### 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 analyse dynamical systems.

Global overview of topics:

• Symbolic dynamics and their use to study chaotic dynamics.
• Topological dynamics and notions to describe attractors, limit sets, recurrence, topological conjugacy.
• Hyperbolic dynamics, stable manifolds, shadowing (finding real orbits near approximate orbits).
• Local and global bifurcations of critical points, periodic points and periodic orbits. (Measurement of) chaos.
• Structural stability.

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