Well Adapted Normal Linearization in Singular Perturbation Problems
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.
KeywordsSmooth normal linearization Vector field Line of singularities Singular perturbations
Mathematics Subject Classification (2000)34C20 34C14
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- 1.Abraham, R., Marsden, J.E.: Foundations of Mechanics. Reading: Benjamin/Cummings Publishing Co. Inc. Advanced Book Program (1978), Second edition, revised and enlarged, with the assistance of Tudor Raţiu and Richard Cushman, MR 515141 (81e:58025)Google Scholar
- 7.Dumortier, F., Roussarie, R.: Canard cycles and center manifolds. Mem. Am. Math. Soc. 121(577), x+100; with an appendix by Cheng Zhi Li (1996), MR 1327208 (96k:34113)Google Scholar
- 9.Takens, F.: A note on the differentiability of centre manifolds. In: Dynamical Systems and Partial Differential Equations (Caracas, 1984), pp. 101–104. Caracas: University of Simon Bolivar (1986), MR 882016 (88f:58089)Google Scholar