Smooth normalization of a vector field near a semistable limit cycle
type: Item
instance of: scholarly article
title: Smooth normalization of a vector field near a semistable limit cycle
zbMATH Open document ID: 0783.58071
author: S. Yakovenko
publication date: 1994-03-20T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75024
review text: We establish a polynomial normal form for a vector field having a limit cycle of multiplicity 2. The smooth classification problem for such fields is closely related to the problem of classification of germs \(\Delta:(\mathbb{R}^ 1,0) \to(\mathbb{R}^ 1,0)\), \(\Delta(x)=x+cx^ 2 + \cdots\), solved by F. Takens in 1973. Such germs appear as the germs of Poincaré return maps for semistable cycles, and a smooth conjugacy between any two such germs may be extended to a smooth orbital equivalence between the original fields. If one deals with smooth conjugacy of flows rather than with the orbital equivalence of the corresponding fields, then two additional real parameters appear. One of them is the period of the cycle, while the second parameter keeps track of the asymmetry of the angular velocity, resulting in a difference between periods of two hyperbolic cycles appearing after perturbation of the given field.
Mathematics Subject Classification ID: 37G05
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 37G15
zbMATH DE Number: 490698
zbMATH Keywords: smooth normal forms
zbMATH Keywords: homotopy method
zbMATH Keywords: homological equation
zbMATH Keywords: limit cycle
MaRDI profile type: Publication
OpenAlex ID: W2059357669
cites work: Q3217281
cites work: EQUIVALENCE AND NORMAL FORMS OF GERMS OF SMOOTH MAPPINGS
cites work: Finitely-smooth normal forms of local families of diffeomorphisms and vector fields
cites work: Q3139743
cites work: Normal forms for certain singularities of vectorfields
cites work: Stability of \(C^ \infty\) mappings. III: Finitely determined map germs
DOI: 10.5802/AIF.1360
published in: Annales de l’institut Fourier
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description: scientific article; zbMATH DE number 490698