Global Dynamics and Bifurcations of Certain Second Order Rational Difference Equation with Quadratic Terms

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• 作者：Sabina Jašarević Hrustić ; M. R. S. Kulenović…
• 关键词：Attractivity ; Basin ; Bifurcation ; Difference equation ; Stable manifold ; Unstable manifold
• 刊名：Qualitative Theory of Dynamical Systems
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：15
• 期：1
• 页码：283-307
• 全文大小：952 KB
• 参考文献：1.Brett, A., Kulenović, M.R.S.: Basins of attraction of equlilibrium points of monotone difference equations. Sarajev. J. Math. 5(18), 211–233 (2009)MATH
  2.Camouzis, E., Ladas, G.: When does local asymptotic stability imply global attractivity in rational equations? J. Differ. Equ. Appl. 12, 863–885 (2006)MathSciNet CrossRef MATH
  3.Dehghan, M., Kent, C.M., Mazrooei-Sebdani, R., Ortiz, N.L., Sedaghat, H.: Dynamics of rational difference equations containing quadratic terms. J. Differ. Equ. Appl. 14, 191–208 (2008)MathSciNet CrossRef MATH
  4.Dehghan, M., Kent, C.M., Mazrooei-Sebdani, R., Ortiz, N.L., Sedaghat, H.: Monotone and oscillatory solutions of a rational difference equation containing quadratic terms. J. Differ. Equ. Appl. 14, 1045–1058 (2008)MathSciNet CrossRef MATH
  5.Drymonis, E., Ladas, G.: On the global character of the rational system \(x_{n+1}=\frac{\alpha _{1}}{A_{1}+B_{1}x_{n}+y_{n}}\) and \(y_{n+1}=\frac{\alpha _{2}+\beta _{2}x_{n}}{A_{2}+B_{2}x_{n}+C_{2}y_{n}}\) . Sarajev. J. Math. 8(21), 293–309 (2012)MathSciNet CrossRef
  6.Garić-Demirović, M., Kulenović, M.R.S., Nurkanović, M.: Basins of attraction of equilibrium points of second order difference equations. Appl. Math. Lett. 25, 2110–2115 (2012)MathSciNet CrossRef MATH
  7.Garić-Demirović, M., Kulenović, M.R.S., Nurkanović, M.: Global dynamics of certain homogeneous second order quadratic fractional difference equation. Sci. World J. Math. Anal. 2013, article ID 210846 (10 pages)
  8.Garić-Demirović, M., Nurkanović, M.: Dynamics of an anti-competitive two dimensional rational system of difference equations. Sarajev. J. Math. 7(19), 39–56 (2011)MathSciNet MATH
  9.Grove, E.A., Hadley, D., Lapierre, E., Schultz, S.W.: On the global behavior of the rational system \(x_{n+1}=\frac{\alpha _{1}}{ x_{n}+y_{n}}\) and \(y_{n+1}=\frac{\alpha _{2}+\beta _{2}x_{n}+y_{n}}{y_{n}}\) . Sarajev. J. Math. 8(21), 283–292 (2012)MathSciNet CrossRef
  10.Grove, E.A., Ladas, G.: Periodicities in Nonlinear Difference Equations. Advances in Discrete Mathematics and Applications, 4th edn. Chapman & Hall/CRC, Boca Raton (2005)MATH
  11.Jašarević, S., Kulenović, M.R.S.: Basins of attraction of equilibrium and boundary points of second order difference equation. J. Differ. Equ. Appl. 20, 947–959 (2014)CrossRef MATH
  12.Kalabušić, S., Kulenović, M.R.S., Pilav, E.: Global dynamics of anti-competitive systems in the plane. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 20, 477–505 (2013)MathSciNet MATH
  13.Kent, C.M., Sedaghat, H.: Global attractivity in a quadratic-linear rational difference equation with delay. J. Differ. Equ. Appl. 15, 913–925 (2009)MathSciNet CrossRef MATH
  14.Kent, C.M., Sedaghat, H.: Global attractivity in a rational delay difference equation with quadratic terms. J. Differ. Equ. Appl. 17, 457–466 (2011)MathSciNet CrossRef MATH
  15.Kulenović, M.R.S., Ladas, G.: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman and Hall/CRC, Boca Raton (2001)CrossRef MATH
  16.Kulenović, M.R.S., Merino, O.: Discrete Dynamical Systems and Difference Equations with Mathematica. Chapman and Hall/CRC, Boca Raton (2002)CrossRef MATH
  17.Kulenović, M.R.S., Merino, O.: Global bifurcations for competitive system in the plane. Discrete Contin. Dyn. Syst. Ser. B 12, 133–149 (2009)MathSciNet CrossRef MATH
  18.Kulenović, M.R.S., Merino, O.: Invariant manifolds for competitive discrete systems in the plane, Internat. J. Bifurn. Chaos Appl. Sci. Eng. 20, 2471–2486 (2010)CrossRef MATH
  19.Kulenović, M.R.S., Pilav, E., Silić, E.: Local dynamics and global attractivity of a certain second order quadratic fractional difference equation. Adv. Differ. Equ. 2014, 32p (2014)CrossRef
  20.Ladas, G., Lugo, G., Palladino, F.J.: Open problems and conjectures on rational systems in three dimensions. Sarajev. J. Math. 8(21), 311–321 (2012)MathSciNet CrossRef MATH
  21.Moranjkić, S., Nurkanović, Z.: Basins of attractionof certain rational anti-competitive system of difference equations in theplane. Adv. Differ. Equ. 2012, 153 (2012)CrossRef
  22.Sedaghat, H.: Global behaviours of rational difference equations of orders two and three with quadratic terms. J. Differ. Equ. Appl. 15, 215–224 (2009)MathSciNet CrossRef MATH
  23.Smith, H.L.: Periodic competitive differential equations and the discrete dynamics of competitive maps. J. Differ. Equ. 64, 163–194 (1986)MathSciNet CrossRef
  24.Smith, H.L.: Planar competitive and cooperative difference equations. J. Differ. Equ. Appl. 3, 335–357 (1998)MathSciNet CrossRef MATH
  
