A classical approach to the analyticity problem of center manifolds
type: Item
instance of: scholarly article
title: A classical approach to the analyticity problem of center manifolds
zbMATH Open document ID: 0564.34046
DOI: 10.1007/BF00949030
published in: ZAMP. Zeitschrift für angewandte Mathematik und Physik
publication date: 1985-01-01T00:00:00Z
review text: The author is concerned with ordinary differential equations of the form (1) \(\dot x=-y+X(x,y,z)\), \(\dot y=x+Y(x,y,z)\), \(\dot z=Az+Z(x,y,z)\), (x,y\(\in {\mathbb{R}}\) and \(z\in {\mathbb{R}}^ n)\), where X, Y and Z are analytic and real-valued functions which together with their first derivatives vanish at the origin 0. The matrix A is assumed to have no purely imaginary eigenvalues. Any center manifold equation corresponding to (1) is a two-dimensional system of the form (2) \(\dot x=-y+X(x,y,c(x,y)),\) \(\dot y=x+Y(x,y,c(x,y)),\) where \(z=c(x,y)\) is a function representing locally a center manifold of (1) passing through 0. It can be proved by means of Poincaré's focal values that if \(z=c(x,y)\) is analytic, then the following alternative holds: ''The zero solution of (2) is either a focus or a center.'' However, the center manifold \(z=c(x,y)\) may be non- analytic in general. The main purpose of this paper is to prove the validity of the above alternative in spite of the lacking analyticity for the center manifold.
Mathematics Subject Classification ID: 34C40
Mathematics Subject Classification ID: 34A25
zbMATH DE Number: 3899333
zbMATH Keywords: analyticity
zbMATH Keywords: center manifold equation
zbMATH Keywords: Poincaré's focal values
zbMATH Keywords: focus
author: Bernd Aulbach
reviewed by: Ding Tongren
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf00949030
OpenAlex ID: W1974616107
description: scientific article; zbMATH DE number 3899333