Local Stability of Period Two Cycles of Second Order Rational Difference Equation

Local Stability of Period Two Cycles of Second Order Rational Difference Equation

We consider the second order rational difference equation   n = 0,1,2,…, where the parameters are positive real numbers, and the initial conditions are nonnegative real numbers. We give a necessary and sufficient condition for the equation to have a prime period-two solution. We show that the peri...

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Main Authors: S. Atawna, R. Abu-Saris, I. Hashim
Format: Article
Language:English
Published: Hindawi Limited 2012-01-01
Series:Discrete Dynamics in Nature and Society
Online Access:http://dx.doi.org/10.1155/2012/969813
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spelling doaj-7c7fc52314504486bc0c30fc2e3dab942020-11-24T23:48:45ZengHindawi LimitedDiscrete Dynamics in Nature and Society1026-02261607-887X2012-01-01201210.1155/2012/969813969813Local Stability of Period Two Cycles of Second Order Rational Difference EquationS. Atawna0R. Abu-Saris1I. Hashim2School of Mathematical Sciences, Universiti Kebangsaan Malaysia, Selangor, 43600 Bangi, MalaysiaDepartment of Basic Sciences, King Saud bin Abdulaziz University for Health Sciences, P.O. Box 22490, Riyadh 11426, Saudi ArabiaSchool of Mathematical Sciences, Universiti Kebangsaan Malaysia, Selangor, 43600 Bangi, MalaysiaWe consider the second order rational difference equation   n = 0,1,2,…, where the parameters are positive real numbers, and the initial conditions are nonnegative real numbers. We give a necessary and sufficient condition for the equation to have a prime period-two solution. We show that the period-two solution of the equation is locally asymptotically stable. In particular, we solve Conjecture 5.201.2 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.3 in Kulenović and Ladas monograph (2002).http://dx.doi.org/10.1155/2012/969813
collection DOAJ
language English
format Article
sources DOAJ
author S. Atawna
R. Abu-Saris
I. Hashim
spellingShingle S. Atawna
R. Abu-Saris
I. Hashim
Local Stability of Period Two Cycles of Second Order Rational Difference Equation
Discrete Dynamics in Nature and Society
author_facet S. Atawna
R. Abu-Saris
I. Hashim
author_sort S. Atawna
title Local Stability of Period Two Cycles of Second Order Rational Difference Equation
title_short Local Stability of Period Two Cycles of Second Order Rational Difference Equation
title_full Local Stability of Period Two Cycles of Second Order Rational Difference Equation
title_fullStr Local Stability of Period Two Cycles of Second Order Rational Difference Equation
title_full_unstemmed Local Stability of Period Two Cycles of Second Order Rational Difference Equation
title_sort local stability of period two cycles of second order rational difference equation
publisher Hindawi Limited
series Discrete Dynamics in Nature and Society
issn 1026-0226
1607-887X
publishDate 2012-01-01
description We consider the second order rational difference equation   n = 0,1,2,…, where the parameters are positive real numbers, and the initial conditions are nonnegative real numbers. We give a necessary and sufficient condition for the equation to have a prime period-two solution. We show that the period-two solution of the equation is locally asymptotically stable. In particular, we solve Conjecture 5.201.2 proposed by Camouzis and Ladas in their book (2008) which appeared previously in Conjecture 11.4.3 in Kulenović and Ladas monograph (2002).
url http://dx.doi.org/10.1155/2012/969813
work_keys_str_mv AT satawna localstabilityofperiodtwocyclesofsecondorderrationaldifferenceequation
AT rabusaris localstabilityofperiodtwocyclesofsecondorderrationaldifferenceequation
AT ihashim localstabilityofperiodtwocyclesofsecondorderrationaldifferenceequation
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