Global behavior of a rational system of difference equations with arbitrary powers

In this work, we are interested in the global behavior of the following second-order rational system of difference equations with the presence of arbitrary powers given by \(\begin{aligned} u_{n+1}=a+\frac{v^{p}_{n-1}}{bv^{p}_n+cv^{p}_{n-1}},\quad v_{n+1}=\left( \frac{\alpha u^{q}_{n}+\beta u^{q}_{n-1}}{\gamma u^{q}_n+\lambda u^{q}_{n-1}}\right) u^{r}_nu^{s}_{n-1},\,n\in \mathbb {N}_0, \end{aligned}\) $$\begin{array}{c}\hfill u_{n+1}=a+\frac{v_{n-1}^p}{b{v}_n^p+c{v}_{n-1}^p},v_{n+1}=\left(\frac{\alpha u_n^q+\beta u_{n-1}^q}{\gamma u_n^q+\lambda u_{n-1}^q}\right)u_n^r{u}_{n-1}^s,n\in {\mathbb{N}}_0,\end{array}$$ where \(p,\,q\in \mathbb {N}\) $p,q\in \mathbb{N}$ , \(r,\,s\in \mathbb {N}_{0}\) $r,s\in {\mathbb{N}}_0$ , the parameters a, b, c, \(\alpha \) $\alpha$ , \(\beta \) $\beta$ , \(\gamma \) $\gamma$ , \(\lambda \) $\lambda$ and the initial values \(u_{-1}\) $u_{-1}$ , \(u_{0}\) $u_0$ , \(v_{-1}\) $v_{-1}$ , \(v_{0}\) $v_0$ are positive real numbers. We establish results on the stability of the unique equilibrium point and the existence of periodic solutions. Numerical examples that confirm the obtained results are presented.