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

Year 2022, Volume: 5 Issue: 2, 78 - 87, 30.06.2022
https://doi.org/10.33434/cams.1028122

Abstract

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.

References

  • [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.
Year 2022, Volume: 5 Issue: 2, 78 - 87, 30.06.2022
https://doi.org/10.33434/cams.1028122

Abstract

References

  • [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.
There are 9 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Mohammad Saleh 0000-0002-4254-2540

Batool Raddad 0000-0002-2716-5740

Publication Date June 30, 2022
Submission Date November 24, 2021
Acceptance Date April 21, 2022
Published in Issue Year 2022 Volume: 5 Issue: 2

Cite

APA Saleh, M., & Raddad, B. (2022). Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-1}}{A+Bx_{n}+C x_{n-1}}$. Communications in Advanced Mathematical Sciences, 5(2), 78-87. https://doi.org/10.33434/cams.1028122
AMA Saleh M, Raddad B. Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-1}}{A+Bx_{n}+C x_{n-1}}$. Communications in Advanced Mathematical Sciences. June 2022;5(2):78-87. doi:10.33434/cams.1028122
Chicago Saleh, Mohammad, and Batool Raddad. “Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-1}}{A+Bx_{n}+C x_{n-1}}$”. Communications in Advanced Mathematical Sciences 5, no. 2 (June 2022): 78-87. https://doi.org/10.33434/cams.1028122.
EndNote Saleh M, Raddad B (June 1, 2022) Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-1}}{A+Bx_{n}+C x_{n-1}}$. Communications in Advanced Mathematical Sciences 5 2 78–87.
IEEE M. Saleh and B. Raddad, “Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-1}}{A+Bx_{n}+C x_{n-1}}$”, Communications in Advanced Mathematical Sciences, vol. 5, no. 2, pp. 78–87, 2022, doi: 10.33434/cams.1028122.
ISNAD Saleh, Mohammad - Raddad, Batool. “Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-1}}{A+Bx_{n}+C x_{n-1}}$”. Communications in Advanced Mathematical Sciences 5/2 (June 2022), 78-87. https://doi.org/10.33434/cams.1028122.
JAMA Saleh M, Raddad B. Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-1}}{A+Bx_{n}+C x_{n-1}}$. Communications in Advanced Mathematical Sciences. 2022;5:78–87.
MLA Saleh, Mohammad and Batool Raddad. “Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-1}}{A+Bx_{n}+C x_{n-1}}$”. Communications in Advanced Mathematical Sciences, vol. 5, no. 2, 2022, pp. 78-87, doi:10.33434/cams.1028122.
Vancouver Saleh M, Raddad B. Dynamics and Bifurcation of $x_{n+1}=\frac{\alpha+\beta x_{n-1}}{A+Bx_{n}+C x_{n-1}}$. Communications in Advanced Mathematical Sciences. 2022;5(2):78-87.

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