Table of Contents Author Guidelines Submit a Manuscript
Discrete Dynamics in Nature and Society
Volume 2017, Article ID 1295089, 8 pages
https://doi.org/10.1155/2017/1295089
Research Article

Global Dynamics of Rational Difference Equations and

1School of Economics and Finance, Xi’an Jiaotong University, Xi’an 710061, China
2School of Mathematics and Statistics, North China University of Water Resources and Electric Power, Zhengzhou 450045, China
3College of Business Administration, Huaqiao University, Quanzhou 362021, China

Correspondence should be addressed to Weizhou Zhong; nc.ude.utjx.liam@uohziew

Received 24 December 2016; Revised 9 March 2017; Accepted 15 March 2017; Published 3 May 2017

Academic Editor: Douglas R. Anderson

Copyright © 2017 Keying Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. Q. Zhang and W. Zhang, “On the system of nonlinear rational difference equations,” International Journal of Mathematical, Computational, Physical and Quantum Engineering, vol. 8, no. 4, pp. 688–691, 2014. View at Google Scholar
  2. Q. Xiao and Q.-H. Shi, “Qualitative behavior of a rational difference equation yn+1=(yn+yn-1)/(p+ynyn-1),” Advances in Difference Equations, vol. 2011, article 6, 2011. View at Publisher · View at Google Scholar · View at Scopus
  3. M. M. El-Dessoky, “Qualitative behavior of rational difference equation of big order,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 495838, 6 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  4. R. Abu-Saris, C. Çinar, and I. Yalçinkaya, “On the asymptotic stability of xn+1=(a+xnxn-k)/(xn+xn-k),” Computers and Mathematics with Applications, vol. 56, no. 5, pp. 1172–1175, 2008. View at Publisher · View at Google Scholar
  5. L. Xianyi and Z. Deming, “Global asymptotic stability in a rational equation,” Journal of Difference Equations and Applications, vol. 9, no. 9, pp. 833–839, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. X. Li and D. Zhu, “Two rational recursive sequences,” Computers and Mathematics with Applications, vol. 47, no. 10-11, pp. 1487–1494, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  7. M. Gümüş and Ö. Öcalan, “Global asymptotic stability of a nonautonomous difference equation,” Journal of Applied Mathematics, vol. 2014, Article ID 395954, 5 pages, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  8. S. Jašarević and M. R. S. Kulenović, “Basins of attraction of equilibrium and boundary points of second-order difference equations,” Journal of Difference Equations and Applications, vol. 20, no. 5-6, pp. 947–959, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. E. J. Janowski and M. R. Kulenović, “Attractivity and global stability for linearizable difference equations,” Computers & Mathematics with Applications, vol. 57, no. 9, pp. 1592–1607, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. C. M. Kent and H. Sedaghat, “Global attractivity in a quadratic-linear rational difference equation with delay,” Journal of Difference Equations and Applications, vol. 15, no. 10, pp. 913–925, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  11. C. M. Kent and H. Sedaghat, “Global attractivity in a rational delay difference equation with quadratic terms,” Journal of Difference Equations and Applications, vol. 17, no. 4, pp. 457–466, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  12. H. Sedaghat, “Global behaviours of rational difference equations of orders two and three with quadratic terms,” Journal of Difference Equations and Applications, vol. 15, no. 3, pp. 215–224, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. S. Jašarević Hrustić, M. R. Kulenović, and M. Nurkanović, “Local dynamics and global stability of certain second-order rational difference equation with quadratic terms,” Discrete Dynamics in Nature and Society, vol. 2016, Article ID 3716042, 14 pages, 2016. View at Publisher · View at Google Scholar · View at MathSciNet
  14. B. Iričanin and S. Stević, “On a class of third-order nonlinear difference equations,” Applied Mathematics and Computation, vol. 213, no. 2, pp. 479–483, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. X. Yang, “On the system of rational difference equations xn=A+yn-1/(xn-pyn-q),yn=A+xn-1(xn-ryn-s),” Journal of Mathematical Analysis and Applications, vol. 307, no. 1, pp. 305–311, 2005. View at Publisher · View at Google Scholar · View at MathSciNet
