Bifurcation theory is the study of qualitative changes of the dynamics as a parameter of the system varies. We will focus on local bifurcations for vector fields. A typical situation is when the vector field admits an equilibrium where an eigenvalue of the linearization crosses the imaginary axis as the parameter varies. With the sign of the (real part of the) eigenvalue changes the stability of the equilibrium. But also other invariant sets and heteroclinic connections might pop up nearby. The nature of the dynamics bifurcating from the reference equilibrium depends on the nonlinearity and the dimension of the parameter. The appearance of an eigenvalue with zero real part at the critical parameter value suggests that center manifolds will play an important role. We will explore the bifurcation zoo and illustrate the theory by examples coming from physics, biology and other fields of applications. Depending on the interests of the audience and the time available, we might make excursions to some of the following topics: bifurcations in discrete dynamical systems, in PDE's, bifurcations and symmetries, global bifurcations, bifurcation without parameter.

The general idea of perturbation theory is to write a complicated system as the sum of a simpler dynamical system for which information about the dyamics is available, and a small perturbation. We will study methods adapted to various type of problems such as averaging, systems with different time scales, geometric singular perturbation theory, analysis of degenerate equilibria via blow-up,...

Bifurcation and perturbation theory are deeply related and often combined. Therefore we strongly recommand to attend both lectures.

Prerequisites are Dynamical systems I and II.