CANCELLED 2019/20: Dynamical Systems

Course content

The course is an introduction to dynamical systems with emphasis on Ordinary Differential Equations (ODEs). The following topics will be discussed:

  • Linear Systems: Existence and Uniqueness Theorem, Continuous Dependence on Initial Conditions and Parameters
  • Stability Theory: Matrix solutions, Lyapunov functions
  • Global Theory of Nonlinear Systems: Hartman-Grobman Theorem, Poincare-Bendixson Theorem
  • Bifurcation theory: One and two dimensional bifurcations
  • Discrete Dynamical Systems: Period doubling bifurcation
  • Applications to chemistry, ecology, epidemiology and mechanics.

MSc Programme in Mathematics

Learning outcome


  • Basic concepts in dynamic system theory: solutions, stability, bifurcation. 
  • Standard methodes to determine the behavior of a dynamical system: Lyapunov function approach, eigenvalues of coefficient matrix
  • Local versus global behavior.



  • Solve 2- and 3-dimensional linear ODEs with constant coefficients
  • Determine uniform / asymptotic stability of equilibria for linear systems 
  • Prove stability / instability of gradient systems by constructing Lyapunov functions
  • Compute stable / unstable manifolds of simple 2-3 dimensional nonlinear systems
  • Identify fixed-point bifurcations and Hopf bifurcation for 1 or 2-dimensional ODEs 
  • Describe bifurcations for simple discrete mappings



  • Apply dynamical systems theory to build models and understand natural phenomena (for instance, Hopf bifurcation)

4 hours lectures and 2 hours exercises per week for 7 weeks.

See Absalon for final course literature. The following is an example of expected course literature

EA Coddington and N. Levinson, Theory of ordinary differential equations. Tata McGraw-Hill Education, 1955.

L. Perko, Differential Equations and Dynamical Systems. 3rd Ed, Springer-Verlag, 2001.

MW Hirsch, S. Smale, RL Devaney, Differential Equations, Dynamic Systems and an Introduction to Chaos. Elsevier, Amsterdam, 2004.



E.g., the course Differential Equations (Diff).

Academic qualifications equivalent to a BSc degree is recommended.

7,5 ECTS
Type of assessment
Written examination, 3 hours
Written exam.
Written aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner.
Criteria for exam assessment

The student must in a satisfactory way demonstrate that he/she has mastered the learning outcome of the course.

Single subject courses (day)

  • Category
  • Hours
  • Exam
  • 3
  • Preparation
  • 161
  • Lectures
  • 28
  • Theory exercises
  • 14
  • English
  • 206