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 ransformation. 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^\infty 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 epsilon to a (k-dependent) suffiiently small neighborhood of the origin.