Dynamics and Bifurcation of a Second Order Quadratic Rational Difference Equation

Year 2020, , 102 - 119, 29.12.2020

Shahd Herzallah Mohammad Saleh

https://doi.org/10.33187/jmsm.748724

Cited By: 2

Abstract

In this paper, we study the dynamics and bifurcation of $$ x_{n+1} = \frac{\alpha+ \beta {x^2}_{n-1}}{A+B {x_n}+C{x^2}_{n-1}}, \ n=0,\ 1, \ 2, \ ... $$ with positive parameters $\alpha,\ \beta, \ A, \ B, \ C, $ and non-negative initial conditions. Among others, we investigate local stability, invariant intervals, boundedness of the solutions, periodic solutions of prime period two and global stability of the positive fixed points.

Keywords

Fixed point, Neimark-Sacker bifurcation, Stability

References

• [1] M. Kulenovic, et al., Naimark-Sacker bifurcation of second order rational difference equation with quadratic terms, J. Nonlinear Sci. Appl., 10(7) (2017), 3477-3489.
• [2] Y. Kostrov, Z. Kudlak, On a second-order rational difference equation with a quadratic term, Int. J. Difference Equ., (2016), 179-202.
• [3] S. Moranjkic, Z. Nurkanovic, Local and global dynamics of certain second-order rational difference equations containing quadratic terms, Adv. Dyn. Syst. Appl., (2017), 123-157.
• [4] M. Abu Alhalawa M, M.Saleh, Dynamics of higher order rational difference equation $x_{n+1} = \frac{ \alpha x_{n}+\ beta x_{n}}{A+Bx_n+Cx_{n-k}}$, Int. J. Nonlinear Anal. Appl. 8(2) (2017), 363-379.
• [5] A. Jafar, M. Saleh, Dynamics of nonlinear difference equation $x_{n+1}=\frac{ \beta x_n+\gamma x_{n-k}}{A+Bx_n+Cx_{n-k}}$, J. Appl. Math. Comput., 57 (2018), 493-522.
• [6] M. Saleh, N. Alkoumi, A. Farhat, 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 Soliton, 96 (2017), 76-84.
• [7] M. Saleh, A. Farhat, 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., 55 (2017), 135-148.
• [8] M. Saleh, A. Asad, Dynamics of kth order rational difference equation, J. Appl. Nonlinear Dynam., (2021), 125-149, DOI 10.5890/JAND.2021.03.008.
• [9] M. Saleh, S.Hirzallah, Dynamics and bifurcation of a second order rational difference equation with quadratic terms, J. Appl. Nonlinear Dynam., (in press).
• [10] C. Wang, X. Fang, R. Li, On the solution for a system of two rational difference equations, J. Comput. Anal. Appl., 20(1) (2016), 175-186.
• [11] C. Wang, X. Fang, R. Li, On the dynamics of a certain four-order fractional difference equations, J. Comput. Anal. Appl., 22(5) (2017), 968-976.
• [12] M. Kulenovic, G. Ladas, Dynamics of Second Order Rational Difference Equations With Open Problems and Conjectures, Chapman. Hall/CRC, Boca Raton, 2002.
• [13] S. Elaydi, An Introduction to Difference Equations, 3rd edition. Springer, 2000.
• [14] S. Elaydi, Discrete Chaos With Applications In Science And Engineering, 2nd edition. Chapman Hall/CRC.
• [15] Y. Kuznetsov, Elements of Applied Bifurcation Theory, 2nd edition, Springer-Verlag, 1998.