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.

References

1.  [1] Alhalawa, M.A. and Saleh, M. (2017), { Dynamics of higher order rational difference equation $x_{n+1} = \frac{\alpha +\beta x_{n}}{A+Bx_{n}+x_{n-k}}$}, ph {Int. J. Nonlinear Anal. Appl.}, 363-379.
2.  [2] Anisimova, A. and Bula, I. (2014), { Some Problems of Second-Order Rational Difference Equations with Quadratic Terms}, International Journal of Difference Equations, 11-21.
3.  [3] Amleh, A. M., Camouzis, E., and Ladas, G. (2008), {On the Dynamics of a Rational Difference Equation, Part 1}, International Journal of Difference Equations (IJDE), 1-35.
4.  [4] Amleh, A. M., Camouzis, E., and Ladas, G. (2008), {On the Dynamics of a Rational Difference Equation, Part 2}, International Journal of Difference Equations (IJDE), 195-225.
5.  [5] GariT-Demirovi\{c}, M., Kulenovi$\acute{c}$, M.R.S., and Nurkanovi\{c}, M. (2013), { Global Dynamics of Certain Homogeneous Second-Order Quadratic Fractional Difference Equation}, The Scientific World Journal.
6.  [6] Jafar, J. and Saleh, M. (2018), {Dynamics of nonlinear difference equation $x_{n+1} = \frac{ \beta x_{n}+\gamma x_{n-k}}{A+Bx_{n}+Cx_{n-k}}$}, ph {J. Appl. Math. Comput}, 493-524.
7.  [7] Saleh, M. and Aloqeili, M. (2005), On the rational difference equation $y_{n+1}=A + \frac{y_{n-k}}{y_{n}}$, Appl. Math. Comput., 171, 862-869.
8.  [8] Saleh, M. and Abu-Baha, S. (2006), Dynamics of higher order Rational Difference Equation, App. Math. Comp., 181(2006), 84-102.
9.  [9] Saleh, M. and Aloqeili, M. (2005), On the rational difference equation $y_{n+1}= A + \frac{y_{n-k}}{y_{n}}$, Appl. Math. Comp., 171(2005), 862-869.
10.  [10] Saleh, M. and Aloqeili, M. (2006), On the rational difference equation $y_{n+1}=A + \frac{y_{n}}{y_{n-k}}$, Appl. Math. Comput., 177, 189-193.
11.  [11] Saleh, M., Alkoumi, N., and Farhat, A. (2017), ph{On the Dynamics of a Rational Difference Equation $x_{n+1}=\frac{ \alpha +\beta x_{n}+\gamma x_{n-k}}{Bx_{n}+Cx_{n-k}}$}, Chaos and Soliton, 76-84.
12.  [12] Saleh, M. and Farhat, A. (2017), ph{Global Asymptotic Stability of the Higher Order Equation $x_{n+1} = \frac{ ax_{n}+bx_{n-k}}{A+Bx_{n-k}}$}, J. Appl. Math. Comput, DOI 10.1007/s12190-016-1029-4, 135-148.
13.  [13] Saleh, M. and Asad, A. (2021), {Dynamics of Kth Order Rational Difference Equation}, ph{J. Applied nonlinear Dynamics}, 125-149, DOI:10.5890/JAND.2021.03.008.
14.  [14] Sebdani, R.M. and Dehghan, M. (2006), Global stability of $y_{n+1}=\frac{p + q y_{n}+ry_{n-k}}{1+y_{n}}$, Appl. Math. Comput., 182, 621-630.
15.  [15] Sebdani, R.M. and Dehghan, M. (2006), The study of a class of rational difference equation, Appl. Math. Comput., 179, 98-107.
16.  [16] Sebdani, R.M. and Dehghan, M. (2006), Dynamics of a non-linear difference equation, Appl. Math. Comput., 178, 250-261.
17.  [17] Yan, X., Li, W.T., and Zhao, Z. (2006), Global asymptotic stability for a higher order nonlinear rational difference equations, Appl. Math. Comput., 182, 1819-1831.
18.  [18] Kuznetsov, A.Y. (2007), Applied Mathematical Sience, Elements of applied bifurcation theory, 2nd edition Journal of Mathematical Analysis and Applications, 331, 230-239.
19.  [19] Zayed, E.M.E. and El-Moneam, M.A. (2006), On the Rational Recursive Sequence, Egypt.
20.  [20] Kulenovic, M.R.S. and Ladas, G. (2002), Dynamics of Second Order Rational Difference Equations With Open Problems and Conjectures, Chapman. Hall/CRC, Boca Raton.
21.  [21] Elaydi, S. (2000), An introduction to difference equations, 3rd edition. Springer.
22.  [22] Elaydi, S. Discrete Chaos With Applications In Science And Engineering, 2nd edition. Chapman Hall/CRC.
23.  [23] Kulenovic, M.R.S. and Ladas, G. (2002), Dynamics of Second Order Rational Difference Equations With Open Problems and Conjectures.
24.  [24] Li, W.T. and Sun, H.R. (2005), Dynamics of a rational difference equation, Appl. Math. Comput., 163, 577-591.
25.  [25] Salas Einarhille, S.L. (1982), Calculus One and Several Variables, fourth edition.