Dynamical Systems

TIF155/FIM770 Lp2 2017-2018 (7.5p)

Useful links

Link to web quiz (Login to room DYNSYS)
Click here if only interested in news and the next course activity
Web based discussion board to ask questions on the problem sets or on the lectures (sign up here using your Chalmers email on the form: CID@student.chalmers.se). This is where you ask questions about the course (rather than emailing the teaching staff).


Wednesday, Nov. 1    The first problem set is now available. Register here to access the problems and the submission forms. You are allowed to collaborate and/or use Mathematica to solve the problems. However, you need to prepare your own figures and answers. You should also consider that problems of similar character may occur in the written exam.
Wednesday, Nov. 1    Due to demand, I uploaded some old exams [Exam 1] [Exam 2] [Exam 3]. Do not feel the pressure to start looking at them yet, it is better to focus on the hand-in problems. Some of the exam questions (or similar questions) will be covered during the excercise classes in the course.
Monday, Nov. 13    The second problem set is now available at openTA. Same rules as for the first problem set applies.
Tuesday, Nov. 14    If you are having problem logging in to openTA: First note that your username is your cid (not your full email address). If you still need help, please mail the support at fim770[at]openta.se. If you do not get openTA to work, mail me and I will find a solution.
Friday, Nov. 24    The second problem set accidently got seven points as a maximum. This means that the course will have a maximum of 37 points this year.
Sunday, Nov. 26    Subtask f) of the damped pendulum in Problem set 2 has been reformulated because the answer was not unique. To simplify the correction, please choose the additive integration constant such that your conserved quantity is equal to -1 when $\phi=\pi/2$ and $\omega=0$.
Saturday, Dec. 2    I have uploaded a test exam to be demonstrated in class on Dec. 7th. It is adviced that you have a look at and try to solve the problems beforehand. The aim of the exam is to test basic understanding of dynamical systems and to test that you have understood the home problems. The actual exam will have questions on the same level as the questions on the test exam (but the questions will not necessarily be the same nor similar). It is likely that I on the actual exam will extend the starting short questions and put more points on them (maybe 4).
Thursday, Dec. 7    I have extended the deadline for the fourth problem set to 20/12. However, if you want to be sure it is corrected before the written exam, you should hand it in before the old deadline (15/12)
Wednesday, Dec. 20    Here are some answers/solutions to the old exams [Exam 1] [Exam 2] [Exam 3].
Friday, Dec. 22    The third and fourth problem sets have been corrected. You can find them on Soliden floor 3 (near the elevator). Merry Christmas everybody and good luck on the final exam!
Monday, Jan. 8    Here are some examples of answers/soluions to the exam.

Aim and content of the course

This course provides an 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 Meibohm, 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 using the OpenTA system. 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-based submission (Problem sets 1 and 2) Give answers to the questions in the problem set using the OpenTA system. 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. 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). If you are looking for collaborators for the problem sets, you can try 'teammates' on Piazza.

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

The following persons are (randomly selected) course representatives:
Louis Deversdevers[at]student.chalmers.se
Henrik Gingsjöhengin[at]student.chalmers.se
Kristoffer Karlssonkrika[at]student.chalmers.se
Tobias Karlssontobka[at]student.chalmers.se
Holger Lindströmholgerl[at]student.chalmers.se
Contact them if you have any comments or suggestions about the course. At the end of the course all students need 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) [Concept test 1.1]
Thursday, Nov. 213:15 (EE) Lecture 2 Bifurcations and Catastrophes (Chapter 3) [Concept test 2.1]
Thursday, Nov. 215:15 (EE) Mathematica tutorial (Jan) 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) [Concept test 3.1]
Tuesday, Nov. 7 This is a good time to start working on Problem set 1
Thursday, Nov. 913:15 (EE) Lecture 4 (Updated 2017-11-09 10.30) Phase plane (Chapter 6) [If you think StreamPlot[] is boring, you can play with a dynamical visualization of your flow]
Thursday, Nov. 915:15 (EE) Problem session (Jan) On problem set 1
Friday, Nov. 1013:15 (HA2) Lecture 5 Closed orbits and Limit cycles (Chapters 6,7) [Concept test 5.1]
Week 46
Tuesday, Nov. 1413:15 (HA2) Lecture 6 Index Theory (Chapter 6.8)
Tuesday, Nov. 14 This is a good time to start working on Problem set 2
Thursday, Nov. 1613:15 (SB-H3) Lecture 7 Two-dimensional bifurcations (Chapter 8) [Strogatz: Video demonstrations of Hopf bifurcations] [Concept test 7.1]
Thursday, Nov. 1615:15 (SB-H3) Problem session (Jan) On problem set 2
Friday, Nov. 1713:15 (HA2) Lecture 8 Relaxation oscillators, Two-timing (Chapter 7)
Week 47
Monday, Nov. 2023.59 Problem set 1 deadline.
Thursday, Nov. 2313:15 (EE) Lecture 9 (Updated 2017-11-24 09.30) Damped driven pendulum. Chaotic billiards (Chapters 6.7, 8.5, 8.6).
Thursday, Nov. 2315:15 (EE) Mathematica tutorial 2
Friday, Nov. 2413:15 (HA2) Lecture 10 Chaos and Lyapunov exponents (Chapter 9)
Friday, Nov. 24 This is a good time to start working on Problem set 3
Week 48
Monday, Nov. 2723.59 Problem set 2 deadline.
Tuesday, Nov. 2813:15 (HA2) Lecture 11 Strange attractors and Lyapunov dimension (Chapters 9, 12)
One example (of many) of Lorenz attractor [Concept test 11.1]
Thursday, Nov. 3013:15 (EE) Problem session (Jan) On problem set 3
Thursday, Nov. 3015:15 (EE) Lecture 12 (Bernhard) Bernhard Mehlig presents research on dynamical systems
Friday, Dec. 113:15 (HA2) Lecture 13 (Updated 2017-12-01 16:00) Fractals and fractal dimensions (Chapter 11) [Concept test 12.1]
Friday, Dec. 1 This is a good time to start working on Problem set 4
Week 49
Tuesday, Dec. 513:15 (EE) Lecture 14 Transition to chaos in dissipative and in Hamiltonian systems
Thursday, Dec. 713:15 (EE) Lecture 15 Preparation for exam. I will go through the solutions to the test exam.
If you have time, it would be good if you try the problems yourself before the lecture.
Thursday, Dec. 715:15 (EE) Problem session (Jan) On problem set 4
Friday, Dec. 812.00 (lunch) Problem set 3 deadline.
Week 50
Friday, Dec. 1512.00 (lunch) Problem set 4 deadline if you want to be sure it is corrected before the written exam.
Week 51
Wednesday, Dec. 2012.00 (lunch) Problem set 4 final deadline.
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.