In this paper, we study the boundedness, persistence, and periodicity of the positive solutions and the global asymptotic stability of the positive equilibrium points of the system of difference equations $$x_{n+1}=A+\frac{x_{n-1}}{z_{n}},\qquad y_{n+1}=A+ \frac{y_{n-1}}{z _{n}},\qquad z_{n+1}=A+\frac{z_{n-1}}{y_{n}},\quad n=0,1,\ldots , $$ where $A\in ( 0,\infty ) $ and the initial conditions $x_{i}$ , $y_{i}$ , $z_{i}\in ( 0,\infty ) $ , $i=-1,0$ .

This paper deals with the form, the stability character, the periodicity and the global behavior of solutions of the following four rational difference equations \[x_{n+1} &=\frac{\pm 1}{x_{n}\left( x_{n-1}\pm 1\right) -1} \\ x_{n+1} &=\frac{\pm 1}{x_{n}\left( x_{n-1}\mp 1\right) +1}\text{.} \].