On a class of nonlinear rational systems of difference equations

In this paper, we construct and formulate the solutions and periodicity character of the following nonlinear rational systems of difference equations:

$$S_{n+1}={\displaystyle \frac{T_n{S}_{n-2}}{S_{n-2}+T_{n-1}}},T_{n+1}={\displaystyle \frac{S_n{T}_{n-2}}{\pm T_{n-2}\pm S_{n-1}}},n=0,1,2,....,(0.1)$$

where the initial conditions $s_{-2},s_{-1},s_0,t_{-2},t_{-1},t_0$ are positive real numbers. Moreover, some mathematical programs are used to support our theoretical results of each system in this paper.