An introduction to bifurcation theory

The aim of this chapter is to introduce tools from bifurcation theory which will be necessary in the following sections for the study of neural field equations (NFE) set in the primary visual cortex. In a first step, we deal with elementary bifurcations in low dimensions such as saddle-node, transcritical, pitchfork and Hopf bifurcations. NFEs are dynamical systems defined on Banach spaces and thus are infinite dimensional. Bifurcation analysis for infinite dimensional systems is subtle and can lead to difficult problems. If it is possible, the idea is to locally reduce the problem to a finite dimensional one. This reduction is called the center manifold theory and it will be the main theoretical result of this chapter. The center manifold theory requires some functional analysis tools which will be recalled, especially the notions of linear operator, spectrum, resolvent, projectors etc... We also present some extensions of the center manifold theorem for paramter-dependent and equivarient differential equations. Directly related to the center manifold theory is the normal form theory which is a canonical way to write differential equations. We conclude this chapter with an overview of bifurcations with symmetry and give as a result the Equivariant Branching Lemma. Most of the theorems of this chapter are taken from the excellent book of Haragus-Iooss [4] (center manifolds and normal forms). The last part on the Equivariant Branching Lemma is taken from the very interesting (but difficult) book of Chossat-Lauterbach [2]. One other complementary reference is the book of Golubitsky-Stewart-Schaeffer [3]. For an elementary review on functional analysis the book of Brezis is recommanded [1]. 1 Elementary bifurcation Definition 1.1. In dynamical systems, a bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden “qualitative” or topological change in its behaviour. Generally, at a bifurcation, the local stability properties of equilibria, periodic orbits or other invariant sets changes.