On solutions of three-dimensional system of difference equations with constant coefficients

Authors : Merve KARA, Ömer AKTAŞ

Pages : 462-481

Doi:10.31801/cfsuasmas.1163955

View : 32 | Download : 51

Publication Date : 2023-06-23

Article Type : Research Paper

Abstract :In this study, we show that the system of difference equations \\begin{align} x_{n}=\\frac{x_{n-2}y_{n-3}}{y_{n-1}\\leftinsert ignore into journalissuearticles values(a+bx_{n-2}y_{n-3} \\right); }, \\nonumber \\\\ y_{n}=\\frac{y_{n-2}z_{n-3}}{z_{n-1}\\leftinsert ignore into journalissuearticles values(c+dy_{n-2}z_{n-3} \\right); },~n\\in\\mathbb{N}_{0}, ~ \\nonumber \\\\ z_{n}=\\frac{z_{n-2}x_{n-3}}{x_{n-1}\\leftinsert ignore into journalissuearticles values(e+fz_{n-2}x_{n-3} \\right); }, \\nonumber \\\\ \\end{align} where the initial values $x_{-i}, y_{-i}, z_{-i}$, $i=\\overline{1,3}$ and the parameters $a$, $b$, $c$, $d$, $e$, $f$ are non-zero real numbers, can be solved in closed form. Moreover, we obtain the solutions of above system in explicit form according to the parameters $a$, $c$ and $e$ are equal $1$ or not equal $1$. In addition, we get periodic solutions of aforementioned system. Finally, we define the forbidden set of the initial conditions by using the acquired formulas.
Keywords : Explicit form, forbidden set, periodicity