

Lecturers
Rob van der Vorst (VU) and Arjen Doelman (UL)

Rules about Homework/Exam
There will be four sets of homework (hand-in) exercises, each counting for 10% towards the final grade, and one written exam, counting for 60% towards the final grade. To pass the course, the total grade must be 5.5/10 or higher. In addition, a minimum score of 5.0/10 for the final exam is required. The hand-in exercises are mandatory in order to participate in the final assessment (final exam). The score for the hand-in exercises remains for the retake.

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). For the second part of the course we use course notes in addition to the book by Robinson.