Stability and Neimark–Sacker Bifurcation of Certain Mixed Monotone Rational Second-Order Difference Equation

• Nurkanovic, Zehra ^[1] ; Nurkanovic, Mehmed ^[1] ; Garic-Demirovic, Mirela ^[1]
  1. [1] University of Tuzla

     University of Tuzla

     Bosnia y Herzegovina

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 20, Nº 3, 2021
• Idioma: inglés
• DOI: 10.1007/s12346-021-00515-4
• Enlaces
  • Texto completo
• Resumen

  • This paper investigates the local and global character of the unique positive equilibrium of certain mixed monotone rational second-order difference equation with quadratic terms. The equation’s corresponding associated map is always decreasing for the second variable and can be either decreasing or increasing for the first variable depending on the corresponding parametric values. In some parametric space regions, we prove that the unique positive equilibrium point’s local asymptotic stability implies global asymptotic stability. Our main tool for studying this equation’s global dynamics is the determination of invariant interval and use so-called “m–M” theorems and semi-cycle analysis. Also, we show that the considered equation exhibits Neimark–Sacker bifurcation under certain conditions.

• Referencias bibliográficas
  • 1. Enciso, E.G., Sontag, E.D.: Global attractivity, I/O monotone small-gain theorems, and biological delay systems. Discrete Contin. Dyn....
  • 2. Gari´c-Demirovi´c, M., Hrusti´c, S., Nurkanovi´c, M.: Stability and periodicity of certain homogeneous second-order fractional difference...
  • 3. Gari´c-Demirovi´c, M., Kulenovi´c, M.R.S., Nurkanovi´c, M.: Global dynamics of certain homogeneous second-order quadratic fractional difference...
  • 4. Gari´c-Demirovi´c, M., Kulenovi´c, M.R.S., Nurkanovi´c, M.: Basins of attraction of certain homogeneous second order quadratic fractional...
  • 5. Gari´c-Demirovi´c, M., Nurkanovi´c, M., Nurkanovi´c, Z.: Stability, periodicity and Neimark–Sacker bifurcation of certain homogeneous fractional...
  • 6. Grove, E.A., Ladas, G.: Periodicities in Nonlinear Difference Equations. Advances in Discrete Mathematics and Applications, Chapman and...
  • 7. Jašarevi´c Hrusti´c, S., Kulenovi´c, M.R.S., Nurkanovi´c, M.: Global dynamics and bifurcations of certain second order rational difference...
  • 8. Kalabuši´c, S., Nurkanovi´c, M., Nurkanovi´c, Z.: Global dynamics of certain mix monotone difference equation. Mathematics 6(10), 13 (2018)....
  • 9. Kostrov, Y., Kudlak, Z.: On a second-order rational difference equation with a quadratic term. Int. J. Differ. Equ. 11(2), 179–202 (2016)
  • 10. Kulenovi´c, M.R.S., Ladas, G.: Dynamics of Second Order Rational Difference Equations with Open Problems and Conjectures. Chapman and...
  • 11. Kulenovi´c, M.R.S., Merino, O.: Discrete Dynamical Systems and Difference Equations with Mathematica. Chapman and Hall/CRC, Boca Raton...
  • 12. Kulenovi´c, M.R.S., Moranjki´c, S., Nurkanovi´c, M., Nurkanovi´c, Z.: Global asymptotic stability and Naimark–Sacker bifurcation of certain...
  • 13. Kulenovi´c, M.R.S., Moranjki´c, S., Nurkanovi´c, Z.: Naimark–Sacker bifurcation of second order rational difference equation with quadratic...
  • 14. Kulenovi´c, M.R.S., Moranjki´c, S., Nurkanovi´c, Z.: Global dynamics and bifurcation of perturbed Sigmoid Beverton–Holt difference equation....
  • 15. Kulenovi´c, M.R.S., Nurkanovi´c, M., Nurkanovi´c, Z.: Global dynamics of certain mix monotone difference equation via center manifold...
  • 16. Kulenovi´c, M.R.S., Nurkanovi´c, M.: Asymptotic behavior of a two dimensional linear fractional system of difference equations. Radovi...
  • 17. Kulenovi´c, M.R.S., Nurkanovi´c, M.: Asymptotic behavior of a system of linear fractional difference equations. J. Inequal. Appl. 2005(2),...
  • 18. Kuznetsov, Y.: Elements of Applied Bifurcation Theory. Springer, New York (1998)
  • 19. Moranjki´c, S., Nurkanovi´c, Z.: Basins of attraction of certain rational anti-competitive system of difference equations in the plane....
  • 20. Robinson, S.: Stability, Symbolic Dynamics and Chaos. CRC Press, Boca Raton (1995)
  • 21. Sedaghat, H.: Global behaviours of rational difference equations of orders two and three with quadratic terms. J. Diffe. Equ. Appl. 15,...
  • 22. Smith, H.L.: Non-monotone systems decomposable into monotone systems with negative feedback. J. Math. Biol. 53, 747–758 (2006)
  • 23. Thomson, G.G.: A proposal for a treshold stock sizeand maximum fishing mortality rate. In: Smith, S.J., Hunt, J.J., Rivard, D. (eds.)...
  • 24. Wiggins, S.: Introduction to Applied Nonlinear Dynamical Systems and Chaos. Text in Applied Mathematics, 2, vol. 2. Springer, New York...
  • 25. Zhong, J., Deng, S.: Two codimension-two bifurcations of a second-order difference equation from macroeconomics. Discrete Contin. Dyn....
  • 26. Zhang, Z., Zhou, Y.: The bifurcation of two invariant closed curves in a discrete model. Discrete Dyn. Nat. Soc. 2018, 1613709 (2018)....