For the dynamical systems over finite fields, a lot of work has been done. In the linear case, the complete phase space structure has been determined from the matrix representation. For nonlinear systems, the phase space structure becomes very difficult to descript. Even for the monomial systems, there is no complete answer for its dynamics. But some progresses have been made to determine the fixed points. In this paper, we first slightly generalize an important reduction result in [3]. Then, we consider the more general monomial system over finite fields but with specific bidirectional cycle dependency graph. We obtain some results about its dynamics. The cardinality of the set of all such systems is also presented. When  we have an efficient way to describe its dynamics by applying the results on linear case in [7] and affine case in [15]. Especially, when the network has odd number of vertices (genes), a complete description is obtained.

finite fields, dynamical system.