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Dynamics and Bifurcation of a Second Order Quadratic Rational Difference Equation

Year 2020, Volume: 3 Issue: 3, 102 - 119, 29.12.2020
https://doi.org/10.33187/jmsm.748724

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.

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.
Year 2020, Volume: 3 Issue: 3, 102 - 119, 29.12.2020
https://doi.org/10.33187/jmsm.748724

Abstract

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.
There are 15 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Shahd Herzallah This is me

Mohammad Saleh 0000-0002-4254-2540

Publication Date December 29, 2020
Submission Date June 6, 2020
Acceptance Date December 18, 2020
Published in Issue Year 2020 Volume: 3 Issue: 3

Cite

APA Herzallah, S., & Saleh, M. (2020). Dynamics and Bifurcation of a Second Order Quadratic Rational Difference Equation. Journal of Mathematical Sciences and Modelling, 3(3), 102-119. https://doi.org/10.33187/jmsm.748724
AMA Herzallah S, Saleh M. Dynamics and Bifurcation of a Second Order Quadratic Rational Difference Equation. Journal of Mathematical Sciences and Modelling. December 2020;3(3):102-119. doi:10.33187/jmsm.748724
Chicago Herzallah, Shahd, and Mohammad Saleh. “Dynamics and Bifurcation of a Second Order Quadratic Rational Difference Equation”. Journal of Mathematical Sciences and Modelling 3, no. 3 (December 2020): 102-19. https://doi.org/10.33187/jmsm.748724.
EndNote Herzallah S, Saleh M (December 1, 2020) Dynamics and Bifurcation of a Second Order Quadratic Rational Difference Equation. Journal of Mathematical Sciences and Modelling 3 3 102–119.
IEEE S. Herzallah and M. Saleh, “Dynamics and Bifurcation of a Second Order Quadratic Rational Difference Equation”, Journal of Mathematical Sciences and Modelling, vol. 3, no. 3, pp. 102–119, 2020, doi: 10.33187/jmsm.748724.
ISNAD Herzallah, Shahd - Saleh, Mohammad. “Dynamics and Bifurcation of a Second Order Quadratic Rational Difference Equation”. Journal of Mathematical Sciences and Modelling 3/3 (December 2020), 102-119. https://doi.org/10.33187/jmsm.748724.
JAMA Herzallah S, Saleh M. Dynamics and Bifurcation of a Second Order Quadratic Rational Difference Equation. Journal of Mathematical Sciences and Modelling. 2020;3:102–119.
MLA Herzallah, Shahd and Mohammad Saleh. “Dynamics and Bifurcation of a Second Order Quadratic Rational Difference Equation”. Journal of Mathematical Sciences and Modelling, vol. 3, no. 3, 2020, pp. 102-19, doi:10.33187/jmsm.748724.
Vancouver Herzallah S, Saleh M. Dynamics and Bifurcation of a Second Order Quadratic Rational Difference Equation. Journal of Mathematical Sciences and Modelling. 2020;3(3):102-19.

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