Investigating dynamical behaviors of the difference equation $x_{n+1}= \frac{Cx_{n-5}}{A+Bx_{n-2}x_{n-5}}$

Ghazel, M. ¹ ; Elsayed, E. M. ² ; Matouk, A. E. ¹ ; Mousallam, A. M. ¹

¹ Mathematics Department, Faculty of Science, University of Hail, Hail 2440, Saudi Arabia
² Mathematics Department, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia;Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Journal of nonlinear sciences and its applications, Tome 10 (2017) no. 9, p. 4662-4679.

Résumé

In this work, we investigate the dynamical behaviors of the rational difference equation%

$ x_{n+1}=\frac{Cx_{n-5}}{A+Bx_{n-2}x_{n-5}}, $

with arbitrary initial conditions, where $A,\ B$, and $C$ are arbitrary constants. A general solution is obtained. Asymptotic behavior and asymptotic stability of the equilibrium points are investigated. The existence of the periodic solutions is discussed. Numerical simulations are carried out to verify the analytical results.

Détail

DOI : 10.22436/jnsa.010.09.09

Classification : 34K13, 34K05, 34K20, 39A10
Keywords: Rational difference equations, asymptotic behavior, infinite products, local stability, periodicity, convergence.

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