A Qualitative Investigation of the Solution of the Difference Equation $\Psi_{m+1}=\frac{\Psi_{m-3}\Psi_{m-5}}{\Psi_{m-1} \left( \pm1\pm \Psi_{m-3}\Psi_{m-5} \right) }$
Abstract
We explore the dynamics of adhering to rational difference formula
\begin{equation*}
\Psi_{m+1}=\frac{\Psi_{m-3}\Psi_{m-5}}{\Psi_{m-1} \left( \pm1\pm \Psi_{m-3}\Psi_{m-5} \right) } \quad m \in \mathbb{N}_{0}
\end{equation*}
where the initials $\Psi_{-5}$, $\Psi_{-4}$, $\Psi_{-3}$,$\Psi_{-2}$, $\Psi_{-1}$, $\Psi_{0}$ are arbitrary nonzero real numbers. Specifically, we examine global asymptotically stability. We also give examples and solution diagrams for certain particular instances.
Keywords
Boundedness, Equilibrium point, Global asymptotic stability, Solution of difference equation, Stability
References
- [1] R. P. Agarwal, Difference Equations and Inequalities. 1st edition, Marcel Dekker, New York, 1992.
- [2] V. L. Kocic, G. Ladas, Global behavior of nonlinear difference equations of higher order with applications, volume 256 of Mathematics and its Applications. Kluwer Academic Publishers Group, Dordrecht, 1993.
- [3] M. A. Radin, Difference Equations for Scientists and Engineering, Interdisciplinary Difference Equations, World Scientific Publishing, October 2019.(https://doi.org/10.1142/11349)
- [4] M. Aloqeili, Dynamics of a rational difference equation, Appl. Math. Comput., 176(2), (2006), 768-774.
- [5] C. Cinar, On the positive solutions of the difference equation $\Psi_{m+1}=\frac{\Psi_{m-1}}{1+\alpha \Psi_{m} \Psi_{m-1}}$, Appl. Math. Comput., 158(3), (2004), 809-812.
- [6] A. Gelisken, On A System of Rational Difference Equations, J. Comput. Anal. Appl., 23(4), (2017), 593-606.
- [7] R. Karatas, C. Cinar, D. Simsek, On Positive Solutions of the Difference Equation $\Psi_{m+1}=\frac{\Psi_{m-5}}{1+\Psi_{m-2}\Psi_{m-5}}$, Int. J. Contemp. Math. Sci., 10(1), (2006), 495-500.
- [8] B. Ogul, D. Şimşek, H. Ogunmez, A.S. Kurbanli, Dynamical behavior of rational difference equation $\Psi_{m+1}= \frac{\Psi_{m-17}}{\pm 1\pm \Psi_{m-2}\Psi_{m-5}\Psi_{m-8}\Psi_{m-11}\Psi_{m-14}\Psi_{m-17}}$, Bol. Soc. Mat. Mexicana, 27(49), (2021). https://doi.org/10.1007/s40590-021-00357-9
- [9] D. Simsek, B. Ogul, F. Abdullayev, Solution of the Rational Difference Equation $\Psi_{m+1}=\frac{\Psi_{m-13}}{1+\Psi_{m-1}\Psi_{m-3}\Psi_{m-5}\Psi_{m-7}\Psi_{m-9}\Psi_{m-11}}$, Appl. Math. Nonlinear Sci., 5(1), (2020), 485-494.
- [10] I. Yalcinkaya, C. Cinar, On the dynamics of difference equation $\Psi_{m+1}=\frac{a \Psi_{m-k}}{b+c_{m}^{p}}$, Fasciculi Mathematici, 42, (2009), 141-148.
