Introduction to Dynamical Systems

Introduction to Dynamical Systems '17 - '18


Lecturer

Arjen Doelman

Assistance

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

Book

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

Time & Place

Fall semester, Mondays, 9.00 - 11.00 am, room 401 (Snellius).

Office hours

Mondays, 2.00 - 5.00 pm, Olfa's office (# 202, Snellius).

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, Poincare 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

Handing in assignments.


PROGRAM


Week 36

  • General introduction & discussion of some basic techniques.

    Week 37

  • Definition of flow and related issues (4.1, 4.2 book).

    Week 38

  • Existence & uniqueness (based on 3.2, 3.3, 3.4 book).
  • Global existence (4.3).

    September 19: Exercise Series I.
    Deadline: Tuesday October 10 at 11 am, directly in Olfa's office/in her mailbox (`postvakje')/in an email to Olfa.

    Week 39

  • A bit more on global existence (4.3).
  • Gronwalls Lemma & smooth dependence on initial conditions (from 3.4).
  • Linearization & linear systems (4.4).
  • Stability in the sense of Lyapunov (4.5).

    Week 40 No lectures

    Week 41

  • The nonlinear stability of a critical point (4.5).
  • Lyapunov functions (4.6).

    Week 42

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

    September 16: Exercise Series II.
    Deadline: Tuesday October 31 at 11 am, directly in Olfa's office/in her mailbox (`postvakje')/in an email to Olfa.

    Week 43

  • Attractors (4.10).
  • Stability of periodic orbits (4.11).

    Week 44

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    Week 45 No lectures

    Week 46

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    Week 47

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    Week 48

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    Week 49

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    Week 50 (last week)

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