August/September
- Introduction/examples
- Implicit function theorem/big O notation
- Saddle-node bifurcation
- Continuity of eigenvalues
- Lyapunov–Schmidt reduction
- Center manifold reduction
- Center-stable and center-unstable manifolds, genericity saddle-node, regularity issues in center manifolds (Tiago, 22 sep)
- Normal form reduction
October
- Normal form reduction: basics
- Preparation student seminars
- Preparation student seminars
- Guest lecture Tiago: graph transform
- Normal form reduction: infinite order, bifurcation parameters
- Normal form reduction: the transpose method
- Bella on infinite dimensional Lyapunov–Schmidt reduction and Fredholm operators
- Ana on the Hopf bifurcation
- Narcicegi on Transversality in families of matrices and the notion of genericity
November
- No class
- No class
- Normal form reduction: the method of semisimple parts and sl2-triples
- Guest lecture Vitor: numerical methods for finding stable and unstable manifolds
- No class
- More on sl2-triples and introduction to symmetry
- The consequences of symmetry in bifurcation theory
- My own research: generic bifurcations in network dynamical systems
Exercises
Lecture notes (handwritten)
Literature
- Bifurcation Theory by Ale Jan Homburg and Jürgen Knobloch
- Centre Manifolds, Normal Forms and Elementary Bifurcations by A. Vanderbauwhede
- Methods of Qualitative Theory in Nonlinear Dynamics (Part I) by L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, and L. O. Chua
- Lectures on Dynamical Systems by O. Lanford
- Stable mappings and their singularities by M. Golubitsky and V. Guillemin
- Ordinary Differential Equations with Applications by C. Chicone
- Singularities and groups in bifurcation theory, Vol 1, by M. Golubitsky and D. G. Schaeffer
- Differential Topology by M. W. Hirsch
See also
Tiago Pereira's page on the course in previous years