Arnold normal form and formal reduction of system of linear difference equations
type: Item
instance of: scholarly article
title: Arnold normal form and formal reduction of system of linear difference equations
zbMATH Open document ID: 0892.39001
DOI: 10.1007/BF02755460
author: Guoting Chen
published in: Aequationes Mathematicae
publication date: 1998-07-19T00:00:00Z
full work available at URL: https://eudml.org/doc/137707
review text: The author presents an algorithm to obtain Turrittin's formal reduction of linear difference equations of the forms \(Y(x+1)=x^cA(x)Y(x)\) or \(Y(x+1)-Y(x)=x^{-r/s}A(x)Y(x)\), where \(c\) is a rational number, \(r\) and \(s\) are two positive integers, and the \(n\times n\) matrix \(A(x)\) admits the formal expansion \(A(x)=A_0+A_1 x^{-1/s}+A_2 x^{-2/s}+\ldots\), with \(A_0\neq0\) [cf. \textit{H. L. Turrittin}, Bol. Soc. Mat. Mex., II. Ser. 5, 255-264 (1960; Zbl 0100.08201)]. In particular, \(\det{A(x)}\neq0\) in the first system. In the nilpotent case, the Arnold normal form of matrices depending on parameters is used [cf. \textit{V. I. Arnold}, Usp. Mat. Nauk 26, No. 2(158), 101-114 (1971; Zbl 0258.15011)]. It is shown that for systems in the second form, the exponential part in a formal fundamental matrix solution \(Y(x)=F(x^{-1/s'})x^C\exp(Q(x^{1/s'})\) depends only on the matrix coefficients \(A_0,\ldots,A_{m(s-r)-1}\). Similarly, for systems in the first form, the \(\Gamma\) part of the formal fundamental matrix solution \(Y(x)=F(x^{-1/s'})x^C\exp(Q(x^{1/s'})R^x\Gamma(x)^M\) depends only on \(A_0,\ldots,A_m\), and, generally, the irregular part depends only on \(A_0,\ldots,A_{m+ns-1}\). These results are important from an algorithmic point of view and the bounds are best possible.
Mathematics Subject Classification ID: 39A10
zbMATH DE Number: 1098631
zbMATH Keywords: system of linear difference equations
zbMATH Keywords: Turittin's formal reduction
zbMATH Keywords: Arnold's normal form
reviewed by: Rémi Vaillancourt
MaRDI profile type: Publication
cites work: ON MATRICES DEPENDING ON PARAMETERS
cites work: Formal reduction theory of meromorphic differential equations: a group theoretic view
cites work: Q3331307
cites work: Q5640134
cites work: Asymptotics of analytic difference equations
cites work: Q3669874
cites work: Q3281277
cites work: Arnold's canonical matrices and the asymptotic simplification of ordinary differential equations
Recommended article: Q3483569
Recommended article: Formal reduction of linear difference systems
Recommended article: Normal forms for difference and differential systems
Recommended article: Effective formal reduction of linear differential systems
Recommended article: A characterization of reduced forms of linear differential systems
Recommended article: Normal and quasinormal forms for systems of difference and differential-difference equations
Recommended article: Q3778298
Recommended article: Form Symmetries and Reduction of Order in Difference Equations
Recommended article: Q3336087
description: scientific article; zbMATH DE number 1098631