Introduction to Dynamical Systems

Vivi Rottschäfer;

Assistent: Olfa Jaïbi; office 202

Book: James D. Meiss `Differential Dynamical Systems', SIAM.

Period: fall semester.

Audience: third year bachelor students and master students.


The analysis courses of the first and second year and some linear algebra. `Analyse 3' (ordinary differential equations) can be seen as an important preparatory course. An equivalent background in calculus-like courses should also be sufficient.

Contents & Description

There are various kinds of dynamical systems: discrete maps, smooth, finite dimensional, ordinary differential equations, and infinite dimensional systems such as partial, functional or stochastic differential equations. This introductory course focuses on the second type, dynamical systems generated by ordinary differential equations. However, the ideas developed in this course are central to all types of dynamical systems. First, some fundamental concepts -- asymptotic stability by linearization, topological conjugacy, omega-limit sets, Poincaré maps -- are introduced, building on a basic background in the field of ordinary differential equations. Next, the existence and character of invariant manifolds -- that play an essential role in the theory of dynamical systems -- will be considered. This will give a starting point for the study of bifurcations. Finally, the concept of `chaos' will be discussed, mostly through the definition and basic properties of Lyapunov exponents.

The field of dynamical systems is driven by the interplay between `pure' mathematics and explicit questions and insights from `applications' -- ranging from (classical) physics and astronomy to ecology and neurophysiology. This is also reflected in the way this course will be taught: it will be a combination of developing mathematical theory and working out explicit example systems.


This course can be seen as a basic ingredient of the program chosen by a student who intends to specialize on analysis. However, it also is a relevant subject for students whose main interests lie in geometry, stochastics or numerical mathematics.

More explicitly, this course can be seen as a natural preparation for the courses `Introduction to Asymptotic Analysis', `Bifurcations and Chaos', and several national master courses (such as `Partial Differential Equations').

Time & Place

Monday, 11.15 - 13.00 am; room 401 (Snellius).

Office hours: Thursday 14:00 - 17:00, office 202


Handing in assignments.


Week 37

  • Introduction & discussion of some basic techniques.

    Week 38

  • Definition of flow and related issues (4.1, 4.2 book).
  • Existence & uniqueness (based on 3.2, 3.3, 3.4 book).    
        Assignment Set 1 
        Deadline: 10 October 2016

    Week 39

  • Global existence (4.3).
  • Gronwalls Lemma (from 3.4).
  • Smooth dependence on initial conditions (from 3.4).
  • Linearization (from 4.4).

    Week 40 No lectures

  • Week 41

  • Some background on linear systems (4.4).
  • Stability in the sense of Lyapunov (4.5).
  • The nonlinear stability of a critical point (4.5).

  • Week 42

  • Lyapunov functions (4.6).

        Assignment Set 2
    31 October
    Week 43

  • Topological equivalence & the Hartman-Grobman Theorem (4.7 & 4.8).
  • Omega limit sets (4.9).

    Week 44

  • Attractors (4.10).
  • The stability of periodic solutions  (4.11).
  •     Assignment Set 3

        Deadline: 28 November
  • Week 45 No lectures.

  • Week 46

  • Floquet theory (2.8).
  • More on the stability of periodic solutions (4.11).
  • The proof of Abels theorem (Chapter 2).

  • Week 47

  • Some examples (4.11).
  • Poincare maps (4.12).
  • Week 48

  • Stable and unstable sets (5.1).
  • Heteroclinic orbits (5.2).

    Week 49

    Assignment Set 4
    Deadline: 19 December 2016

    Assignment Set 5
    Deadline: 20 January 2017 17:00 in mailbox Olfa