Well Adapted Normal Linearization in Singular Perturbation Problems

Abstract

We provide smooth local normal forms near singularities that appear in planar singular perturbation problems after application of the well-known family blow up technique. The local normal forms preserve the structure that is provided by the blow-up transformation. In a similar context, C k -structure-preserving normal forms were shown to exist, for any finite k. In this paper, we improve the smoothness by showing the existence of a C ∞ normalizing transformation, or in other cases by showing the existence of a single normalizing transformation that is C k for each k, provided one restricts the singular parameter ε to a (k-dependent) sufficiently small neighborhood of the origin.