On the Poincaré-Lyapunov centre theorem
type: Item
instance of: scholarly article
title: On the Poincaré-Lyapunov centre theorem
zbMATH Open document ID: 0959.34021
author: Marco Brunella
author: Massimo Villarini
published in: Boletín de la Sociedad Matemática Mexicana. Third Series
publication date: 2001-04-22T00:00:00Z
review text: The classical Poincaré-Lyapunov centre theorem says that a centre-type singularity of an analytic vector field on the plane has an analytic first integral, provided that the linear part of the vector field at the singular point generates a nontrivial rotation. The authors consider the multidimensional analog of the planar centre dynamics, and give a geometric proof and a slight generalization of a result of \textit{M.~Urabe} and \textit{Y.~Sibuya} [J. Sci. Hiroshima Univ., Ser. A 19, 87-100 (1955; Zbl 0073.31001)], stating that an analytic multicentre, with linear part at the singular point which generates a multirotation, is always linearizable. The main tools in the proof are the analytic extension up to the origin of the period function, obtained through the analysis of the complexification of the vector field, and the use of the Bochner theorem on the linearization of compact group actions in a neighbourhood of a fixed point. As a byproduct of these arguments, a still another proof of the Poincaré-Lyapunov centre theorem is obtained.
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 34M15
Mathematics Subject Classification ID: 34D20
Mathematics Subject Classification ID: 57S15
Mathematics Subject Classification ID: 37C10
zbMATH DE Number: 1501464
zbMATH Keywords: linearization of multicentres
zbMATH Keywords: centre-type singularity of an analytic vector field on the plane
zbMATH Keywords: analytic multicentre
zbMATH Keywords: multirotation
zbMATH Keywords: complexification of the vector field
zbMATH Keywords: linearization of compact group actions
zbMATH Keywords: Poincaré-Lyapunov centre theorem
zbMATH Keywords: singularities of vector fields
zbMATH Keywords: stability
zbMATH Keywords: periodic orbits
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1501464