MAGIC020: Dynamical Systems

Course details

A core MAGIC course


Spring 2024
Monday, January 29th to Friday, March 22nd; Monday, April 22nd to Friday, May 3rd


Live lecture hours
Recorded lecture hours
Total advised study hours


10:05 - 10:55 (UK)

Course forum

Visit the MAGIC020 forum


This course provides a graduate-level introduction to the qualitative theory of Dynamical Systems, including bifurcation theory for ODEs and chaos for maps. 
The format will involve only 10 hours of MAGIC lectures most of which will be in the format of an examples class/tutorial.

Most of the new material will be presented as short video podcasts that will be linked from the MAGIC website. Hence you will not make much sense of the course if you only attend the timetable lectures - there are at least 10 more hours of lectures that you will need to study in your own time. 
The lectures are posted on: 
NB: This course is equivalent to a 20 MAGIC hour course as there are 10 hours of recorded lectures that you will need to study in addition to the schedule live MAGIC sessions!


Students should have a good understanding of real linear algebra and ordinary differential equations as well as a basic understanding of the topology of subsets of the real line.

Some modelling experience and previous experience of phase plane analysis will be helpful. 


  • Asymptotic Behaviour: Asymptotic behaviour of IVPs for autonomous and non-autonomous ODEs. Omega- and alpha- limit sets. Stability of invariant sets. [1 week] 
  •  Linear and nonlinear systems: Phase space and stability of linear and non-linear equilibria. Near-identity transformations and linearization. Structural stability. [2 weeks] 
  •  Oscillations: Periodic orbits, Poincare index. Statement of Poincare-Bendixson theorem. [1 week] 
  •  Bifurcation: Bifurcation from equilibria for ODEs. Normal forms. Centre manifolds. Statement of Hopf bifurcation theorem. Examples. [2 weeks] 
  •  Chaotic systems: Chaotic ODEs and maps. Iterated maps and orbits. Horseshoes and chaos. Period doubling. Cantor set, shift map and symbolic dynamics. Sharkovskii theorem. Examples of ergodic properties. [3 weeks] 
  •  Further examples (if time allows). [1 week] 


  • Professor Peter Ashwin

    Professor Peter Ashwin

    University of Exeter


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The assessment for this course will be released on Monday 13th May 2024 at 00:00 and is due in before Saturday 25th May 2024 at 11:00.

Assessment for all MAGIC courses is via take-home exam which will be made available at the release date (the start of the exam period).

You will need to upload a PDF file with your own attempted solutions by the due date (the end of the exam period).

If you have kept up-to-date with the course, the expectation is it should take at most 3 hours’ work to attain the pass mark, which is 50%.

Please note that you are not registered for assessment on this course.


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