Geometric Analysis of the Singularly Perturbed Planar Fold

Abstract

The geometric approach to singular perturbation problems is based on powerful methods from dynamical systems theory. These techniques have been very successful in the case of normally hyperbolic critical manifolds. However, at points where normal hyperbolicity fails, e.g. fold points or points of self-intersection of the critical manifold, the well developed geometric theory does not apply. We present a method based on blow-up techniques which leads to a rigorous geometric analysis of these problems. The blow-up method leads to problems which can be analysed by standard methods from the theory of invariant manifolds and global bifurcations. The presentation is in the context of a planar singularly perturbed fold. The blow-up used in the analysis is closely related to the rescalings used in the classical analysis based on matched asymptotic expansions. The relationship between these classical results and our geometric analysis is discussed.