The rational recursive sequence \(x_{n+1}=(\beta x^ 2_ n)/(1+x^ 2_{n-1})\)
type: Item
instance of: scholarly article
title: The rational recursive sequence \(x_{n+1}=(\beta x^ 2_ n)/(1+x^ 2_{n-1})\)
zbMATH Open document ID: 0806.39002
DOI: 10.1016/0898-1221(94)00091-3
author: G. Ladas
author: I. W. Rodrigues
author: Elias Camouzis
author: Sam Northshield
publication date: 1994-09-21T00:00:00Z
review text: The authors consider the global behavior of the rational sequence satisfying the equation (1) \(x_{n+1}=(\beta x^ 2_ n)/(1 + x^ 2_{n-1})\), \(n \in Z_ + :=\{0,1,2, \dots\}\), where \(\beta \in (0,\infty)\) is a parameter, and the initial values \(x_{-1}\), \(x_ 0\) are arbitrary positive numbers. The results obtained can be summarized as follows: A. Every solution of equation (1) with \(\beta \in (0,\infty)\) is bounded; B. When \(0<\beta<2\), the unique equilibrium point \(\overline x_ 1=0\) of (1) is globally asymptotically stable; C. When \(\beta=2\), equation (1) has two equilibria \(\overline x_ 1=0\) and \(\overline x_ 2=1\), where \(\overline x_ 1=0\) is locally asymptotically stable and \(\overline x_ 2=1\) is unstable. There exists a solution of (1) which is strictly increasing to \(\overline x_ 2=1\). Any solution of (1) which is neither identically equal to one nor strictly increasing to 1 obeys \(x_ n \to 0\) as \(n \to \infty\); D. When \(\beta>2\), equation (1) has three equilibria, namely, \(\overline x_ 1=0\), \(\overline x_ 2={1 \over 2} (\beta-\sqrt {\beta^ 2-4})\) and \(\overline x_ 3={1 \over 2} (\beta + \sqrt {\beta^ 2-4})\), where \(\overline x_ 1\) is locally asymptotically stable, \(\overline x_ 2\) is a saddle point, and \(\overline x_ 3\) is a repellor (hence they are all unstable). There exist two solutions of equation (1) such that one of them is strictly increasing to the smallest positive equilibrium \(\overline x_ 2\) and the other obeys \(x_{n-1} > \overline x_ 2\) for all \(n \in Z_ +\). The method of proof used is quite elementary.
Mathematics Subject Classification ID: 39A10
Mathematics Subject Classification ID: 37C70
zbMATH DE Number: 641429
zbMATH Keywords: recursive sequence
zbMATH Keywords: local asymptotic stability
zbMATH Keywords: global asymptotic stability
zbMATH Keywords: attractor
zbMATH Keywords: global behavior
zbMATH Keywords: rational sequence
zbMATH Keywords: repellor
MaRDI profile type: Publication
full work available at URL: 0898-1221(94)00091-3
OpenAlex ID: W2034370727
cites work: Q4005852
cites work: Oscillations and stability of a discrete delay logistic model
cites work: On rational recursive sequences
cites work: Global Attractivity in Nonlinear Delay Difference Equations
cites work: Q3137926
cites work: Oscillations and global attractivity in a discrete delay logistic model
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 641429