On a close to symmetric system of difference equations of second order

Abstract

Closed form formulas of the solutions to the following system of difference equations: xn=yn−1yn−2xn−1(an+bnyn−1yn−2),yn=xn−1xn−2yn−1(αn+βnxn−1xn−2),n∈N0,$$x_{n}=\frac{y_{n-1}y_{n-2}}{x_{n-1}(a_{n}+b_{n}y_{n-1}y_{n-2})},\qquad y_{n}=\frac{x_{n-1}x_{n-2}}{y_{n-1}(\alpha _{n}+\beta _{n}x_{n-1}x_{n-2})},\quad n\in \mathbb {N}_{0}, $$ where an$a_{n}$, bn$b_{n}$, αn$\alpha _{n}$, βn$\beta _{n}$, n∈N0$n\in \mathbb {N}_{0}$, and initial values x−i$x_{-i}$, y−i$y_{-i}$, i∈{1,2}$i\in\{1,2\}$ are real numbers, are found. The domain of undefinable solutions to the system is described. The long-term behavior of its solutions is studied in detail for the case of constant an$a_{n}$, bn$b_{n}$, αn$\alpha _{n}$ and βn$\beta _{n}$, n∈N0$n\in \mathbb {N}_{0}$.