Periodic solutions and stability of eighth order rational difference equations
Volume 26, Issue 4, pp 405417
http://dx.doi.org/10.22436/jmcs.026.04.08
Publication Date: January 14, 2022
Submission Date: September 18, 2021
Revision Date: December 26, 2021
Accteptance Date: January 01, 2022
Authors
M. B. Almatrafi
 Department of Mathematics, Faculty of Science, Taibah University, Saudi Arabia.
M. M. Alzubaidi
 Department of Mathematics, College of Duba, University of Tabuk, Saudi Arabia.
Abstract
Some real life problems are modeled using difference equations. Extracting the exact solutions of such equations is an active topic for some scientists. This paper investigates the equilibrium points, stability, boundedness, periodicity, and some exact solutions for eighth order rational difference equations. The exact solutions are obtained using the iterations method. We also present some 2D figures to show the validity of the obtained results. The used methods can be applied for other nonlinear difference equations.
Share and Cite
ISRP Style
M. B. Almatrafi, M. M. Alzubaidi, Periodic solutions and stability of eighth order rational difference equations, Journal of Mathematics and Computer Science, 26 (2022), no. 4, 405417
AMA Style
Almatrafi M. B., Alzubaidi M. M., Periodic solutions and stability of eighth order rational difference equations. J Math Comput SCIJM. (2022); 26(4):405417
Chicago/Turabian Style
Almatrafi, M. B., Alzubaidi, M. M.. "Periodic solutions and stability of eighth order rational difference equations." Journal of Mathematics and Computer Science, 26, no. 4 (2022): 405417
Keywords
 Equilibrium points
 stability
 boundedness
 exact solution
 numerical solution
MSC
 39A10
 39A33
 39A30
 39A23
 39A22
 39A06
References

[1]
H. S. Alayachi, M. S. M. Noorani, A. Q. Khan, M. B. Almatrafi, Analytic solutions and stability of sixth order difference equations, Math. Probl. Eng., 2020 (2020), 12 pages

[2]
M. B. Almatrafi, Solutions structures for some systems of fractional difference equations, Open J. Math. Anal., 3 (2019), 5161

[3]
M. B. Almatrafi, Exact solutions and stability of sixth order difference equations, Electron. J. Math. Anal. Appl., 10 (2022), 209225

[4]
M. B. Almatrafi, M. M. Alzubaidi, Analysis of the qualitative behaviour of an eighthorder fractional difference equation, Open J. Discrete Appl. Math., 2 (2019), 4147

[5]
M. B. Almatrafi, M. M. Alzubaidi, Qualitative analysis for two fractional difference equations, Nonlinear Eng., 9 (2020), 265272

[6]
M. B. Almatrafi, E. M. Elsayed, Solutions and formulae for some systems of difference equations, MathLAB J., 1 (2018), 365369

[7]
M. B. Almatrafi, E. M. Elsayed, F. Alzahrani, Qualitative behavior of two rational difference equations, Fund. J. Math. Appl., 1 (2018), 194204

[8]
M. B. Almatrafi, E. M. Elsayed, F. Alzahrani, Qualitative behavior of a quadratic secondorder rational difference equation, Int. J. Adv. Math., 2019 (2019), 14 pages

[9]
M. A. AlShabi, R. AboZeid, Global asymptotic stability of a higher order difference equation, Appl. Math. Sci. (Ruse), 4 (2010), 839847

[10]
A. M. Amleh, E. Drymonis, Eventual monotonicity in nonlinear difference equation, Int. J. Difference Equ., 11 (2016), 123137

[11]
S. Elaydi, An introduction to Difference Equations: Third ed., Springer, New York (2005)

[12]
E. M. Elsayed, Dynamics and behavior of a higher order rational difference equation, J. Nonlinear Sci. Appl., 9 (2016), 14631474

[13]
GarićDemirović, M. Nurkanović, Z. Nurkanović, Stability, periodicity and NeimarkSacker bifurcation of certain homogeneous fractional difference equations, Int. J. Difference Equ., 12 (2017), 2753

[14]
M. Ghazel, E. M. Elsayed, A. E. Matouk, A. M. Mousallam, Investigating dynamical behaviors of the difference equation $x_{n+1}=Cx_{n5}/(A+Bx_{n2}x_{n5})$, J. Nonlinear Sci. Appl., 10 (2017), 46624679

[15]
M. Gümüş, R. AboZeid, Ö. Özkan, Dynamical behavior of a thirdorder difference equation with arbitrary powers, Kyungpook Math. J., 57 (2017), 251263

[16]
V. L. Kocic, G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, Kluwer Academic Publishers Group, Dordrecht (1993)

[17]
Y. Kostrov, Z. Kudlak, On a secondorder rational difference equation with a quadratic term, Int. J. Difference Equ., 11 (2016), 179202

[18]
M. R. S. Kulenovic, G. Ladas, Dynamics of second order rational difference equations with open problems and conjectures, Chapman and Hall/CRC Press, New York (2001)

[19]
K. Liu, P. Li, F. Han, W. Z. Zhong, Global dynamics of nonlinear difference equation $x_{n+1}=A+x_{n}/x_{n1}x_{n2}$, J. Comput. Anal. Appl., 24 (2018), 11251132

[20]
S. Moranjkić, Z. Nurkanović, Local and global dynamics of certain secondorder rational difference equations containing quadratic terms, Adv. Dyn. Syst. Appl., 12 (2017), 123157

[21]
J. D. Murray, Mathematical biology, 3rd Ed., SpringerVerlag, Berlin (2001)

[22]
M. Saleh, N. Alkoumi, Aseel Farhat, On the dynamic of a rational difference equation $x_{n+1}=\alpha+\beta x_{n}+\gamma x_{nk}/B x_{n}+C x_{nk} $, Chaos Solitons Fractals, 96 (2017), 7684