MAGIC020: Dynamical Systems

Course details

A core MAGIC course

Semester

Autumn 2021
Monday, October 4th to Friday, December 10th

Hours

Live lecture hours
10
Recorded lecture hours
10
Total advised study hours
80

Timetable

Wednesdays
09:05 - 09:55 (UK)

Description

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!
   
There are a variety of apps/applications to enable exploration of dynamical systems, for example xpp/xppaut, available from 

Prerequisites

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. 

Syllabus

  • 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] 

Lecturer

  • Professor Peter Ashwin

    Professor Peter Ashwin

    University
    University of Exeter

Bibliography

Follow the link for a book to take you to the relevant Google Book Search page

You may be able to preview the book there and see links to places where you can buy the book. There is also link marked 'Find this book in a library' - this sometimes works well, but not always - you will need to enter your location, but it will be saved after you do that for the first time.

Assessment

The assessment for this course will be released on Monday 10th January 2022 at 00:00 and is due in before Sunday 23rd January 2022 at 23:59.

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.

Files

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Lectures

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