Dynamical Systems (MAGIC020)
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This course is part of the MAGIC core.
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: http://empslocal.ex.ac.uk/people/staff/pashwin/pa-ds.html An alternative location for the lectures is: https://www.youtube.com/channel/UCfFBi14HFxr-4W1btT8c1qw/playlists NB: This course is effectively 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!
Autumn 2018 (Monday, October 8 to Friday, December 14)
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]
Clicking on the link for a book will take you to the relevant Google Book Search page. You may be able to preview the book there. On the right hand side you will 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 will be via Take-Home exam. You will be given 4 equally weighted questions to attempt: to pass the course you need to attain 50%.
MAGIC020 Dynamical Systems Exam, Jan 2019
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