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.
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Dedicated to the memory of Jack Hale.
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Bonckaert, P., De Maesschalck, P. & Dumortier, F. Well Adapted Normal Linearization in Singular Perturbation Problems. J Dyn Diff Equat 23, 115–139 (2011). https://doi.org/10.1007/s10884-010-9191-0
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DOI: https://doi.org/10.1007/s10884-010-9191-0