Course syllabus

Course-PM

TIF155 / FYM155 / FIM770 Dynamical systems lp2 HT25 (7.5 hp)

The course is offered by the department of Physics

Teachers

Examiner, lectures: Kristian Gustafsson, Room S3013, tel: +46-700502211 (kristian.gustafsson@physics.gu.se)
Problem sessions: Thorsteinn Freygardsson, Room S3012B (thorsteinn.freygardsson@physics.gu.se)
Guest lecturer: Bernhard Mehlig, will give a lecture on research in dynamical systems

Course purpose

This course provides an introduction to dynamical systems and the methods used to analyze them. The aim of the course is to, without too much mathematical rigorousness, provide an understanding of theoretical concepts and practical aspects arising in the description of dynamical systems. While dynamical systems are applied in a wide range of disciplines, including mathematics, physics, biology, chemistry, electronics and economics, most examples in the course are based on simple physics problems.

Schedule

TimeEdit

Course literature

Textbook: Nonlinear Dynamics and Chaos, third edition, by Stephen H. Strogatz (Possible to buy at Chalmers STORE; first and second edition also work)
Lecture notes will be made available on the schedule on the start page. Some lecture notes starts with a section 'Requirements'. This is content you are expected to be familiar with from your earlier studies. If needed, please review any material you may have missed or forgotten.
Recordings
There is a recorded Mathematica introduction to help you get started with Mathematica.
On the start page I will upload old recordings for the first four lectures (note that the course has changed somewhat so references to the lecture notes may be wrong).
You can also check out Strogatz's blackboard lectures on Youtube.
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.

Course design

Teaching The course outline is found on the start page. The teaching is divided into lectures and problem sessions. In most lectures we cover the material in the lecture notes. The purpose of the problem sessions is to show how to solve dynamical systems problems. Similar problems will appear in hand-in problems in five problem sets. Before each session, it helps to try solving the problems yourself to better follow the steps and ask questions.

Examples Students are encouraged to calculate many example problems as this is often the best way to understand the details. There are plenty of good examples and exercises in the course book. The start page lists a few exercises from the book related to the content of each lecture (problems marked with * are harder). These could be viewed as recommended exercises for understanding the course material, but since the participating students have different backgrounds, it is up to each student to judge in which area they need training. There are also the hand-in problems and old exam problems.

Lecture quizzes During lectures ungraded quizzes are given as new material is covered. The aim is for both the students and the teacher to check whether the students have understood the material. The questions will be of varying level, where some are simple questions on definitions and other require heavy thinking and a mature understanding of the subject. Moreover, on OpenTA you can find additional ungraded questions under the section 'Study Quizzes' to test your understanding of the content of different lectures.

Communication The Piazza discussion forum is used to ask questions rather than emailing the teaching staff (common questions may already be answered in the FAQ, so check there first). You are advised to ask questions about lectures, the course book, example problems, problem sets, old exams and other course material. There is an option to post anonymously if you prefer so. To encourage students to answer each other's questions, the teachers will typically not answer immediately (but we will check that answers given by fellow students are reasonable). You can of course also ask questions directly to the teachers during, or directly after the lectures. Alternatively, if you find something confusing when reading the lecture notes prior to the lecture, you can post on the forum asking the teacher to explain it in more detail in the lecture.

Software: You are encouraged to learn Mathematica in this course. The problem sessions demonstrate both how to solve problems by hand and by Mathematica. The basics of Mathematica needed to solve dynamical systems problems is covered in the Mathematica introduction (there you also find instructions to download Mathematica). All instructions and support in the course are given as if all students use Mathematica. However, if you would rather use other software than Mathematica (such as SymPy or Maple) you are free to do so. The problem sets and the written examination will not test Mathematica-specific abilities.

Changes made since the last occasion

Minor changes to the course content.

Learning objectives

The course follows the study plan on Studieportalen.

After successfully completing this course the students shall be able to:

• understand and explain key concepts in regular dynamical systems
• perform linear stability analysis, and understand its limitations
• analyze qualitative changes in the system as control parameters change (bifurcations)
• understand and explain the key concepts used in describing deterministic chaos in non-linear systems
• efficiently simulate dynamical systems on a computer
• numerically compute Lyapunov exponents and fractal dimensions
• efficiently search for periodic orbits and determine their stabilities
• recognize and analyze chaotic dynamics in initially unfamiliar contexts
• present numerical results graphically in a clear and concise manner
• communicate results and conclusions in a clear and logical fashion

Examination

The examination of the course consists of five problem sets and a written exam which are all graded to give the final grade. The aim of the problem sets is to encourage students to start working by themselves early in the course and to complement the written exam in assessing the learning goals. The problem sets are published on openTA when the corresponding content has been covered in the lectures (green dates in the schedule on the start page). The problem sets must be handed in before the deadlines (yellow times in the schedule on the start page), otherwise they are not corrected. Note that deadlines are set with a far horizon to avoid conflicts with other courses. Avoid waiting until the last minute; it is best to complete the problem sets whenever you are free after the green dates. The problem sessions demonstrate how to solve problems similar to those in the problem sets. They also provide good opportunity to ask your accumulated questions on the problem sets. 

Web-based submission You hand in the answers to the questions in the problem sets electronically using the OpenTA system (linked in the course navigation pane). For each submitted solution, please also provide documentation proving that you have solved the problem, for example photos of your hand-written solution or of your Mathematica code saved as pdf. Leaving this out could lead to zero points on the problem. You can upload photos and pdfs using the camera and pdf buttons in OpenTA (preferably one pdf file per problem [ 1.1, 1.2, ...]).

Each sub-task [ a), b), c), ... ] of a problem requires either a text input or a figure upload, which is  corrected by the TA as correct or wrong. For the text inputs, follow the formatting instructions carefully and ensure that the data is accepted by the system. In most cases, the system will give feedback on whether your answer is correct.

For subtasks where you answer with a figure, always include a title, axis labels, and, when appropriate, explanatory labels. If your figure(s) are clearly identified as your answer(s), it is sufficient to include them in the code you submit to demonstrate that you solved the problem (see above). Before saving, make sure that the notebook has been evaluated so all outputs are visible.

Format of written examination The exam covers the material in the lecture notes as well as in the homework problems. At least one problem and two short questions will be taken (perhaps slightly modified) from earlier exams. No books, lecture notes, personal notes, or calculators are allowed. The only allowed material are pens, erasers, rulers, dictionaries (not electronic), and Mathematics Handbook for Science and Engineering, Lennart Råde and Bertil Westergren (available at Chalmers STORE). Any edition of this handbook is allowed.

Grading principles Five problem sets are graded during the course, giving a maximum of 50 points. The number of points for each task is quoted in the problem formulation. The written exam gives a maximum of 50 points. Each problem gives a stated number of points that are approximately equally distributed among the subtasks a), b), etc. The total combined score on the four problem sets and the written exam determine the grade according to the scales:

• Chalmers: 3: 40-59.5p, 4: 60-79.5p, 5: 80-100p
• GU: G: 40-69.5p, VG: 70-100p
• ECTS: C: 40-59.5p, B: 60-79.5p, A: 80-100p

In addition, to be able to pass the course, a score of at least 20 points must be achieved on the combined problem sets, and at least 20 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:

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.