Renormalization of planar analytic saddles
type: Item
instance of: scholarly article
title: Renormalization of planar analytic saddles
zbMATH Open document ID: 1256.34027
published in: Journal of Dynamics and Differential Equations
publication date: 2012-10-31T00:00:00Z
full work available at URL: http://hdl.handle.net/1942/13110
review text: Let \(X=\lambda_1x_1\frac{\partial}{\partial x_1}+ \lambda_2x_2\frac{\partial}{\partial x_2}+O(|x|^2)\) be a planar analytic vector field near \(x=(0,0)\). It is assumed that the linear part of \(X\) has real eigenvalues \(\lambda_1\), \(\lambda_2\) whose ratio \(\eta =-\frac{\lambda_1}{\lambda_2}\) is a positive irrational number. In [the first author and \textit{P. De Maesschalck}, Discrete Contin. Dyn. Syst., Ser. B 10, No. 2--3, 377--400 (2008; Zbl 1153.37328)], it was shown that any analytic saddle can be conjugated analytically to a form `as close as desired' to the formal normal form. In the present paper, these conjugacies are iterated and renormalized. The iteration of this procedure is strongly connected with the diophantine nature of \(\eta\). The convergence of this process is established. The main result of the paper is the following theorem. Let \(\left(\frac{q_n}{p_n}\right)_{n\in\mathbb N}\) be continued fraction approximants of \(\eta\). Suppose that \[ \sum_{n\geq 0}\frac{|\ln (C_{n+1})|}{p_{n+1}+q_{n+1}}<\infty , \text{ where } C_{n+1}=\frac{|\lambda_1p_{n+1}+\lambda_2q_{n+1}|} {p_{n+1}+q_{n+1}+1}. \] Then the vector field \(X\) can be linearized to \(\lambda_1x_1\frac{\partial}{\partial x_1}+ \lambda_2x_2\frac{\partial}{\partial x_2}\) by means of a convergent transformation. From this result, the two-dimensional version of the classical linearization theorem of A. D. Bruno follows.
reviewed by: Eugene Ershov
Mathematics Subject Classification ID: 34C20
zbMATH DE Number: 6100825
zbMATH Keywords: dynamical systems
zbMATH Keywords: normal forms
zbMATH Keywords: renormalization
zbMATH Keywords: majorant method
zbMATH Keywords: Bruno linearization theorem
zbMATH Keywords: planar analytic saddles
author: Patrick Bonckaert
author: Freek Verstringe
MaRDI profile type: Publication
OpenAlex ID: W2012607150
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DOI: 10.1007/S10884-012-9253-6
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description: scientific article; zbMATH DE number 6100825