This work deals with the dynamical features of the system for three-dimensional difference equations

$$\{\begin{array}{ll}{u}_{n+1}=\alpha +\frac{u_{n-1}^q}{v_n^q},& \\ & \\ v_{n+1}=\alpha +\frac{v_{n-1}^q}{w_n^q},& n=0,1,\cdots ,\\ & \\ w_{n+1}=\alpha +\frac{w_{n-1}^q}{u_n^q},\end{array}$$

where the initial values $u_i,v_i,w_i\in (0,\mathrm{\infty }),i\in \{-1,0\}$, and the parameters $\alpha >0,q\ge 1$. In detail, the local asymptotical stability of the positive equilibrium point, boundedness, persistence, and oscillation behavior of the positive solution for the systems are obtained. Furthermore, using Matlab software, we give some examples to show the validity of theoretic analysis.