MAGIC059: Dynamical Systems I (Flows)

Course details

A core MAGIC course


Autumn 2011
Monday, October 10th to Friday, December 16th


Live lecture hours
Recorded lecture hours
Total advised study hours


09:05 - 09:55


Four of the scheduled lectures will have to be re-arranged. The lecture scheduled for Friday 21st will be at 1pm on Friday 28th instead. Those scheduled for the 2nd, 9th and 16th December will also be rearranged (probably for 1-2pm on Fridays 4, 11, 18 November).

The material for the 10 lectures is now on the web, as is example sheet 1 (due 14 November).


Many problems in Applied Mathematics are nonlinear and described by nonlinear ordinary (or partial) differential equations. This course aims to introduce students to the tools and techniques needed to understand the dynamics that might be found in such systems. The emphasis will be on concepts and examples rather than theorems and proofs, and will include a brief survey of useful numerical methods and packages. Students will be invited to submit examples of their own for possible discussion.


No prerequisites information is available yet.


Outline syllabus
* Definition of a flow (ordinary differential equation), invariant sets, limit sets
* The Poincaré Map
* Equilibria, linearisation, stability of equilibria, periodic orbits and other invariant sets
* Structural stability, Hartman-Grobman Theorem, stable and unstable manifolds
* Centre manifold theorem, local bifurcations of equilibria and periodic orbits, Birkhoff normal form transformations for equilibria
* Example: the saddle-node-Hopf bifurcation (or other examples according to suggestions)
* Global bifurcations in two dimensions: derivation of the Poincaré map, leading on to Dynamical Systems II (maps). Discussion of the three-dimensional case and chaos
* Brief discussion of dynamics of dissipative PDEs (partial differential equations), pattern formation and the role of symmetry (probably will be omitted in 2011, but see the course on hydrodynamic stability theory, MAGIC014)
* Numerical and symbolic methods for ODEs and a mention of packages available. Continuation and the implicit function theorem


  • AR

    Professor Alastair Rucklidge

    University of Leeds


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