• 作者单位：Sabina Jašarević Hrustić (1)
  M. R. S. Kulenović (2)
  M. Nurkanović (1)
  
  1. Department of Mathematics, University of Tuzla, 75000, Tuzla, Bosnia and Herzegovina
  2. Department of Mathematics, University of Rhode Island, Kingston, RI, 02881-0816, USA
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Mathematics
  Dynamical Systems and Ergodic Theory
  Difference and Functional Equations
  
• 出版者：Birkh盲user Basel
• ISSN：1662-3592

文摘

We investigate global dynamics of the equation $$\begin{aligned} x_{n+1}=\frac{x_{n-1}}{ax_{n}^{2}+ex_{n-1}+f},\quad n=0,1,2,\ldots , \end{aligned}$$where the parameters a, e and f are nonnegative numbers with condition \(a+e+f>0\) and the initial conditions \(x_{-1},x_{0}\) are arbitrary nonnegative numbers such that \(x_{-1}+x_{0}>0\). The global dynamics of this equation consists of three bifurcations, two exchange of stability bifurcations and one global period doubling bifurcation. Keywords Attractivity Basin Bifurcation Difference equation Stable manifold Unstable manifold Mathematics Subject Classification 39A10 39A23 39A28 39A30 Page %P Close Plain text Look Inside Reference tools Export citation EndNote (.ENW) JabRef (.BIB) Mendeley (.BIB) Papers (.RIS) Zotero (.RIS) BibTeX (.BIB) Add to Papers Other actions Register for Journal Updates About This Journal Reprints and Permissions Share Share this content on Facebook Share this content on Twitter Share this content on LinkedIn Related Content Supplementary Material (0) References (24) References1.Brett, A., Kulenović, M.R.S.: Basins of attraction of equlilibrium points of monotone difference equations. Sarajev. J. Math. 5(18), 211–233 (2009)MATH2.Camouzis, E., Ladas, G.: When does local asymptotic stability imply global attractivity in rational equations? J. Differ. Equ. Appl. 12, 863–885 (2006)MathSciNetCrossRefMATH3.Dehghan, M., Kent, C.M., Mazrooei-Sebdani, R., Ortiz, N.L., Sedaghat, H.: Dynamics of rational difference equations containing quadratic terms. J. Differ. Equ. Appl. 14, 191–208 (2008)MathSciNetCrossRefMATH4.Dehghan, M., Kent, C.M., Mazrooei-Sebdani, R., Ortiz, N.L., Sedaghat, H.: Monotone and oscillatory solutions of a rational difference equation containing quadratic terms. J. Differ. Equ. Appl. 14, 1045–1058 (2008)MathSciNetCrossRefMATH5.Drymonis, E., Ladas, G.: On the global character of the rational system \(x_{n+1}=\frac{\alpha _{1}}{A_{1}+B_{1}x_{n}+y_{n}}\) and \(y_{n+1}=\frac{\alpha _{2}+\beta _{2}x_{n}}{A_{2}+B_{2}x_{n}+C_{2}y_{n}}\). Sarajev. J. Math. 8(21), 293–309 (2012)MathSciNetCrossRef6.Garić-Demirović, M., Kulenović, M.R.S., Nurkanović, M.: Basins of attraction of equilibrium points of second order difference equations. Appl. Math. Lett. 25, 2110–2115 (2012)MathSciNetCrossRefMATH7.Garić-Demirović, M., Kulenović, M.R.S., Nurkanović, M.: Global dynamics of certain homogeneous second order quadratic fractional difference equation. Sci. World J. Math. Anal. 2013, article ID 210846 (10 pages)8.Garić-Demirović, M., Nurkanović, M.: Dynamics of an anti-competitive two dimensional rational system of difference equations. Sarajev. J. Math. 7(19), 39–56 (2011)MathSciNetMATH9.Grove, E.A., Hadley, D., Lapierre, E., Schultz, S.W.: On the global behavior of the rational system \(x_{n+1}=\frac{\alpha _{1}}{ x_{n}+y_{n}}\) and \(y_{n+1}=\frac{\alpha _{2}+\beta _{2}x_{n}+y_{n}}{y_{n}}\). Sarajev. J. Math. 8(21), 283–292 (2012)MathSciNetCrossRef10.Grove, E.A., Ladas, G.: Periodicities in Nonlinear Difference Equations. Advances in Discrete Mathematics and Applications, 4th edn. Chapman & Hall/CRC, Boca Raton (2005)MATH11.Jašarević, S., Kulenović, M.R.S.: Basins of attraction of equilibrium and boundary points of second order difference equation. J. Differ. Equ. Appl. 20, 947–959 (2014)CrossRefMATH12.Kalabušić, S., Kulenović, M.R.S., Pilav, E.: Global dynamics of anti-competitive systems in the plane. Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal. 20, 477–505 (2013)MathSciNetMATH13.Kent, C.M., Sedaghat, H.: Global attractivity in a quadratic-linear rational difference equation with delay. J. Differ. Equ. Appl. 15, 913–925 (2009)MathSciNetCrossRefMATH14.Kent, C.M., Sedaghat, H.: Global attractivity in a rational delay difference equation with quadratic terms. J. Differ. Equ. Appl. 17, 457–466 (2011)MathSciNetCrossRefMATH15.Kulenović, M.R.S., Ladas, G.: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman and Hall/CRC, Boca Raton (2001)CrossRefMATH16.Kulenović, M.R.S., Merino, O.: Discrete Dynamical Systems and Difference Equations with Mathematica. Chapman and Hall/CRC, Boca Raton (2002)CrossRefMATH17.Kulenović, M.R.S., Merino, O.: Global bifurcations for competitive system in the plane. Discrete Contin. Dyn. Syst. Ser. B 12, 133–149 (2009)MathSciNetCrossRefMATH18.Kulenović, M.R.S., Merino, O.: Invariant manifolds for competitive discrete systems in the plane, Internat. J. Bifurn. Chaos Appl. Sci. Eng. 20, 2471–2486 (2010)CrossRefMATH19.Kulenović, M.R.S., Pilav, E., Silić, E.: Local dynamics and global attractivity of a certain second order quadratic fractional difference equation. Adv. Differ. Equ. 2014, 32p (2014)CrossRef20.Ladas, G., Lugo, G., Palladino, F.J.: Open problems and conjectures on rational systems in three dimensions. Sarajev. J. Math. 8(21), 311–321 (2012)MathSciNetCrossRefMATH21.Moranjkić, S., Nurkanović, Z.: Basins of attractionof certain rational anti-competitive system of difference equations in theplane. Adv. Differ. Equ. 2012, 153 (2012)CrossRef22.Sedaghat, H.: Global behaviours of rational difference equations of orders two and three with quadratic terms. J. Differ. Equ. Appl. 15, 215–224 (2009)MathSciNetCrossRefMATH23.Smith, H.L.: Periodic competitive differential equations and the discrete dynamics of competitive maps. J. Differ. Equ. 64, 163–194 (1986)MathSciNetCrossRef24.Smith, H.L.: Planar competitive and cooperative difference equations. J. Differ. Equ. Appl. 3, 335–357 (1998)MathSciNetCrossRefMATH About this Article Title Global Dynamics and Bifurcations of Certain Second Order Rational Difference Equation with Quadratic Terms Journal Qualitative Theory of Dynamical Systems Volume 15, Issue 1 , pp 283-307 Cover Date2016-04 DOI 10.1007/s12346-015-0148-x Print ISSN 1575-5460 Online ISSN 1662-3592 Publisher Springer International Publishing Additional Links Register for Journal Updates Editorial Board About This Journal Manuscript Submission Topics Mathematics, general Dynamical Systems and Ergodic Theory Difference and Functional Equations Keywords Attractivity Basin Bifurcation Difference equation Stable manifold Unstable manifold 39A10 39A23 39A28 39A30 Authors Sabina Jašarević Hrustić (1) M. R. S. Kulenović (2) M. Nurkanović (1) Author Affiliations 1. Department of Mathematics, University of Tuzla, 75000, Tuzla, Bosnia and Herzegovina 2. 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