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Well Adapted Normal Linearization in Singular Perturbation Problems

  • P. Bonckaert
  • P. De Maesschalck
  • F. Dumortier
Article

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.

Keywords

Smooth normal linearization Vector field Line of singularities Singular perturbations 

Mathematics Subject Classification (2000)

34C20 34C14 

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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Hasselt University, Campus DiepenbeekDiepenbeekBelgium

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