These are lecture notes (slides) for a 10-hour course delivered as part of the MAGIC suite of graduate-level courses in nonlinear dynamics and dynamical systems. The course was all about flows; there is a follow-on course on maps and another on equivariant bifurcation theory.

Topics covered (in 9 lectures) were:

Flows and the Poincare map; Linearisation; Stability of equilibrium points, periodic orbits and other invariant sets; Local and global bifurcation theory; Centre Manifold Theorem; Birkhoff normal form; Local bifurcations of periodic orbits; An in-depth example (the saddle-node--Hopf bifurcation); Homoclinic bifurcations.

There was then an option in the last lecture for students to hear about PDEs, patterns, and the role of symmetry (leading in to a further 10 lectures on equivariant bifurcation theory by Pete Ashwin, or on hydrodynamic stability theory), or Numerical methods for dynamical systems (Symbolic algebra, Integrating ODEs and Continuation).