Dynamical Systems

TIF155/FIM770 Lp2 2017-2018 (7.5p)

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Aim and content of the course

This course provides and introduction to the subject of chaos in dynamical systems. The course plan can be found [here]. Please note that most of the content concerning discrete maps mentioned in the course plan are instead introduced in Computational Biology A, FFR110 (the course plan will be updated the next time the course is given).

Examiner, lectures: Kristian Gustafsson, Room S3013, tel: +46-700502211 (kristian.gustafsson@physics.gu.se)
Problem sessions: Jan Meibom, Room S3015
Guest lecturer: Bernhard Mehlig

Textbook: Nonlinear Dynamics and Chaos, by Stephen H. Strogatz (Possible to buy at Cremona, first and second edition works equally well)
Other literature:
A more mathematically advanced book is Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields by Guckenheimer and Holmes.
A more extensive and advanced online resource ChaosBook.org

Software: Mathematica. You are encouraged to learn Mathematica in this course: Two tutorial sessions on Mathematica are given and all instructions and support in the course is given as if all students use Mathematica. However, if you would rather use other software than Mathematica you are free to do so. The problem sets and the written examination will not test Mathematica-specific abilities.


The examination of the course consists of four problem sets and a written exam which are all graded to give your final grade. The problem sets are handed out when the corresponding content has been covered in the lectures (green dates in the schedule below). For each problem set a problem session is given. The aim of the problem sessions is to answer your accumulated questions, you are supposed to have worked on the problem set before the problem session is given. The problem sets must be handed in before the deadlines (yellow times in the schedule below), otherwise they are not corrected.

The answers to Problem sets 1 and 2 you hand in electronically via web forms. For Problem sets 3 and 4 you hand in written solutions in the lectures, or alternatively in the box on Soliden floor 3 (near the elevator). You also need to mail a pdf copy to kristian.gustafsson.gu@analys.urkund.se
Note special instructions for the urkund email: Subject = [FIM770] (GU) or Subject = [TIF155] (Chalmers),
then attach the pdf file with the filename in the format firstname-lastname-hw1.pdf

Web-form submission (Problem sets 1 and 2) Give answers to the questions in the problem set using simple text input (no latex). Make sure that you follow the instructions on how to format the input data and make sure that the input data is accepted by the system (rejected entries will be marked in red). For questions where you answer with a figure, make sure you always give a title, axis labels, and, when appropriate, explanatory labels in the figure. Each sub-task of each problem is corrected as either correct or wrong.

Format of written solutions (Problem sets 3 and 4) In the written solutions you must explain/describe what you have done and clearly state your answers/results to the questions, as well as your conclusions. If appropriate you should discuss possible errors and inaccuracies in your results. If you are asked to plot results/make graphs, you do this in a figure with axis labels. All symbols and lines must be explained in the figure or in a caption. Program code must be appended as text at the end of your report in the web submission (urkund), but not to the paper submission.
Group work You are encouraged to collaborate and you are allowed to hand in the solutions to Problem sets 3 and 4 in groups of two persons with the restriction that you must have different partners for the two submissions (of course it is also fine to submit one by yourself and one with a partner).
Later, a web form will be made available for people who want to search for collaborators for problem sets 3 and 4.

Format of written examination The exam covers the material in the lecture notes as well as in the homework problems. No books, lecture notes, personal notes, or calculators are allowed. The only allowed material is Mathematics Handbook for Science and Engineering, Lennart Råde and Bertil Westergren (available at Cremona). Any edition of this handbook is allowed.

Grading principles Four problem sets are graded during the course. Each problem set gives a maximum of 6 points, making 24 points the maximum number of points for the problem sets. The number of points for each task/subtask is quoted in the problem formulation. The written exam gives a maximum of 12 points. The total combined score on the four problem sets and the written exam determine the grade according to the scales: In addition, to be able to pass the course, a score of at least 10 points must be achieved on the combined problem sets, and at least 5 points on the written exam.

Late problems Deadlines for the problem sets are sharp. Both electronic vesion and paper version must be handed in before the time of deadline (usually 12.00). Thus, you might as well turn in what you have at the appointed time. If you are on travel and absolutely cannot turn in a paper copy on time, you can email me your work, pdf only. It must however still be received by deadline.

Course representatives

Course representatives are randomly selected when the course has started. Contact them if you have any comments or suggestions about the course. At the end of the course everyone needs to fill out a course evaluation form.

Preliminary schedule 2017

Week 44
Tuesday, Oct. 3113:15 (HA2) Lecture 1 Introduction (Chapter 1); Flows on the line (Chapter 2)
Thursday, Nov. 213:15 (EE) Lecture 2 Bifurcations and Catastrophes (Chapter 3)
Thursday, Nov. 215:15 (EE) Mathematica tutorial Please try to download and install Mathematica before the tutorial
Bring your laptop (with mathematica installed) if you have one
Instructions on how to download and install mathematica
Week 45
Tuesday, Nov. 713:15 (HA2) Lecture 3 Linear flows (Chapter 5)
Thursday, Nov. 913:15 (EE) Lecture 4 Phase plane (Chapter 6)
Thursday, Nov. 915:15 (EE) Problem session (Jan) On problem set 1
Friday, Nov. 1013:15 (HA2) Lecture 5 Limit cycles (Chapter 7)
Week 46
Tuesday, Nov. 1413:15 (HA2) Lecture 6 Two-dimensional bifurcations (Chapter 8)
Thursday, Nov. 1613:15 (SB-H3) Problem session (Jan) On problem set 2
Friday, Nov. 1713:15 (HA2) Lecture 7 Relaxation oscillators, Two-timing (Chapter 7)
Week 47
Tuesday, Nov. 2113:15 (EE) Lecture 8 Damped driven pendulum. Chaotic billiards.
Thursday, Nov. 2313:15 (EE) Mathematica tutorial 2
Friday, Nov. 2413:15 (HA2) Lecture 9 Chaos and Lyapunov exponents (Chapter 9)
Week 48
Tuesday, Nov. 2813:15 (HA2) Lecture 10 Strange attractors and fractals (Chapters 11-12)
Thursday, Nov. 3013:15 (EE) Problem session (Jan) On problem set 3
Friday, Dec. 113:15 (HA2) Lecture 11 Fractals and fractal dimensions (Chapter 11)
Week 49
Tuesday, Dec. 513:15 (EE) Lecture 12 Multifractals
Thursday, Dec. 713:15 (EE) Lecture 13 (Bernhard) Bernhard Mehlig presents research on dynamical systems
Thursday, Dec. 715:15 (EE) Problem session (Jan) On problem set 4
Week 50
Tuesday, Dec. 1213:15 (HA2) Lecture 14 Transition to chaos in dissipative and in Hamiltonian systems
Thursday, Dec. 1413:15 (EE) Lecture 15 Preparation for exam
Week 02
Monday, Jan. 808:30 Written exam. You must sign up before Dec. 20
Week 14
Friday, Apr. 614:00 Written re-examination.
Week 34
Wednesday, Aug. 2208:30 Written re-examination.