In this work, we provide a detailed study on the behavior of solutions of a $k$-dimensional system of rational difference equations, extending some existing results in the literature. We use the linearized stability theory to establish conditions for the local stability of the corresponding unique equilibrium point, and to show its global stability, we prove that the equilibrium point is a global attractor. Also, conditions for the existence of periodic solutions are provided. Our obtained results are confirmed by some examples.

We explicitly solve the following second-order system of difference equations:

$$x_{n+1}=\frac{a{x}_n{y}_{n-1}}{y_n-\beta y_{n-1}-\gamma }+b{x}_n+c,y_{n+1}=\frac{\alpha y_n{x}_{n-1}}{x_n-b{x}_{n-1}-c}+\beta y_n+\gamma ,$$

where $n\in {\mathbb{N}}_0$, $a$, $\alpha$, and the initial conditions $x_{-1}$, $y_{-1}$, $x_0$, and $y_0$ are nonzero real numbers, while the remaining parameters $b$, $c$, $\beta$, and $\gamma$ are real numbers. A detailed analysis of the solutions of our system when $\alpha =a$ with respect to the form, the periodicity and the limiting behavior is presented. To support and illustrate our theoretical results, numerical examples are provided. Our study, considerably generalizes some existing results in the literature.