Dynamics and general form of the solutions of rational difference equations

Volume 34, Issue 4, pp 424--437 https://dx.doi.org/10.22436/jmcs.034.04.08

Publication Date: April 11, 2024 Submission Date: December 14, 2023 Revision Date: February 19, 2024 Accteptance Date: February 27, 2024

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Authors

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. F. A. Al-Rakhami - Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. - Department of Mathematics, Faculty of Science, Thamar University, Dhamar, Yemen. N. M. Seyam - Mathematics Sciences Department, College of Applied Science, Umm al-Qura University, Saudi Arabia.

Abstract

The main objective of this article is to find the general solution to some special cases of the fractional recursive equation% \[ \Psi_{n+1}=\frac{\alpha \Psi_{n-1}\Psi_{n-5}}{\Psi_{n-3}(\beta+\delta \Psi _{n}\Psi_{n-1}\Psi_{n-4}\Psi_{n-5})},~ n=0,1,2,\ldots, \] where \(\alpha,\beta\) and \(\delta\) are arbitrary real numbers. Furthermore, the solution's qualitative behavior is explored, such as local and global stability. For some situations, we have discovered periodic solutions. We also offered numerical examples to demonstrate our results.

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Keywords

• Solutions expressions of difference equation
• local and global stability
• periodic solution

MSC

•  39A10

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