On the solutions and periodicity of some nonlinear systems of difference equations

We investigate the expressions of solutions and the periodicity nature of the following system of rational difference equations of order four xn+1=zn−3a1+b1znyn−1xn−2zn−3,yn+1=xn−3a2+b2xnzn−1yn−2xn−3,zn+1=yn−3a3+b3ynxn−1zn−2yn−3, where the initial conditions x − 3 ; x − 2 ; x − 1 ; x 0 , y − 3 ; y − 2 ; y − 1 ; y 0 ; z − 3 ; z − 2 ; z − 1 and z 0 are arbitrary real numbers and a 1 ; b 1 ; a 2 ; b 2 ; a 3 ; b 3 are integers.