On the global behavior and the periodicity of the solutions of a $ k $-dimensional system of difference equations

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.