Introduction to Dynamical Systems

Introduction to Dynamical Systems '19 - '20

Lecturer

Arjen Doelman

Assistance

Olfa Jaibi (email o.jaibi "a" math.leidenuniv.nl)

Book

James D. Meiss `Differential Dynamical Systems' (Revised Edition), SIAM.

Time & Place

Fall semester, Mondays, 11.15-13.00 h., room 174 (Snellius).

Office hours

Mondays, 14.00-16.00 h.

Audience

Third year bachelor students and master students.

Prerequisites

For math students: the analysis courses of the first and second year and some linear algebra; the course Ordinary Differential Equations can be seen as an important preparatory course. For non-math students: an equivalent background in calculus-like courses should also be sufficient.

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.

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.

Examination

Homework + Exam: End grade = 0.4 (Total homework) + 0.6 (Exam)

PROGRAM

Week 36

  • General introduction & discussion of some basic techniques.

    Week 37 - by Olfa

  • Definition of flow and related issues (4.1, 4.2 book).
  • Existence & uniqueness (based on 3.2, 3.3, 3.4 book).

    Week 38

  • Global existence (4.3).
  • Gronwall's Lemma (from 3.4).
  • Smooth dependence on initial conditions (from 3.4).

    Week 39

  • Linearization & linear systems (4.4).
  • Stability in the sense of Lyapunov (4.5).
  • The nonlinear stability of a critical point (4.5).
  • Lyapunov functions (4.6).

    September 23: Exercise Series I.
    Deadline: Tuesday October 8 at 11 am, directly in Olfa's office/in her mailbox (`postvakje')/in an email to Olfa.

    Week 40 No lectures

    Week 41

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

    Week 42

  • More on omega limit sets (4.9).
  • Attractors (4.10).

    October 14: Exercise Series II.
    Deadline: Tuesday November 5 at 11 am, directly in Olfa's office/in her mailbox (`postvakje')/in an email to Olfa.

    Week 43

  • More on attractors (4.10).
  • The stability of periodic orbits: a first introduction (4.11).

    Week 44

  • Floquet's Theorem and its proof (2.8).
  • Properties of Floquet exponents (4.11).

    Week 45 No lectures

    Week 46

  • Recap stability of periodic orbits.
  • The proof of Abel's Theorem (2.8).
  • Some examples (4.11).

    November 12: Exercise Series III.
    EXTENDED deadline: Thursday December 5 at 11 am, directly in Olfa's office/in her mailbox (`postvakje')/in an email to Olfa.

    Week 47

  • Poincaré maps (4.12).

    Week 48

  • Stable & unstable manifolds (5.1).
  • Heteroclinic orbits (5.2).

    Week 49

  • Stable manifolds (5.3).

    Week 50 (last week)

  • The local stable manifold theorem (5.4)
  • Center manifolds (5.6).

    December 14: Exercise Series IV.
    Deadline: Wednesday January 15 at 11 am, directly in Olfa's office/in her mailbox (`postvakje')/in an email to Olfa.

    Exam material

  • Chapter 2: 2.8.
  • Chapter 3: 3.2, 3.3, 3.4, 3.5.
  • Chapter 4: 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 4.8, 4.9, 4.10, 4.11, 4.12.
  • Chapter 5: 5.1, 5.2, 5.3, 5.4, 5.6.
    Exercise material

  • Exam last year.
  • Exercise series 2017-2018: Series I, Series II, Series III, Series IV.
  • Chapter 2, section 2.9: 19.
  • Chapter 3, section 3.6: 6, 7, 12, 14 (first edition book: 4, 5, 10, 12).
  • Chapter 4, section 4.13: 2 (without numerics), 7, 8, 9, 10, 12, 13, 15, 17 (= 16 in first edition book).
  • Chapter 5, section 5.6: 1, 3, 4, 5, 7, 8 (without numerics), 9 (without numerics), 10 (without numerics).
    Note: the text in the book contains many worked-out examples that also can be used as exercise material.
    Exam

  • The exam will be of `open book' type.
  • Time: Wednesday January 22 2020, 10:15 - 13:15.
  • Place: rooms 412, 407, 409, Snellius building.
  • Question hour: Friday January 17 2020, 11:30 - 13:00 in room 408.