Chaotic Dynamical Systems

6 EC

Semester 1, period 1, 2

5122CHDS6Y

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

Course manual 2018/2019

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

 After this course, the student

- has a basic knowledge of nonlinear dynamical systems;
- understands the mechanisms that cause chaos in 1-dimensional maps and is able to investigate these maps;
- understands mechanisms that cause chaos in 2-dimensional maps and is able to apply techniques to investigate these maps;
- is able to compute and recognize important nonlinear bifurcations and appreciates their importance for dynamics;
- 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 6.4, an other arrangement will be proposed in consultation with the study advisor.

Additional requirements for this course:

 Grades for the group project do not count for the resit, but particpation in the group project is required for taking the resit.

Assessment

Item and weight Details

Final grade

30%

Tussentoets

50%

Tentamen

20%

Groepsproject

 Calculators and literature are not allowed for the tests

Assignments

Group project

  • Takes place in small groups. Graded by report and presentation

Homework exercises

  • Not assessed

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, Elementary definitions, Hyperbolicity, The quadratic family

 Corresponding exercises: 1.3.3, 1.3.10, 1.4.5, 1.4.6, 1.5.3, 1.5.4, 1.5.5, 1.5.10

 

  1.3, 1.4, 1.5
2

 Symbolic dynamics, topological conjugacy (without proofs of Th 7.2, Th 7.3)

Corresponding exercises: 1.6.2, 1.6.4, 1.6.6, 1.7.1, 1.7.2, 1.7.3

 

Homework: 1.6.6, 1.7.2 (a)  Hand in at exercise class next week.

  1.6, 1.7
3

Topological conjugacy: proofs of Th. 7.2 and Th 7.3, Chaos

Corresponding exercises: 1.8.1, 1.8.4, 1.8.5, 1.8.9

 Homework: 1.8.1

 1.7, 1.8
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

  • Ale Jan Homburg