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This page contains information on the MasterMath course "Dynamical Systems" (Fall 2015).

Classes will be on Tuesdays, 14:00-16:45, at VU University, most of times in room WN-S655 (except on Oct 20 and Dec 15 when they are in M129), and consist of two 45 minute lectures and one 45 minute exercise session.

For this course we shall make use of the book "Introduction to Dynamical Systems" by M. Brin and G. Stuck. The first half of this course will be taught by myself and the second half by Ale Jan Homburg.

Your final grade for this course will depend on two personal and written hand-in assignments, each of which counting for 25%, and a written exam, worth 50%.

Practical information concerning covered material, excercises, hand-ins and the exam, will appear on this site

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8 September: Today, I discussed paragraphs 1.1/2/6/11 and 7.1 (the statement of Theorem 7.1.1, the statement and proof of Prop. 7.1.3 and the statement of Theorem 7.1.9). Exercises; 1.1.1/2, 1.2.1/2, 1.6.1/2 and 7.1.2/4.

15 September: We discussed paragraphs 1.2/3/4. Exercises: 1.3.1/2/3/4 and 1.4.1/2/5.

22 September: We discussed paragraphs 1.7/8/12. Exercises: 1.7.1/2/3/4, 1.8.1/2/3, 1.12.1/2.

29 September: I finished Chapter 1. More precisely, we discussed paragraphs 1.9/13 and 7.3 (until Lemma 7.3.2 and the discussion below it on "period three implies chaos"). Exercises: 1.9.1/2/3/4, 1.13.1/3 and 7.3.1/3.

6 October: We covered sections 2.1/2/3 (except Prop. 2.1.3 and Prop. 2.2.2). Exercises: 2.1.1/5/9, 2.2.2/3/5 and 2.3.1/2/3/4.

13 October: I discussed sections 2.5/6 (though not Lemmas 2.6.2 and 2.6.3 nor the proof of Prop. 2.6.4). Exercises: 2.51/2/4/5/7 and 2.6.1/2.

20 October: I discussed sections 3.1/2, Prop. 3.4.1 and sections 3.6/7. Exercises: 3.1.2/4, 3.2.1/3/4, 3.4.1, 3.7.1.

Hand-in: The first hand-in is due on Tue Nov 3 (during the lecture). It can be downloaded here.

Starting Oct 27, the lectures will be given by Ale Jan Homburg. You can find his website here.

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COURSE DESCRIPTION

Aim

The aim of this course is to introduce the student to concepts, examples, results and techniques for studying smooth dynamical systems generated by ordinary differential equations or maps.

The student learns to apply techniques from topology and analysis to study properties of
dynamical systems.


Description

We provide a broad introduction to the subject of dynamical systems.
In particular we develop theory of topological dynamics, symbolic dynamics and
hyperbolic dynamics. Several examples are used to illustrate the theory and clarify the development
of the theory.

An aim of dynamical systems theory is to describe asymptotic properties of orbits for typical initial points. The strength and beauty of the theory lies herein that techniques to do so work not only for special examples but for large classes of dynamical systems.
The focus of the course will always be on learning techniques to analyse dynamical systems
without relying on explicit formulas for the dynamical system.

As an example, the hyperbolic torus automorphism
$(x,y) \mapsto (2  1 // 1  1) (x,y) \mod 1$ on the torus $R^2/Z^2$ is a topologically transitive dynamical system for which most orbits lie dense in the torus.
What makes the example relevant is that small perturbations of it share its relevant properties.
The automorphism is for instance $C^1$-structurally stable, so that a $C^1$ small perturbation
is also topologically transitive.
To see this requires much more advanced techniques than needed to study the linear automorphism.
These techniques rely on the construction of stable and unstable manifolds.

The stable manifold theorem is among the highlights of the course.
Another central result we cover is the structural stability theorem for hyperbolic sets.

A topical description of contents  

-- Topological dynamics. Notions to describe attractors, limit sets and chaotic dynamics
such as recurrence, topological transitivity, topological mixing.

-- Symbolic dynamics and their use to study chaotic dynamics. Full shift. Subshift of finite type. Topological Markov chain. 
 
-- Aspects of bifurcation theory

-- Examples of chaotic dynamical systems such as hyperbolic torus automorphisms, the Smale horseshoe map and the solenoid.

-- Hyperbolic dynamics.  Stable manifolds. Shadowing (finding real orbits near approximate orbits).

-- Structural stability and its relation with hyperbolicity. Shadowing  as a technique to study structural stability.  

 
Organisation

2x45 min lectures  + 45 min exercise session per week

Examination

Two larger sets of homework exercises will be given.
The end grade is determined from these homework sets and an individual written exam, both counting for half the grade.

Literature

Michael Brin and Garrett Stuck
Introduction to Dynamical Systems
Cambridge University Press

Prerequisites

Prerequisite is material covered in a standard bachelor program in mathematics,
containing in particular a bachelor course on ordinary differential equations and topology.

In dynamical systems theory, results for dynamical systems generated by maps or differential equations are developed in parallel. Our focus will be on dynamical systems generated by maps.
A bachelor course on differential equations treats how a differential equation gives rise to a flow, i.e. a dynamical system, and starts a study of its qualitative properties.

Notions and techniques from topological dynamical systems are used throughout the course and require knowledge of topology as taught in a bachelor programme.