On-Central Manifold and Normal Forms Theorem in Local Bifurcations

Abstract

The aim of this study is to introduce tools from local bifurcation theory which will be necessary in the following sections for the study of neural field equations. In a first step, we deal with a basic manifold, elementary bifurcations in low dimensions such as saddle-node, trans critical, pitchfork and Hopf bifurcations. 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 study. The center manifold theory requires some functional analysis tools which will be recalled, especially the notions of linear operator, spectrum, resolve, projectors etc... We also present some extensions of the center manifold theorem for parameter-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 study by some applications.