 ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Dynamics and Bifurcation of a Second Order Rational Difference Equation with Quadratic Terms

Journal of Applied Nonlinear Dynamics 10(3) (2021) 563--578 | DOI:10.5890/JAND.2021.09.014

Department of Mathematics, Birzeit University, West Bank

Abstract

We study some results concerning dynamics and bifurcation of a special case of a second order rational difference equations with quadratic terms. We consider the second order, quadratic rational difference equation $$x_{n+1} = \frac{\alpha+ \beta x_{n-1}}{A+B {x^2}_n+C x_{n-1}}, \ n=0,\ 1, \ 2, \ ...$$ with positive parameters $\alpha$, $\beta$, $A$, $B$, $C,$ and non-negative initial conditions. We investigate local stability, invariant intervals, boundedness of the solutions, periodic solutions of prime period two and global stability of the positive fixed points. And we study the types of bifurcation exist where the change of stability occurs. Then, we give numerical examples with figures to support our results.

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