Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-1}}{A+Bx_{n}+C x_{n-1}}$

Yıl 2022, Cilt: 5 Sayı: 2, 78 - 87, 30.06.2022

Mohammad Saleh Batool Raddad

https://doi.org/10.33434/cams.1028122

Öz

The main goal of this paper is to study the bifurcation of a second order rational difference equation
$$x_{n+1}=\frac{\alpha+\beta x_{n-1}}{A+Bx_{n}+Cx_{n-1}}, ~~n=0, 1, 2, \ldots$$
with positive parameters $\alpha, \beta, A, B, C$ and non-negative initial conditions $\{x_{-k}, x_{-k+1}, \ldots, x_{0}\}$. We study the dynamic behavior and the direction of the bifurcation of the period-two cycle. Numerical discussion with figures are given to support our results.

Anahtar Kelimeler

stability, bifurcation, chaos

Kaynakça

• [1] E. Camouzis and G. Ladas, Dynamics of Third-Order Rational Difference EquationsWith Open Problems And Conjectures. Chapman. Hall/CRC, Boca Raton, (2008).
• [2] S. Elaydi, Discrete Chaos With Applications In Science And Engineering, 2nd edition. Chapman Hall/CRC.
• [3] S. Herzallah, M. Saleh, Dynamics and Bifurcation of a Second Order Quadratic Rational Difference Equation, Journal of mathematical sciences and modeling, 3(3) (2020), 102 - 119
• [4] L. Hu, W. Li, H. Xu, Global Asymptotical Stability of a Second Order Rational Difference Equation. Computers and mathematics with applications 54 (2007) 1260-1266.
• [5] M. Kulenovic, G. Ladas, Dynamics of second order rational difference equations with open problems and conjectures. Chapman. Hall/CRC, Boca Raton, (2002).
• [6] B. Raddad, M. Saleh, Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-2}}{A+Bx_{n}+C x_{n-2}}$, Journal of mathematical sciences and modeling, 4(1) (2021), 25 - 37
• [7] M. Saleh, A. Asad, Dynamics of Kth order rational difference equation, Journal of Applied Nonlinear Dynamics, 10(1) (2021) 125-149
• [8] M. Saleh, S. Herzallah, Dynamics and Bifurcation of A second Order Rational Difference Equation with Quadratic Terms, Journal of Applied Nonlinear Dynamics, 10(3) (2021) 561-576
• [9] G. Tang, L. Hu, and G. Ma, Global Stability of a Rational Difference Equation. Discrete Dynamics in Nature and Society, 2010.