Basin of Attraction through Invariant Curves and Dominant Functions
We study a second-order difference equation of the form zn+1=znF(zn-1)+h, where both F(z) and zF(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...
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| Format: | Article |
| Language: | English |
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Wiley
2015-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2015/160672 |
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| author | Ziyad AlSharawi Asma Al-Ghassani A. M. Amleh |
| author_facet | Ziyad AlSharawi Asma Al-Ghassani A. M. Amleh |
| author_sort | Ziyad AlSharawi |
| collection | DOAJ |
| description | We study a second-order difference equation of the form zn+1=znF(zn-1)+h, where both F(z) and zF(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. |
| format | Article |
| id | doaj-art-c505e6a7499b4d33a7c5d34e3c852982 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-c505e6a7499b4d33a7c5d34e3c8529822025-02-03T01:07:25ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2015-01-01201510.1155/2015/160672160672Basin of Attraction through Invariant Curves and Dominant FunctionsZiyad AlSharawi0Asma Al-Ghassani1A. M. Amleh2Department of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Alkhoud, 123 Muscat, OmanDepartment of Mathematics and Statistics, Sultan Qaboos University, P.O. Box 36, Alkhoud, 123 Muscat, OmanSciences and Engineering, Paris-Sorbonne University Abu Dhabi, P.O. Box 38044, Abu Dhabi, UAEWe study a second-order difference equation of the form zn+1=znF(zn-1)+h, where both F(z) and zF(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.http://dx.doi.org/10.1155/2015/160672 |
| spellingShingle | Ziyad AlSharawi Asma Al-Ghassani A. M. Amleh Basin of Attraction through Invariant Curves and Dominant Functions Discrete Dynamics in Nature and Society |
| title | Basin of Attraction through Invariant Curves and Dominant Functions |
| title_full | Basin of Attraction through Invariant Curves and Dominant Functions |
| title_fullStr | Basin of Attraction through Invariant Curves and Dominant Functions |
| title_full_unstemmed | Basin of Attraction through Invariant Curves and Dominant Functions |
| title_short | Basin of Attraction through Invariant Curves and Dominant Functions |
| title_sort | basin of attraction through invariant curves and dominant functions |
| url | http://dx.doi.org/10.1155/2015/160672 |
| work_keys_str_mv | AT ziyadalsharawi basinofattractionthroughinvariantcurvesanddominantfunctions AT asmaalghassani basinofattractionthroughinvariantcurvesanddominantfunctions AT amamleh basinofattractionthroughinvariantcurvesanddominantfunctions |