One dimensional invariant manifolds of Gevrey type in real-analytic maps
type: Item
instance of: scholarly article
title: One dimensional invariant manifolds of Gevrey type in real-analytic maps
zbMATH Open document ID: 1160.37007
author: I. Baldomaá
author: A. Haro
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2008-10-30T00:00:00Z
review text: The authors consider the map \(F:U\subset \mathbb{R}\times \mathbb{R}^d\rightarrow \mathbb{R}\times \mathbb{R}^d, z=(x,y)\rightarrow F(x,y)=f(x,y),g(x,y))\) defined in an open neighborhood \(\mathcal{U}\) of \(0=(0,0)\) giving a discrete dynamical system \[ \begin{matrix} \overline{x}=x-a x^N+ f_N(x,y)+f_{\geq N+1}(x,y)\\ \overline{y}= Ay+g_{\geq 2} (x,y), \end{matrix} \eqno(1) \] where \(a\) is a nonzero constant; \(1 \not \in\sigma (A); N\geq 2, \widehat{f}_{N}(x,y)\) is an \(N\)-degree homogeneous polynomial \(\widehat{f} _N (x,0)=0; f_{\geq N+1}\) has the order \(N+1;g\geq 2\) has the order \(2\) and the map \(F\) is real-analytic, i.e., it can be extended to a holomorphic function defined in a complex neighborhood \(\mathcal{U}_c\) of \(\mathcal{U}\). The questions of existence, uniqueness, differentiability, analyticity and computability of parabolic invariant manifold (center manifold) tangent to the simple eigenvalue are discussed. The parametrization method reduces the dynamics on the parabolic manifold to a polynomial. Under additional assumptions it is proved that the asymptotic expansion of this manifold is of Gevrey type.
Mathematics Subject Classification ID: 37D10
zbMATH DE Number: 5359475
zbMATH Keywords: invariant manifolds
zbMATH Keywords: parabolic points
zbMATH Keywords: asymptotic series
zbMATH Keywords: Gevrey type manifold
reviewed by: Boris V. Loginov
MaRDI profile type: Publication
full work available at URL: dcdsb.2008.10.295
OpenAlex ID: W2322003944
DOI: 10.3934/DCDSB.2008.10.295
description: scientific article; zbMATH DE number 5359475