In this work, we provide a procedure that allows us to transform certain kinds of deterministic Boolean networks on minterm or maxterm functions into symmetric ones, so inferring that such symmetrizable networks can present only periodic points of periods 1 or 2. In particular, we deal with generalized parallel (or synchronous) dynamical systems (GPDS) over undirected graphs, i.e., discrete parallel dynamical systems over undirected graphs where some of the self-loops may not appear. We also study the class of anti-symmetric GPDS (which are non-symmetrizable), proving that their periodic orbits have period 4. In addition, we introduce a class of non-symmetrizable systems which admit periodic orbits with arbitrary large periods.

In (Veliz-Cuba and Stigler, 2011) the authors proposed a Boolean model for the lac operon in Escherichia coli that is capable of predicting the operon being ON, OFF and bistable when the update schedule is the parallel one. We complement this work by using theoretical and algorithmic tools that allow us to know which are the configurations that converge to a fixed point or limit cycle (set namely attractor basin) for each deterministic update schedule. We show that, when bistability appears, about 70% of the dynamics have only the steady states ON and OFF. This latest having an attractor basin of an average size about 8 times bigger than that of ON. In the other 30%, the proportion is balanced between ON/OFF basins but the basins of limit cycles sum up, in average, about 5 times more than that of ON and OFF respectively. The techniques presented in this work are general and can be used to analyze other Boolean models.