Asymptotic behavior of a reduced conjunctive Boolean network

A conjunctive Boolean network (CBN) is a discrete-time finite state dynamical system, whose variables take values from a binary set, and the value update rule for each variable is a Boolean function consisting only of logic AND operations. Since a CBN is a finite state dynamical system, every trajectory generated by the system will enter a periodic orbit. We characterize in this paper the asymptotic behavior of a special class of weakly connected CBNs where the strongly connected components of their dependency graphs are all cycles of positive lengths. Given an initial condition of such a CBN, we characterize a periodic orbit which the system enters with the given initial condition.