Investigating dynamical behaviors of the difference equation \(x_{n+1}= \frac{Cx_{n-5}}{A+Bx_{n-2}x_{n-5}}\)
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Authors
M. Ghazel
- Mathematics Department, Faculty of Science, University of Hail, Hail 2440, Saudi Arabia.
E. M. Elsayed
- 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.
A. E. Matouk
- Mathematics Department, Faculty of Science, University of Hail, Hail 2440, Saudi Arabia.
A. M. Mousallam
- Mathematics Department, Faculty of Science, University of Hail, Hail 2440, Saudi Arabia.
Abstract
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.
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ISRP Style
M. Ghazel, E. M. Elsayed, A. E. Matouk, A. M. Mousallam, Investigating dynamical behaviors of the difference equation \(x_{n+1}= \frac{Cx_{n-5}}{A+Bx_{n-2}x_{n-5}}\), Journal of Nonlinear Sciences and Applications, 10 (2017), no. 9, 4662--4679
AMA Style
Ghazel M., Elsayed E. M., Matouk A. E., Mousallam A. M., Investigating dynamical behaviors of the difference equation \(x_{n+1}= \frac{Cx_{n-5}}{A+Bx_{n-2}x_{n-5}}\). J. Nonlinear Sci. Appl. (2017); 10(9):4662--4679
Chicago/Turabian Style
Ghazel, M., Elsayed, E. M., Matouk, A. E., Mousallam, A. M.. "Investigating dynamical behaviors of the difference equation \(x_{n+1}= \frac{Cx_{n-5}}{A+Bx_{n-2}x_{n-5}}\)." Journal of Nonlinear Sciences and Applications, 10, no. 9 (2017): 4662--4679
Keywords
- Rational difference equations
- asymptotic behavior
- infinite products
- local stability
- periodicity
- convergence.
MSC
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