Naimark-Sacker bifurcation of second order rational difference equation with quadratic terms

Volume 10, Issue 7, pp 3477--3489 http://dx.doi.org/10.22436/jnsa.010.07.11

Publication Date: July 22, 2017 Submission Date: January 31, 2015

Download PDF

Download XML

• Export citations
  • CITATIONS & REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • ARTICLE CITATION
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)
  • REFERENCES
  • EndNote (.ENW)
  • Mendeley, JabRef (.BIB)
  • Papers, Zotero, Reference Manager, RefWorks (.RIS)

• 2219 Downloads
• 3953 Views

Authors

M. R. S. Kulenovic - Department of Mathematics, University of Rhode Island, Kingston, RI 02881, USA. S. Moranjkic - Department of Mathematics, University of Tuzla, 75350 Tuzla, Bosnia and Herzegovina. Z. Nurkanovic - Department of Mathematics, University of Tuzla, 75350 Tuzla, Bosnia and Herzegovina.

Abstract

We investigate the global asymptotic stability and Naimark-Sacker bifurcation of the difference equation \[x_{n+1} =\frac{F}{bx_nx_{n-1} + cx^2_{n-1} + f} , n = 0, 1, ... ,\] with non-negative parameters and nonnegative initial conditions \(x_{-1}, x_0\) such that \(bx_0x_{-1} + cx^2_{-1} + f > 0\). By using fixed point theorem for monotone maps we find the region of parameters where the unique equilibrium is globally asymptotically stable.

Share and Cite

Keywords

• Attractivity
• bifurcation
• difference equation
• invariant
• Naimark-Sacker bifurcation
• periodic solution.

MSC

•  39A10
•  39A28
•  39A30

References

• [1] A. M. Amleh, E. Camouzis, G. Ladas, On the dynamics of a rational difference equation, I, Int. J. Difference Equ., 3 (2008), 1–35.
  • MathSciNet


• [2] A. M. Amleh, E. Camouzis, G. Ladas, On the dynamics of a rational difference equation, II, Int. J. Difference Equ., 3 (2008), 195–225.
  • MathSciNet
  • Google Scholar


• [3] J. K. Hale, H. Kocak, Dynamics and bifurcations, Texts in Applied Mathematics, Springer-Verlag, New York (1991)
  • Google Scholar


• [4] E. A. Janowski, M. R. S. Kulenović, Attractivity and global stability for linearizable difference equations, Comput. Math. Appl., 57 (2009), 1592–1607.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [5] C. M. Kent, H. Sedaghat, Global attractivity in a quadratic-linear rational difference equation with delay, J. Difference Equ. Appl., 15 (2009), 913–925.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [6] C. M. Kent, H. Sedaghat, Global attractivity in a rational delay difference equation with quadratic terms, J. Difference Equ. Appl., 17 (2011), 457–466.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [7] M. R. S. Kulenović, G. Ladas, Dynamics of second order rational difference equations, With open problems and conjectures, Chapman & Hall/CRC, Boca Raton, FL (2001)
  • Google Scholar


• [8] M. R. S. Kulenović, O. Merino, Discrete dynamical systems and difference equations with Mathematica, Chapman & Hall/CRC, Boca Raton, FL (2002)
  • View Article
  • MathSciNet
  • Google Scholar


• [9] M. R. S. Kulenović, O. Merino, A global attractivity result for maps with invariant boxes, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), 97–110.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [10] M. R. S. Kulenović, O. Merino, Global bifurcation for discrete competitive systems in the plane, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 133–149.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [11] M. R. S. Kulenović, O. Merino, Invariant manifolds for competitive discrete systems in the plane, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 20 (2010), 2471–2486.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [12] M. R. S. Kulenović, E. Pilav, E. Silić, Naimark-Sacker bifurcation of a certain second order quadratic fractional difference equation, J. Math. Comput. Sci., 4 (2014), 1025–1043.
  • Google Scholar


• [13] Y. A. Kuznetsov, Elements of applied bifurcation theory, Second edition. Applied Mathematical Sciences, Springer- Verlag, New York (1998)
  • MathSciNet
  • Google Scholar


• [14] C. Robinson, Stability, symbolic dynamics, and chaos, Stud. Adv. Math. Boca Raton, CRC Press, FL (1995)
  • Google Scholar


• [15] H. Sedaghat, Global behaviours of rational difference equations of orders two and three with quadratic terms, J. Difference Equ. Appl., 15 (2009), 215–224.
  • View Article
  • MathSciNet
  • Google Scholar
  • MATH


• [16] S. Wiggins, Introduction to applied nonlinear dynamical systems and chaos, Second edition, Texts in Applied Mathematics, Springer-Verlag, New York (2003)
  • MathSciNet
  • Google Scholar