The homogenization of an ordinary differential equation with a chessboard structure
type: Item
instance of: scholarly article
title: The homogenization of an ordinary differential equation with a chessboard structure
zbMATH Open document ID: 0734.34013
author: Stefano Mortola
author: Roberto Peirone
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 1991-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/244194
review text: Let f(x,y) denote the function taking value \(\alpha\) (resp. \(\beta\)) on black (resp. white) squares of the infinite chessboard and let y denote any solution of the differential equation \(y'=f(x,y)\). It is known from a theorem of L. Piccinini that \(\gamma (\alpha,\beta)=\lim_{x\to \infty}y(x)/x\) exists and is independent of y. In this note, the authors show that the function \(\gamma\) is continuous and increasing but very intricated, despite of the linearity of f in \(\alpha\) and \(\beta\). The more remarkable result is the description of the contour line \(\gamma (\alpha,\beta)=c\) that highly depends on arithmetic properties of c. Proofs are not given but ``will appear in a forthcoming paper''.
Mathematics Subject Classification ID: 34A25
Mathematics Subject Classification ID: 34A99
zbMATH DE Number: 4214394
zbMATH Keywords: homogenization
zbMATH Keywords: infinite chessboard
reviewed by: Gilles Christol
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4214394