Introduction to Dynamical Systems


Vivi Rottschäfer; vivi@math.leidenuniv.nl

Assistent: Corine Meerman; cmeerman@math.leidenuniv.nl; office 204

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

Period: fall semester.

Audience: third year bachelor students and master students.


Prerequisites

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.

Remarks:

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', `Bifurcations and Chaos', and several national master courses (such as `Partial Differential Equations').

Time & Place

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

Examination

Handing in assignments.

PROGRAM


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: Wednesday 8 October , 9:00


    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

  • Lyapunov functions (4.6).

        Assignment Set 2
        Deadline: Wednesday 29 October , 9:00


    Week 42

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

    Week 43

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

    Week 44

  • Floquet theory (2.8).
  • More on the stability of periodic solutions (4.11).
  • The proof of Abels theorem (Chapter 2).
  •     Assignment Set 3
        Deadline:
    Wednesday 26 November , 9:00

    Week 45 No lectures.

  • Week 46


  • Some examples (4.11).
  • Poincare maps (4.12).
  • Week 47


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

    Week 48 

  •  
    Assignment Set 4
    Deadline:
    Wednesday  17 December , 9.00, 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: Monday 19 January 17:00,  in mailbox Corine