Course manual 2019/2020

Course content

Modern dynamical systems theory originates with the work of Poincare, who revolutionized the study of dynamical systems by introducing qualitative techniques of geometry and topology to discuss global properties of solutions. The study of chaotic dynamical systems from the 1960s on lead to a breakthrough in science and an explosion of interest in the field of dynamical systems.

This course investigates nonlinear dynamical systems and explains basic ideas of the field in low dimensional settings of iterated maps on the line and in the plane. Important results and ideas are explained in this context, such as symbolic dynamics, "period three implies chaos", period doubling route to chaos, the Smale horseshoe map and bifurcations of periodic points.

Study materials

Literature

Devaney, Robert L.

'An introduction to chaotic dynamical systems'

Objectives

The student can explain mechanisms causing chaos in 1- and 2-dimensional map.
The student is able to apply techniques to investigate chaotic maps.
The student is able to compute and recognize important nonlinear bifurcations and appreciates their importance for dynamics.
The student has used the theory of dynamical systems in an application, and has communicated his experience to his/her peers.

Teaching methods

Zelfstandig werken aan bijv. project/scriptie
Presentatie/symposium
Hoorcollege
Werkcollege
Lecture
Self-study
Exercise class
Working independently on e.g. a project or thesis

Learning activities

Activiteit	Aantal uur
Tentamen	3
Tussentoets	3
Hoorcollege	22
Werkcollege	22
Groepsproject	12
Zelfstudie	106

Attendance

Programme's requirements concerning attendance (OER-B):

Each student is expected to actively participate in the course for which he/she is registered.
If a student can not be present due to personal circumstances with a compulsory part of the programme, he / she must report this as quickly as possible in writing to the relevant lecturer and study advisor.
It is not allowed to miss obligatory parts of the programme's component if there is no case of circumstances beyond one's control.
In case of participating qualitatively or quantitatively insufficiently, the examiner can expel a student from further participation in the programme's component or a part of that component. Conditions for sufficient participation are stated in advance in the course manual and on Canvas.
In the first and second year, a student should be present in at least 80% of the seminars and tutor groups. Moreover, participation to midterm tests and obligatory homework is required. If the student does not comply with these obligations, the student is expelled from the resit of this course. In case of personal circumstances, as described in OER-A Article A-6.4, an other arrangement will be proposed in consultation with the study advisor.

Additional requirements for this course:

Grades for homework and the group project do not count for the resit. Particpation in the group project is required for taking the resit.

Assessment

Item and weight	Details
Final grade
20% Tussentoets
50% Tentamen
20% Project
10% Huiswerk

Calculators and literature are not allowed for the tests

Assignments

Group project

Takes place in small groups. Graded by report and presentation

Homework exercises

graded

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Weeknummer	Topics (in italics is preliminary schedule)	Assignments	Sections
1	Introduction. You should be familiar with the material in section 1.2. I discussed Proposition 4.4 and started with Section 1.5.		1.4, 1.5
2	Section 1.5, 1.6 Exercises: 1.5.3, 1.5.4, 1.5.8, 1.5.10, 1.6.1, 1.6.2, 1.6.3, 1.6.6	Homework: 1.6.4 Hand in at exercise class next week.	1.5, 1.6
3	I discussed Section 1.8 and Theorem 10.1 in Section 1.10 Exercises: 1.7.1, 1.7.2, 1.7.3, 1.8.1, 1.8.11	Homework: 1.8.4	1.8, 1.10
4	Sarkovskii's theorem, Another view of period three Corresponding exercises: 1.10.1, 1.13.3-6	Homework: 1.13.4	1.10 (without the proof of Theorem 10.2), 1.13 (up to Theorem 13.7; read the rest yourself)
5	Structural stability, Bifurcation theory Corresponding exercises: 1.9.2, 1.9.5, 1.9.7, 1.9.9, 1.9.10, 1.9.11, 1.12.1	Homework: 1.9.15, 1.12.1(e) (take lambda near -1, x near 0)	1.9, 1.12 (I did not finish the proof of Theorem 12.7, that will be done next week)
6	Bifurcation theory, Maps of the circle Corresponding exercises: 1.14.1, 1.14.3, 1.14.4	Homework: 1.14.1,4	Theorem 12.7, 1.14 up to definition rotation number (its properties next week)
7	Maps of the circle		1.14
8	Test on material from weeks 1-7. Questions will be similar to the exercises and homework.
9	The horseshoe map Corresponding exercises: 2.3.1, 2.3.2, 2.3.3, 2.3.4, 2.3.5, 2.3.6, 2.3.7, 2.3.11	Homework: 2.3.3	2.3
10	Hyperbolic torus automorphisms Corresponding exercises 2.4.1, 2.4.2, 2.4.4		2.4 except the construction of Markov partitions
11	Hyperbolic torus automorphisms Corresponding exercises: 2.4.5, 2.4.6		2.4 Markov partitions
12	(group project) Possibility to work on the project during class
13	(group project) Possibility to work on the project during class
14	(group project) Possibility to work on the project during class on Monday.
15	(group project) Presentations

Timetable

The schedule for this course is published on DataNose.

Honours information

There is no honours extension of this course.

Processed course evaluations

Below you will find the adjustments in the course design in response to the course evaluations.

Contact information

Coordinator

prof. dr. Ale Jan Homburg

Owner	Bachelor Wiskunde
Coordinator	prof. dr. Ale Jan Homburg
Part of	Exchange Programme Faculty of Science, specialisation BSc Mathematics, year 1 Bachelor Wiskunde, year 3 Dubbele bachelor Wis- en Natuurkunde, year 3

Chaotic Dynamical Systems