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! There are a variety of apps/applications to enable exploration of dynamical systems, for example xpp/xppaut, available from http://www.math.pitt.edu/~bard/xpp/xpp.html.
Autumn 2020 (Monday, October 5 to Friday, December 11)
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]
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