  16. A. Q. Khan, M. N. Qureshi, and Q. Din, “Global dynamics of some systems of higher-order rational difference equations,” Advances in Difference Equations, vol. 2013, article 354, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  17. B. Sroysang, “Dynamics of a system of rational higher-order difference equation,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 179401, 5 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  18. Q. Din, M. N. Qureshi, and A. Q. Khan, “Dynamics of a fourth-order system of rational difference equations,” Advances in Difference Equations, vol. 2012, article 215, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  19. A. Q. Khan, “Global dynamics of two systems of exponential difference equations by Lyapunov function,” Advances in Difference Equations, vol. 2014, article 297, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. M. R. S. Kulenović and G. Ladas, Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2002.
  21. E. Camouzis and G. Lada, Dynamics of Third-Order Rational Difference Equations with Open Problems and Conjectures, Chapman & Hall/CRC Press, Boca Raton, Fla, USA, 2008.
  22. C. Wang and M. Hu, “On the solutions of a rational recursive sequence,” Journal of Mathematics and Informatics, vol. 1, pp. 25–33, 2013. View at Google Scholar
  23. M. Aloqeili, “Dynamics of a kth order rational difference equation,” Applied Mathematics and Computation, vol. 181, no. 2, pp. 1328–1335, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  24. R. Abo-Zeid, “On the oscillation of a third order rational difference equation,” Journal of the Egyptian Mathematical Society, vol. 23, no. 1, pp. 62–66, 2015. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. S. Stevi, B. Iričanin, and Z. Šmarda, “Boundedness character of a fourth-order system of difference equations,” Advances in Difference Equations, vol. 2015, article 315, 2015. View at Publisher · View at Google Scholar
  26. M. DiPippo, E. J. Janowski, and M. R. Kulenović, “Global asymptotic stability for quadratic fractional difference equation,” Advances in Difference Equations, vol. 2015, article 179, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. D. Chen, X. Li, and Y. Wang, “Dynamics for nonlinear difference equation xn+1=(αxn-k)/(β+γxn-p),” Advances in Difference Equations, vol. 2009, Article ID 235691, 2009. View at Publisher · View at Google Scholar
  28. H. Shojaei, S. Parvandeh, T. Mohammadi, Z. Mohammadi, and N. Mohammadi, “Stability and convergence of a higher order rational difference equation,” Australian Journal of Basic and Applied Sciences, vol. 5, no. 11, pp. 72–77, 2011. View at Google Scholar · View at Scopus
  29. M. Dehghan and N. Rastegar, “Stability and periodic character of a third order difference equation,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 2560–2564, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  30. K. Liu, Z. Zhao, X. Li, and P. Li, “More on three-dimensional systems of rational difference equations,” Discrete Dynamics in Nature and Society, vol. 2011, Article ID 178483, 9 pages, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  31. K. Liu, Z. Wei, P. Li, and W. Zhong, “On the behavior of a system of rational difference equations xn+1=xn-1/(ynxn-1-1), yn+1=yn-1/(xnyn-1-1),zn+1=1/(xnzn-1),” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 105496, 9 pages, 2012. View at Publisher · View at Google Scholar
  32. K. Liu, P. Li, and W. Zhong, “On a system of rational difference equations,” Fasciculi Mathematici, vol. 51, pp. 105–114, 2013. View at Google Scholar
  33. K. Liu, P. Li, F. Han, and W. Zhong, “Behavior of the difference equations xn+1=xnxn-1-1,” Journal of Computational Analysis and Application, vol. 23, no. 7, pp. 1361–1370, 2017. View at Google Scholar
  34. M. Garić-Demirović, M. R. Kulenović, and M. Nurkanović, “Basins of attraction of equilibrium points of second order difference equations,” Applied Mathematics Letters, vol. 25, no. 12, pp. 2110–2115, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  35. M. R. S. Kulenović and O. Merino, “Invariant manifolds for competitive discrete systems in the plane,” International Journal of Bifurcation and Chaos, vol. 20, no. 8, pp. 2471–2486, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  36. M. R. Kulenović and M. Nurkanović, “Global asymptotic behavior of a two-dimensional system of difference equations modeling cooperation,” Journal of Difference Equations and Applications, vol. 9, no. 1, pp. 149–159, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  37. E. Camouzis, M. R. S. Kulenovic, G. Ladas, and O. Merino, “Rational systems in the plane,” Journal of Difference Equations and Applications, vol. 15, no. 3, pp. 303–323, 2009. View at Google Scholar
  38. M. B. Bekker, M. J. Bohner, and H. D. Voulov, “Global attractor of solutions of a rational system in the plane,” Discrete Dynamics in Nature and Society, vol. 2015, Article ID 195247, 6 pages, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  39. S. J. Hrustić, M. R. Kulenović, and M. Nurkanović, “Global dynamics and bifurcations of certain second order rational difference equation with quadratic terms,” Qualitative Theory of Dynamical Systems, vol. 15, no. 1, pp. 283–307, 2016. View at Publisher · View at Google Scholar · View at MathSciNet