NMAK19008U  Dynamical Systems

Volume 2019/2020

MSc Programme in Mathematics


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.
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)

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.
4 hours lectures and 2 hours exercises per week for 7 weeks.
7,5 ECTS
Type of assessment
Written examination, 3 hours
Written exam.
Exam registration requirements

To qualify for the exam two written assignments must be approved (passed/not passed)

Written aids allowed
Marking scale
7-point grading scale
Censorship form
No external censorship
One internal examiner.

Same as regular exam. If the required written assignments were not approved before the ordinary exam they must be resubmitted. They have to be approved no later than three weeks before the beginning of the re-exam week in order to qualify for the re-exam.

Criteria for exam assesment

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

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