Course syllabus

Course-PM

TIF155 / FIM770 Dynamical systems lp2 HT19 (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: Jan Meibohm, Room S3047
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, 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.
Lecture notes will be made available on the schedule on the start page.

Course design

Teaching The teaching is divided into lectures and problem sessions. In the lectures the course material is introduced and in the problems session example problems are solved and the homework problems are discussed. The course outline is found on the start page. There you will also find the lecture notes as they become available.

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 in the course book. There are also the hand-in problems and old exam problems. No specific list of problems is provided, since the participating students have different backgrounds, it is up to each student to judge in which area they need training.

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.

Communication The discussion board is used to ask questions (rather than emailing the teaching staff). You are advised to ask questions about lectures, the course book, example problems, problem sets, old exams and other course material. To encourage students to answer each others 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.

Software: Mathematica (Instructions to download Mathematica). You are encouraged to learn Mathematica in this course: The first two problem sessions are devoted to solving dynamical systems problems using Mathematica. 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.

Changes made since the last occasion

Migrated the course to Canvas. Minor changes to 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 analyse chaotic dynamics in initially unfamiliar contexts
• communicate results and conclusions in a clear and logical fashion
• present numerical results graphically in a clear and concise manner

Examination

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 on the start page). 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 on the start page), otherwise they are not corrected.  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.

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 that 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.

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:

• Chalmers: 3: 18-25.5p, 4: 26-30.5p, 5: 31-36p
• GU: G: 18-27.5p, VG: 28-36p
• ECTS: C: 18-25.5p, B: 26-30.5p, A: 31-36p

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:

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.