We study the kth order system of two rational difference equations

$$x_n=\frac{\beta_k x_{n-k} +\gamma_k y_{n-k}} {1+\sum_{j=1}^{k-1}B_j x_{n-j}
+ \sum_{j=1}^{k-1}C_{j}y_{n-j}}, n\in\mathbb{N},$$

$$y_{n}=\frac{\delta_k x_{n-k} +\epsilon_k y_{n-k}} {1+\sum_{j=1}^{k-1}D_j
x_{n-j} + \sum_{j=1}^{k-1}E_j y_{n-j}}, n\in\mathbb{N},$$ with nonnegative
parameters and nonnegative initial conditions.

We establish the existence of periodic tetrachotomy behavior which depends on
the matrix $$(\begin{array}{cc} \beta_{k} & \gamma_k

\delta_k & \epsilon_{k} \end{array}).$$

We provide some partial results for similar systems of three or more rational
difference equations.