Dynamical Systems - M1 - 8EC

Prerequisites

Prerequisite is material covered in most standard bachelor programs in mathematics. Specifically, we expect the student to have taken BSc-level courses on

  1. Ordinary differential equations (including the existence and uniqueness theorem for ODEs). For example, from the book “An introduction to ordinary differential equations” by Robinson.

  2. Topology (including open, closed and compact sets, the definition of a topology, continuous maps and homeomorphisms). For example, chapters 1, 2 and 3 from the book “Topology” by Munkres.

  3. Linear algebra (including the Jordan normal form). For example, from the book “Linear algebra and its applications” by Lay, Lay and McDonald.

  4. Calculus and multi-variable calculus (including the implicit function theorem). For example, from the book “Calculus: a complete course” by Adams and Essex).

Some familiarity with the language of differential geometry (manifolds, tangent spaces) and measure theory (sigma-algebra, measure) can be useful but is not required.

Aim of the course

Students will learn examples, results and techniques for studying smooth dynamical systems generated by ordinary differential equations and maps. They will be given a broad introduction to the subject of dynamical systems - both discrete (first half of the course) and continuous (second half of the course) in time. We cover local and global techniques, and we discuss general results as well as their implications for concrete examples.

In the course, these examples will vary from simple circle rotations to high-dimensional chaotic differential equations. We study their local behavior near stationary states and periodic orbits, but also their global chaotic and ergodic properties. To do so, we define concepts such as limit sets, recurrence, entropy, invariant manifolds, hyperbolic sets, attractors, and bifurcations. A particular aim of the course is to describe asymptotic properties of orbits for typical initial points and how these depends on varying parameters.

Rules about Homework/Exam

The final grade for the course is determined by several components:

  1. Two written hand-in assignments, each counting for 10% towards the final grade. 

  2. A 10–15-minute oral presentation, during class, of solutions to a homework exercise. This component counts for 10% towards the final grade. Students can select an exercise to present a week before. Mathematical level, clarity of argumentation, and presentation skills will all count towards the grade for this component, in a 2:2:1 ratio.

  3. A final written exam worth 70% of the final grade. To pass the course, it is required that the student obtains a minimum grade of 5.5 for this written exam.

The resit exam consists of a written exam counting for 100%. In case this is beneficial for the student, it can also count for 70%, and the two hand-in assignments and the oral presentation for 10% each.

Workload: during the 15-week semester, students are expected to spend approximately 10 hours per week on attending lectures, studying the theory, and doing homework exercises. Doing the two graded written hand-in assignments and preparing for the oral presentation and the final written exam comes on top of this.

Lecture notes/Literature

During the first half of the course, which covers discrete dynamics, we will use “Introduction to dynamical systems” by Michael Brin and Garrett Stuck, Cambridge University Press, 2003

During the second half of the course, which covers continuous dynamics, we will use “Dynamical Systems: Stability, Symbolic Dynamics, and Chaos” by Clark Robinson, second edition, CRC Press, 1999