Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation
We investigate the global asymptotic stability of the following second order rational difference equation of the form xn+1=Bxnxn-1+F/bxnxn-1+cxn-12, n=0,1,…, where the parameters B, F, b, and c and initial conditions x-1 and x0 are positive real numbers. The map associated with this equation is alw...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Wiley
2018-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2018/7052935 |
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| author | M. R. S. Kulenović S. Moranjkić M. Nurkanović Z. Nurkanović |
| author_facet | M. R. S. Kulenović S. Moranjkić M. Nurkanović Z. Nurkanović |
| author_sort | M. R. S. Kulenović |
| collection | DOAJ |
| description | We investigate the global asymptotic stability of the following second order rational difference equation of the form xn+1=Bxnxn-1+F/bxnxn-1+cxn-12, n=0,1,…, where the parameters B, F, b, and c and initial conditions x-1 and x0 are positive real numbers. The map associated with this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric space. In some cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. Also, we show that considered equation exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period. |
| format | Article |
| id | doaj-art-1c676a8b43df4fb7af009e5bff923158 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-1c676a8b43df4fb7af009e5bff9231582025-08-20T03:54:51ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2018-01-01201810.1155/2018/70529357052935Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference EquationM. R. S. Kulenović0S. Moranjkić1M. Nurkanović2Z. Nurkanović3Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USADepartment of Mathematics, University of Tuzla, 75000 Tuzla, Bosnia and HerzegovinaDepartment of Mathematics, University of Tuzla, 75000 Tuzla, Bosnia and HerzegovinaDepartment of Mathematics, University of Tuzla, 75000 Tuzla, Bosnia and HerzegovinaWe investigate the global asymptotic stability of the following second order rational difference equation of the form xn+1=Bxnxn-1+F/bxnxn-1+cxn-12, n=0,1,…, where the parameters B, F, b, and c and initial conditions x-1 and x0 are positive real numbers. The map associated with this equation is always decreasing in the second variable and can be either increasing or decreasing in the first variable depending on the parametric space. In some cases, we prove that local asymptotic stability of the unique equilibrium point implies global asymptotic stability. Also, we show that considered equation exhibits the Naimark-Sacker bifurcation resulting in the existence of the locally stable periodic solution of unknown period.http://dx.doi.org/10.1155/2018/7052935 |
| spellingShingle | M. R. S. Kulenović S. Moranjkić M. Nurkanović Z. Nurkanović Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation Discrete Dynamics in Nature and Society |
| title | Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation |
| title_full | Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation |
| title_fullStr | Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation |
| title_full_unstemmed | Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation |
| title_short | Global Asymptotic Stability and Naimark-Sacker Bifurcation of Certain Mix Monotone Difference Equation |
| title_sort | global asymptotic stability and naimark sacker bifurcation of certain mix monotone difference equation |
| url | http://dx.doi.org/10.1155/2018/7052935 |
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