Maximum number of periodic orbits in parallel dynamical systems

For parallel dynamical systems over undirected graphs with a Boolean maxterm or minterm functions as global evolution operators, it is known that every periodic orbit has period less than or equal to two. In fact, periodic orbits of different periods cannot coexist and a fixed point theorem, based in the uniqueness of a fixed point, is also known. In this paper, we complete the study of the periodic structure of such systems, providing a 2-periodic orbit theorem, and giving an upper bound for the number of fixed points and also for the number of 2-periodic orbits. Actually, we provide examples where these bounds are attained, demonstrating that they are the best possible ones.