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Classes will be on ** Tuesdays, 14:00-16:45, **at VU University, most of times in room WN-M655 (except on Oct 21 and Dec 16 when they are in M143), 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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**September 9:** I covered sections 1.1/2/3/4 in class. Representative exercises are 1.1.1/2, 1.2.1/2, 1.3.1/2/3/4 and 1.4.1.

Here's a hint for exc 1.2.2: when k is a natural number, then the number 2^n starts with k if and only if k*10^q \leq 2^n < (k+1)*10^q for some integer q. Now take the logarithm with base 10.

**September 16:** I discussed sections 1.5/6/7 and 1.8 (partially) in class. Exercises: 1.4.5, 1.5.4/6, 1.6.1, 1.7.1/2/3/4 and 1.8.1.

**September 23:** I discussed sections 1.8/9/13 in class. Read paragraph 1.10 yourself. Exercises: 1.8.2, 1.9.2, 1.10.2/3, 1.13.1/3.

**September 30:** I discussed sections 1.11/12 and 2.0/1/2 in class. Exercises: 1.11.1/2, 1.12.3, 2.1.1/5/9, 2.2.2/3 and 2.3.2.

**October 7:** I discussed sections 2.1/2/3/4/5 in class (section 2.5 only half, I will continue it next time). Exercises: 2.2.5/6, 2.3.1/4, 2.4.2, 2.5.1/2/4/5.

**October 14:** I discussed sections 2.5/6 in class. Exercises: 2.5.6/7, 2.6.1/2/4.

**October 21:** We talked about sections 3.1/2 in class. Exercises: 3.1.1/2/4, 3.2.1/3/4, 3.3.1/2, 3.4.1. Note: I will announce the homework problem set during my last class next week. You will have 2 weeks to make these exercises.

**October 28: **I discussed sections 3.2/4/5/6/7/8 in class. There are no exercises this week**.**

**Homework problems:** Click here for the set of homework exercises. This homework is due on November 11, at the start of the lecture

From now on, the lectures will be given by Ale Jan Homburg. Information about the course will appear on his website.

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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.