Dynamic behavior of a recursive sequence
type: Item
instance of: scholarly article
title: Dynamic behavior of a recursive sequence
zbMATH Open document ID: 1069.39025
published in: Applied Mathematics and Computation
publication date: 2004-11-23T00:00:00Z
review text: The authors study the behavior of solutions of the difference equation (\(*\)) \(x_{n+1}=\alpha-x_{n-1}/x_n\) (\(n=0,1,2,\dots\)), where \(\alpha\in{\mathbb R}\) is a given number. According to the value of \(\alpha\) and the initial conditions, the authors show that the negative equilibrium point \(\bar x:=\alpha-1\) is globally asymptotically stable (for \(\alpha<-1\)), or every negative solution converges to a two-cycle (for \(\alpha=-1\)), or (\(*\)) has unbounded solutions (for \(-1<\alpha\leq1\)), or (\(*\)) is chaotic (for \(1<\alpha<3\)), or the positive equilibrium point \(\bar x=\alpha-1\) is a global attractor with a restricted basin (for \(\alpha>3\)). Similar results were recently obtained for the difference equation (\(*\)) but with the plus instead of the minus sign.
reviewed by: Roman Šimon Hilscher
Mathematics Subject Classification ID: 39A20
Mathematics Subject Classification ID: 39A11
zbMATH DE Number: 2117758
zbMATH Keywords: rational difference equation
zbMATH Keywords: global attractivity
zbMATH Keywords: global asymptotic stability
zbMATH Keywords: period-two solution
zbMATH Keywords: permanence
zbMATH Keywords: chaotic behavior
zbMATH Keywords: negative solutions
zbMATH Keywords: unbounded solutions
author: Xing-Xue Yan
author: Wan-Tong Li
MaRDI profile type: Publication
full work available at URL: j.amc.2003.08.107
OpenAlex ID: W2070953424
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DOI: 10.1016/J.AMC.2003.08.107
description: scientific article; zbMATH DE number 2117758