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