Basin of attraction through invariant curves and dominant functions
type: Item
instance of: scholarly article
title: Basin of attraction through invariant curves and dominant functions
zbMATH Open document ID: 1418.39013
DOI: 10.1155/2015/160672
author: Ziyad AlSharawi
author: A. M. Amleh
author: Asma S. Al-Ghassani
published in: Discrete Dynamics in Nature and Society
publication date: 2019-02-19T00:00:00Z
review text: Summary: We study a second-order difference equation of the form \(z_{n + 1} = z_n F(z_{n - 1}) + h\), where both \(F(z)\) and \(z F(z)\) are decreasing. We consider a set of invariant curves at \(h = 1\) and use it to characterize the behaviour of solutions when \(h > 1\) and when \(0 < h < 1\). The case \(h > 1\) is related to the Y2K problem. For \(0 < h < 1\), we study the stability of the equilibrium solutions and find an invariant region where solutions are attracted to the stable equilibrium. In particular, for certain range of the parameters, a subset of the basin of attraction of the stable equilibrium is achieved by bounding positive solutions using the iteration of dominant functions with attracting equilibria.
Mathematics Subject Classification ID: 39A30
Mathematics Subject Classification ID: 39A20
zbMATH DE Number: 7025240
Wikidata QID: Q59105637
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2015/160672
OpenAlex ID: W1581241190
cites work: Q3137926
cites work: Q3994920
cites work: Q4120386
cites work: A note on difference-delay equations
cites work: Global stability in discrete population models with delayed-density dependence
cites work: The dynamics of some discrete models with delay under the effect of constant yield harvesting
cites work: A global attractor in some discrete contest competition models with delay under the effect of periodic stocking
cites work: The Beverton–Holt model with periodic and conditional harvesting
cites work: First integrals for difference equations
cites work: Q4650355
cites work: Q4381451
cites work: Invariants and Liapunov functions for nonautonomous systems
cites work: Open Problems and Conjectures
cites work: Global stability, two conjectures and Maple
cites work: Q5474854
cites work: Global attractors for difference equations dominated by one-dimensional maps
cites work: The Y2K problem revisited
cites work: Global attractivity of the equilibrium of a difference equation: An elementary proof assisted by computer algebra system
cites work: Dynamics of second order rational difference equations. With open problems and conjectures
description: scientific article; zbMATH DE number 7025240