Topological equivalence of planar vector fields and their generalised principal part
type: Item
instance of: scholarly article
title: Topological equivalence of planar vector fields and their generalised principal part
zbMATH Open document ID: 0971.34030
author: Vesna Županović
published in: Journal of Differential Equations
publication date: 2001-10-19T00:00:00Z
review text: In the space of \(C^{\infty}\) planar vector fields are considered subspaces \(V\) of vector fields with isolated not monodromic (not a center nor a focus) singularities and a fixed Newton diagram. For \(X \in V\) the author defines generalized, nondegenerate generalized and minimal generalized principal parts of the vector field by a filtration and some other properties of the Newton diagram. The main result of the paper is that \(X\) is locally topologically equivalent to the minimal generalized principal part. The method of proof is induction on \(n,\) where \(n\) is a number of useful sides (sides that are not entirely contained in \(\{x \leq 1 \} \cup \{y \leq 1 \}\)) of the Newton diagram. During induction the normal form method and blowing-up method are used. This result generalizes previous results by \textit{F. S. Berezovskaja} [Russ. Math. Surveys 33, No. 2, 227-228 (1978; Zbl 0397.34019)] and \textit{M. Brunella} and \textit{M. Miari} [J. Differ. Equations 85, No. 2, 338-366 (1990; Zbl 0704.58038)] and the result by the author [Glas. Mat., III. Ser. 31, No. 2, 333-351 (1996; Zbl 0888.32018)].
Mathematics Subject Classification ID: 34C41
zbMATH DE Number: 1529991
zbMATH Keywords: vector field
zbMATH Keywords: singularity
zbMATH Keywords: Newton diagram
zbMATH Keywords: topological equivalence
zbMATH Keywords: blowing-up
reviewed by: Nikolaj M. Glazunov
MaRDI profile type: Publication
full work available at URL: jdeq.2000.3810
OpenAlex ID: W2137030938
cites work: A TOPOLOGICAL NORMAL FORM FOR A SYSTEM OF TWO DIFFERENTIAL EQUATIONS
cites work: Topological equivalence of a plane vector field with its principal part defined through Newton polyhedra
cites work: Singularities of vector fields on the plane
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Q4697155
cites work: Generalised principal part of some planar vector fields
DOI: 10.1006/JDEQ.2000.3810
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description: scientific article; zbMATH DE number 1529991