Dynamical Analysis and Solutions of Nonlinear Difference Equations of Thirty Order

Year 2024, Volume: 7 Issue: 3, 111 - 120

Burak Oğul , Dağıstan Şimşek

https://doi.org/10.32323/ujma.1483974

Abstract

Discrete-time systems are sometimes used to explain natural phenomena that happen in nonlinear sciences. We study the periodicity, boundedness, oscillation, stability, and certain exact solutions of nonlinear difference equations in this paper. Using the standard iteration method,
exact solutions are obtained. Some well-known theorems are used to test the stability of the equilibrium points. Some numerical examples are also provided to confirm the theoretical work’s validity. The numerical component is implemented with Wolfram Mathematica. The method presented may be simply applied
to other rational recursive issues. \par
In this paper, we explore the dynamics of adhering to rational difference formula
\begin{equation*}
x_{n+1}=\frac{x_{n-29}}{\pm1\pm x_{n-5}x_{n-11}x_{n-17}x_{n-23}x_{n-29}},
\end{equation*}
where the initials are arbitrary nonzero real numbers.

Keywords

Difference equation, Local stability, Periodicity

References

• V. L. Kocic, G. Ladas, Global Behavior of Nonlinear Difference Equations of Higher Order with Applications} Mathematics and Its Applications, Kluwer Academic Publishers Group, Dordrecht (1993).
• M.R.S Kulenovic, G. Ladas, G., W.S. Sizer, On the recursive sequence $\frac{\alpha x_{n}+\beta x_{n-1}}{\chi x_{n}+\beta x_{n-1}}$, Math. Sci. Res. Hot-Line, 2(5) (1998), 1-16.
• R. DeVault, G. Ladas, S.W. Schultz, On the recursive sequence $x_{n+1}=\frac{A}{x_{n}}+\frac{1}{x_{n-2}}$, Proc.Amer. Math. Soc., 126(11) (1998), 3257-3261.
• A.M. Amleh, G.A. Grove, G. Ladas, Georgiou, D.A. On the recursive sequence $x_{n+1}=\alpha + \frac{x_{n-1}}{x_{n}}$, J. of Math. Anal. App., 233 (1999), 790-798.
• C.H. Gibbons, M.R.S. Kulenovic, G. Ladas, On the recursive sequence $\frac{\alpha+\beta x{n-1}}{\chi+\beta x{n-1}}$, Math. Sci. Res. Hot-Line, 4(2) (2000), 1-11.
• S. Elaydi, An Introduction to Difference Equations, 3rd Ed., Springer, USA, (2005).
• R. Karatas, C. Cinar, D. Simsek, On positive solutions of the difference equation $x_{n+1}=\frac{x_{n-5}}{1+x_{n-2}x_{n-5}}$, Int. J. Contemp. Math. Sci, 10(1) (2006), 495-500 .
• M. Aloqeili, Dynamics of a rational difference equation, Appl. Math. Comput., 176(2) (2006), 768-774.
• S. Stevic, A note on periodic character of a higher order difference equation, Rostock. Math. Kolloq., 61 (2006), 2-30.
• E.M. Elsayed, On the difference equation $x_{n+1}=\frac{x_{n-5}}{-1+x_{n-2}x_{n-5}}$, Inter. J. Contemp. Math. Sci., 3(33) (2008), 1657-1664.
• R.P. Agarwal, E.M. Elsayed, On the solution of fourth-order rational recursive sequence, Adv. Stud. Contemp. Math. , 20(2) (2010), 525-545.
• S. Stevic, B. Iricanin, Z. Smarda, On a product-type system of difference equations of second order solvable in closed form, J. Inequal. Appl., 2015(1) (2012), 327-334.
• O. Karpenko, O. Stanzhytskyi, The relation between the existence of bounded solutions of differential equations and the corresponding difference equations, J. Difference Equ. Appl., 19(12) (2013), 1967-1982. https://doi.org/10.1080/10236198.2013.794795
• M. Bohner, O. Karpenko, O. Stanzhytskyi, Oscillation of solutions of second-order linear differential equations and corresponding difference equations, J. Difference Equ. Appl., 20(7) (2014), 1112-1126. https://doi.org/10.1080/10236198.2014.893297
• M.B. Almatrafi, M.M. Alzubaidi, Analysis of the qualitative behaviour of an eighth-order fractional difference equation, Open J. Discrete Appl. Math., 2(1), (2019), 41-47.
• A. Sanbo, A. E.M. Elsayed, Some properties of the solutions of the difference equation $x_{n+1}= \alpha x_{n} + (b x_{n}x_{n-4}) /(c x_{n-3}+d x_{n-4})$, Open J. Discrete Appl. Math., 2(2) (2019), 31–47.
• A. F. Yeniçerioğlu, C. Yazıcı, V. Yazıcı, Stability behaviour in functional differential equations of the neutral type, Univers. J. Math. Appl., 4(1) (2021), 33-40. https://doi.org/10.32323/ujma.711881
• A.M. Ahmed, A.M. Samir, and L.S. Aljoufi, Expressions and dynamical behavior of solutions of a class of rational difference equations of fifteenth-order, J. Math. Computer Sci., 25 (2022), 10-22.
• M. Berkal, R. Abo-zeid, On a rational (P+1)th order difference equation with quadratic term, Univers. J. Math. Appl., 5(4) (2022), 136-144. https://doi.org/10.32323/ujma.1198471
• B. Oğul, D. Şimşek, T.F. Ibrahim, A qualitative investigation of the solution of the difference equation $\Psi_ {m+ 1}=\frac {\Psi_ {m-3}\Psi_ {m-5}}{\Psi_ {m-1}\left (\pm1\pm\Psi_ {m-3}\Psi_ {m-5}\right)}$, Commun. Adv. Math. Sci., 6(2) (2023), 78-85.
• B. Oğul, D. Şimşek, H. Öğünmez, A.S. Kurbanlı, Dynamical behavior of rational difference equation $ x_ {n+ 1}=\frac {x_ {n-17}}{\pm 1\pm x_ {n-2} x_ {n-5} x_ {n-8} x_ {n-11} x_ {n-14} x_ {n-17}}$, Bol. Soc. Mat. Mex., 27(2) (2021), 1-20.