Introduction to Dynamical Systems

Vivi Rottschäfer

Assistent: Corine Meerman;; office 204

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 Pattern Formation', `Applied Analysis', and several national master courses (such as `Partial Differential Equations').

Time & Place

Thursday, 11.15 - 13.00 am; room 412 (Snellius).


Handing in assignments.


Week 36

  • Introduction & discussion of some basic techniques.

    Week 37

  • 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: Thursday 10 October , 11:15

    Week 38

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

    Week 39

  • 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 40 No lectures

  • Week 41

  • The proof of theorem 4.6 (4.5).
  • Lyapunov functions (4.6).

        Assignment Set 2
        Deadline: Thursday 31 October , 11:15

    Week 42

  • A bit more on Lyapunov functions (4.6).
  • Topological equivalence & the Hartman-Grobman Theorem (4.7 & 4.8).
  • Omega limit sets (4.9).

    Week 43

  • A bit more on omega limit sets (4.9).
  • Attractors (4.10).
  • The stability of periodic solutions -- a beginning (4.11).

    Week 44 

  • The stability of periodic solutions (4.11).
  • Floquet theory (2.8).

        Assignment Set 3
        Deadline: Thursday 28 November , 11:15

    Week 45 No lectures.

  • Week 46

  • More on the stability of periodic solutions (4.11).
  • The proof of Abels theorem (Chapter 2).
  • Some examples (4.11).

    Week 47

  • Poincare maps (4.12).
  • Stable and unstable sets (5.1).
  • Heteroclinic orbits (5.2).

    Week 48

  • The explicit construction of a stable manifold (5.3).
    Assignment Set 4
    Deadline: Thursday 19 December , 11:15, in mailbox Corine

  • Week 49

  • The local stable manifold theorem and its proof (5.4).

    Week 50 (last week)

  • The proof of the local stable manifold theorem: some final remarks (5.4).
  • Center manifolds (5.6).

    Assignment Set 5
    Deadline: Wednesday 22 January , 17:00, in mailbox Corine