Introduction to Dynamical Systems

Introduction to Dynamical Systems

By Arjen Doelman & Martin van der Schans.

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, Poincar\'e 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', `Applied Analysis', and several national master courses (such as `Partial Differential Equations').

Time & Place

Fridays, 9-11 am; room 401 (Snellius).

Examination

Handing in assignments.

PROGRAMMA

Week 36

  • Inleiding & opfrissen basistechnieken.

    Week 37

  • Definitie `flow' (4.1, 4.2 boek).
  • Kort: existentie & uniciteit, Lipschitz continuiteit (uit 3.2, 3.3, 3.4 boek).

    Week 38

  • Globale existentie; het Lemma van Gronwall (4.3).
  • Continue afhankelijkheid van beginvoorwaarden (uit 3.4).
  • Linearisatie (4.4).

    18 september: OPGAVESERIE I.
    Deadline: 2 oktober.

    Week 39

  • Stabiliteit (4.5).
  • De nietlineaire stabiliteit van een kritiek punt (4.5).

    Week 40

  • Lyapunov functies (4.6).
  • Topologische equivalentie & de Hartman-Grobman stelling (kort, uit 4.7 & 4.8).

    2 oktober: OPGAVESERIE II.
    Deadline: 16 oktober, 17.00 u., postvakje Martin.

    Week 41

  • Omega-limietverzamelingen (4.9).
  • Attractors (4.10).

    Week 42

  • Attractors, een kort voorbeeld (4.10).
  • De stabiliteit van periodieke oplossingen (4.11).
  • Floquet theorie (2.8).

    16 oktober: OPGAVESERIE III.
    Deadline: WOENSDAG 4 november, 17.00 u., postvakje Martin.

    Week 43: Geen college

    Week 44

  • Vervolg `stabiliteit van periodieke oplossingen' (4.11).
  • De stelling van Abel.
  • Toepassingen in Hamilton systemen.

    Week 45

  • Poincare afbeeldingen (4.12).
  • Stabiele en instabiele verzamelingen (5.1).
  • Heterocliene banen (5.2)

    Week 46 (College verzorgd door Martin)

  • Stabiele varieteiten (5.3).
  • Locale en globale stabiele varieteiten - kort (5.3, 5.4).
  • Centrumvarieteiten (5.5).

    13 november: OPGAVESERIE IV.
    Deadline: maandag 7 december, 17.00 u., postvakje Martin.

    Week 47: Geen college

    Week 48

  • Afronden centrumvarieteiten (5.5).
  • Chaos: gevoelige afhankelijkheid van beginvoorwaarden (7.1).

    Week 49

    Ingelast vragenuur: woensdagochtend 2 december, 9-11 uur, locaal 401.

  • Chaos: transitiviteit van een `flow' (7.1).
  • Inleiding Lyapunov exponenten (7.2).

    Week 50 (laatste college)

  • Lyapunov exponenten: definitie & eigenschappen (7.2).
  • Strange attractors: fractale dimensies (7.3)

    11 december: OPGAVESERIE V. Uit het boek, pp. 265, 266, opgaves: 3, 5, 6, 8, 9.

    OPMERKINGEN BIJ OPGAVE 6:
    - DE HINT IS ALLEEN MAAR VERWARREND. Denk zelf na.
    - Als de opgave is z'n algemeenheid wat lastig is, beschouw dan eerst eens een speciaal geval, bijvoorbeeld: een vlak (i.e. twee-dimensionaal) systeem, of, de onderliggende baan is periodiek, etc.

    Deadline: vrijdag 22 januari, 17.00 u., postvakje Martin.

    Vragenuur: woensdagochtend 20 januari, 11-13 uur, locaal 401.