Dynamical Systems

TIF155/FIM770 Lp2 2018-2019 (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). To encourage students to answer each others questions, the teachers will typically not answer immediately (but we will check that the answers are reasonable). You can of course also ask questions directly to the teachers during, or directly after the lectures.


Friday, Nov. 9    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, where you will not have Mathematica or collaborators to discuss with.
Sunday, Nov. 11    Reading instructions outlining the parts we will not cover in Strogatz
Tuesday, Nov. 20    The second problem set is available on openTA. Same rules as for the first problem set applies.
Thursday, Nov. 22    Since it seems quite a few students have gotten a slow start on the first problem set, we have moved the deadlines from Friday to the next Monday for the first two problem sets. Keep in mind that this may give you less time to work on the third and fourth problem sets. Also, no other deadlines will be moved, we do not want you to work on Christmas eve..
Friday, Nov. 30    The third problem set is available on openTA. We have decided that all the problem sets should be handed in using the openTA system. This means that you work on and submit problem sets 3 and 4 in the same way as problem sets 1 and 2. No report needs to be submitted. Each student need to make an individual submission, you cannot submit together in pairs (as was initially suggested for the reports of problem sets 3 and 4)
Friday, Dec. 7    The fourth problem set is available on openTA.
Wednesday, Dec. 12    I have uploaded a test exam to be demonstrated in class on Dec. 21st. It is adviced that you have a look at and try to solve the problems beforehand. 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).
Friday, Dec. 14    All masters students are given the opportunity to contribute to research on academic english literacy and earn a free cinema ticket. More information is given here and in the last few minutes of the lecture on Tuesday.
Saturday, Dec. 22    Here are some answers/solutions to the old exams [Exam 1] [Exam 2] [Exam 3] [Exam 4] [Exam 5] [Exam 6].
Wednesday, Jan. 2    Here are some slides with the course content and a sample of the quizzes.
Thursday, Jan. 10    Here are answers to the multiple choice questions.
Sunday, Jan. 14    Here are some examples of answers/solutions to the exam.
Friday, Jan. 25    I have corrected the exams. If you want to see your corrected exam, you can do that on Thursday January 31 between 14.00-16.00. I will sit in the kitchen in soliden level 3. I also have the exams for GU students.

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] (you need to select the current study year manually).

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
A visual introduction to the subject without equations or exhaustive text is Dynamics: The Geometry Of Behavior by Abraham and Shaw.

Software: Mathematica. You are encouraged to learn Mathematica in this course: Two tutorial sessions on Mathematica are given. The tutorials are on the form of demonstrations, so it is not necessary to bring your own laptop with Mathematica installed, but it may be helpful to have your own laptop to test the commands and to do experimentation on your own. 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 published on openTA 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.

Web-based submission The answers to the questions in the problem sets you hand in electronically 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 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.
Examples of old exams: [Exam 1] [Exam 2] [Exam 3] [Exam 4] [Exam 5] [Exam 6].
Solutions: [Exam 1] [Exam 2] [Exam 3] [Exam 4] [Exam 5] [Exam 6].

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. The openTA system does not accept any changes made after the deadline. Note that you can make as many changes to your answers as you want until the deadline.

Course representatives

The following persons are (randomly selected) course representatives:
Anders Elfversonphoquslgen[at]gmail.com
Nadejda Grochevaianadejda[at]student.chalmers.se
Luzia Knödlerluzia.knoedler[at]web.de
Dante Landa Vegadante.vega.3[at]gmail.com
Linn Lysterlyster[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 2018

Week 45
Tuesday, Nov. 613:15 (HA2) Lecture 1 Introduction (Chapter 1); Geometrical approach (Chapter 2)
Thursday, Nov. 813:15 (SB-H3) Lecture 2 Analytical approach; Bifurcations (Chapters 2 and 3)
Thursday, Nov. 815:15 (SB-H3) Mathematica tutorial
(Use "save link as")
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
Friday, Nov. 913:15 (HA2) Lecture 3 Catastrophes (Chapter 3); Linear flows (Chapter 5)
Friday, Nov. 9 This is a good time to start working on Problem set 1
Week 46
Tuesday, Nov. 1313:15 (HA2) Lecture 4 Phase plane (Chapter 6) [If you think StreamPlot[] is boring, you can play with a dynamical visualization of your flow]
Thursday, Nov. 1513:15 (SB-H3) Lecture 5 Closed orbits and Limit cycles (Chapters 6,7)
Thursday, Nov. 1515:15 (SB-H3) Problem session (Jan)
(Use "save link as")
On problem set 1
Friday, Nov. 1613:15 (HA2) Lecture 6 Index Theory (Chapter 6.8)
Week 47
Tuesday, Nov. 2013:15 (HA2) Lecture 7 Two-dimensional bifurcations (Chapter 8)
Tuesday, Nov. 20 This is a good time to start working on Problem set 2
Thursday, Nov. 2213:15 (SB-H3) Lecture 8 Relaxation oscillators, Two-timing (Chapter 7)
Thursday, Nov. 2215:15 (SB-H3) Problem session (Jan) On problem set 2
Friday, Nov. 2313:15 (HA2) Lecture 9 Damped driven pendulum. Chaotic billiards (Chapters 6.7, 8.5, 8.6).
Week 48
Monday, Nov. 2623.59 Problem set 1 deadline (OBS: Moved 2018-11-22).
Tuesday, Nov. 2713:15 (HA2) Lecture 10 Chaos and Lyapunov exponents (Chapter 9)
Thursday, Nov. 2913:15 (SB-H3) Mathematica tutorial 2
Friday, Nov. 3013:15 (HA2) Lecture 11 Strange attractors and Lyapunov dimension (Chapters 9, 12)
Friday, Nov. 30 This is a good time to start working on Problem set 3
Week 49
Monday, Dec. 323.59 Problem set 2 deadline (OBS: Moved 2018-11-22).
Tuesday, Dec. 413:15 (HA2) Lecture 12 (Bernhard) Guest lecture: Bernhard Mehlig presents research on dynamical systems
Thursday, Dec. 613:15 (SB-H3) Lecture 13 Fractals and fractal dimensions (Chapter 11)
Thursday, Dec. 615:15 (SB-H3) Problem session (Jan) On problem set 3
Friday, Dec. 713:15 (HA2) Lecture 14 Generalized fractal dimensions. Transition to chaos in dissipative systems
Friday, Dec. 7 This is a good time to start working on Problem set 4
Week 50
Thursday, Dec. 1313:15 (SB-H3) Lecture 15 Transition to chaos in Hamiltonian systems (lecture notes in DynSysLecture13.pdf)
Thursday, Dec. 1315:15 (SB-H3) Problem session (Jan) On problem set 4
Friday, Dec. 1423.59 Problem set 3 deadline.
Week 51
Tuesday, Dec. 1813:15 (HA2) Lecture 16 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.
Friday, Dec. 2123.59 Problem set 4 deadline.
Week 03
Monday, Jan. 1408:30 Written exam. You must sign up before Dec. 20
Week 17
Friday, Apr. 2614:00 Written re-examination.
Week 34
Wednesday, Aug. 2108:30 Written re-examination.