A generalized normal form and formal equivalence of systems of differential equations with zero characteristic numbers
type: Item
instance of: scholarly article
title: A generalized normal form and formal equivalence of systems of differential equations with zero characteristic numbers
zbMATH Open document ID: 1078.34018
DOI: 10.1023/A:1025144730356
author: Xianqiang Yang
published in: Differential Equations
publication date: 2005-03-15T00:00:00Z
review text: Normal forms (analytic or formal) are important in the theory of systems of ordinary differential equations (ODEs). Systems in normal forms are sometimes referred to as ``resonance normal forms''. The existence of transformations that reduce a differential equation to a simple form, a normal form, has been investigated by many authors. In particular, following a previous work by \textit{A. D. Bryuno} [Tr. Mosk. Mat. Obshch. 25, 119--262 (1971; Zbl 0263.34003) and Math. Notes 14, 844--848 (1973); translation from Mat. Zametki 14, No. 4, 499--507 (1973; Zbl 0298.34004)], in the paper under review, divided in two parts, the first one being devoted to a general study on the autonomous system of ODEs: \[ \dot x_i= \lambda_i x_i+ \sigma_i x_{i-1}+ x_i(x),\quad i= 1,2,\dots, n,\tag{1} \] whose right-hand sides do not contain free terms, the matrix of the linear part is reduced to a Jordan form and the nonlinearities are power series, either formal or convergent at zero. By suitable changes of variables (or substitutions) system (1) can be reduced to other more simplified normal forms. The author considers a singular case occurring when all eigenvalues \(\lambda_1,\lambda_2,\dots, \lambda_n\) of the matrix of the linear part of (1) are zero and by virtue of the classification of normal forms depending on the arrangement of the eigenvalues \(\lambda_i\) on the complex plane realized by A. D. Bryuno, it is derived that any system (1) has in this case a form which is a normal form itself and that can also be rewritten as \[ \dot x_i= P_{\gamma i}^{[k]}(x)+ X_i(x), \quad(i=1,2,\dots, n,\tag{2} \] where \(P_{\gamma i}^{[k]}(x)\) stands for a form \((\not\equiv 0\) for any \(i)\) of degree \(k\) with weight \(\gamma\), referred as to the zero approximation or not perturbed part of system (2) and \(X\) being the perturbation of this system. A new formal substitution reduces (2) to another system which is referred to as a generalized normal form (GNF). The simplification of the pre…
reviewed by: Nácere Hayek
Mathematics Subject Classification ID: 34C20
Mathematics Subject Classification ID: 37G05
Mathematics Subject Classification ID: 34C41
zbMATH DE Number: 2146330
zbMATH Keywords: normal form
zbMATH Keywords: generalized normal form
zbMATH Keywords: zero approximation
zbMATH Keywords: equivalence of systems
zbMATH Keywords: resonance equations
zbMATH Keywords: quasihomogeneous polynomial
MaRDI profile type: Publication
full work available at URL: a:1025144730356
OpenAlex ID: W248576587
description: scientific article; zbMATH DE number 2146330