Table of Contents Author Guidelines Submit a Manuscript
The Scientific World Journal
Volume 2014, Article ID 819290, 8 pages
http://dx.doi.org/10.1155/2014/819290
Research Article

Birkhoff Normal Forms and KAM Theory for Gumowski-Mira Equation

1Department of Mathematics, University of Rhode Island, Kingston, RI 02881-0816, USA
2Department of Mathematics, University of Tuzla, 75000 Tuzla, Bosnia and Herzegovina
3Department of Mathematics, University of Sarajevo, 71000 Sarajevo, Bosnia and Herzegovina

Received 29 August 2013; Accepted 19 October 2013; Published 16 January 2014

Academic Editors: Z. Guo, Z. Huang, X. Song, and Y. Xia

Copyright © 2014 M. R. S. Kulenović 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. I. Gumowski and C. Mira, Recurrences and Discrete Dynamic Systems, Lecture Notes in Mathematics, Springer, Berlin, Germany, 1980.
  2. G. Bastien and M. Rogalski, “On the algebraic difference equations un+2un=ψ(un+1) in *+, related to a family of elliptic quartics in the plane,” Advances in Difference Equations, vol. 2005, no. 3, pp. 227–261, 2005. View at Publisher · View at Google Scholar · View at Scopus
  3. G. Bastien and M. Rogalski, “On the algebraic difference equations un+2+un=ψ(un+1) in ℝ, related to a family of elliptic quartics in the plane,” Journal of Mathematical Analysis and Applications, vol. 326, no. 2, pp. 822–844, 2007. View at Publisher · View at Google Scholar · View at Scopus
  4. F. Beukers and R. Cushman, “Zeeman's monotonicity conjecture,” Journal of Differential Equations, vol. 143, no. 1, pp. 191–200, 1998. View at Publisher · View at Google Scholar · View at Scopus
  5. A. Cima, A. Gasull, and V. Maňosa, “Dynamics of rational discrete dynamical systems via first integrals,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, pp. 631–645, 2006. View at Google Scholar
  6. A. Cima, A. Gasull, and V. Maňosa, “Non-autonomous two-periodic Gumowski-Mira difference equation,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 16, 14 pages, 2012. View at Google Scholar
  7. C. A. Clark, E. J. Janowski, and M. R. S. Kulenović, “Stability of the Gumowski-Mira equation with period-two coefficient,” Journal of Mathematical Analysis and Applications, vol. 307, no. 1, pp. 292–304, 2005. View at Publisher · View at Google Scholar · View at Scopus
  8. E. J. Janowski, M. R. S. Kulenović, and Z. Nurkanović, “Stability of the kTH order lyness' equation with a period-k coefficient,” International Journal of Bifurcation and Chaos, vol. 17, no. 1, pp. 143–152, 2007. View at Publisher · View at Google Scholar · View at Scopus
  9. M. R. S. Kulenović, “Invariants and related Liapunov functions for difference equations,” Applied Mathematics Letters, vol. 13, no. 7, pp. 1–8, 2000. View at Google Scholar · View at Scopus
  10. M. R. S. Kulenović and O. Merino, Discrete Dynamical Systems and Difference Equations with Mathematica, Chapman and Hall/CRC, Boca Raton, Fla, USA, 2002.
  11. R. S. MacKay, Renormalization in Area-Preserving Maps, World Scientific, River Edge, NJ, USA, 1993.
  12. E. C. Zeeman, Geometric Unfolding of A Difference Equation, Hertford College, Oxford, UK, 1996.
  13. M. Gidea, J. D. Meiss, James, I. Ugarcovici, and H. Weiss, “Applications of KAM theory to population dynamics,” Journal of Biological Dynamics, vol. 5, no. 1, pp. 44–63, 2011. View at Google Scholar
  14. V. L. Kocic, G. Ladas, G. Tzanetopoulos, and E. Thomas, “On the stability of Lyness' equation,” Dynamics of Continuous, Discrete and Impulsive Systems, vol. 1, pp. 245–254, 1995. View at Google Scholar
  15. M. R. S. Kulenović and Z. Nurkanović, “Stability of Lyness' equation with period-two coeffcient via KAM theory,” Journal of Concrete and Applicable Mathematics, vol. 6, pp. 229–245, 2008. View at Google Scholar
  16. G. Ladas, G. Tzanetopoulos, and A. Tovbis, “On May's host parasitoid model,” Journal of Difference Equations and Applications, vol. 2, pp. 195–204, 1996. View at Google Scholar
  17. M. Tabor, Chaos and Integrability in Nonlinear Dynamics. An Introduction, Wiley-Interscience, New York, NY, USA, 1989.
  18. C. Siegel and J. Moser, Lectures on Celestial Mechanics, Springer, New York, NY, USA, 1971.
  19. J. K. Hale and H. Kocak, Dynamics and Bifurcation, Springer, New York, NY, USA, 1991.
  20. G. Bastien and M. Rogalski, “Level sets lemmas and unicity of critical points of invariants, tools for local stability and topological properties of dynamical systems,” Sarajevo Journal of Mathematics, vol. 21, pp. 273–282, 2012. View at Google Scholar
  21. J. Duistermaat, Discrete Integrable Systems. QRT Maps and Elliptic Surfaces, Springer Monographs in Mathematics, Springer, New York, NY, USA, 2010.