  A Boolean network is a finite dynamical system, whose variables take values
from a binary set. The value update rule for each variable is a Boolean
function, depending on a selected subset of variables. Boolean networks have
been widely used in modeling gene regulatory networks. We focus in this paper
on a special class of Boolean networks, termed as conjunctive Boolean networks.
A Boolean network is conjunctive if the associated value update rule is
comprised of only AND operations. It is known that any trajectory of a finite
dynamical system will enter a periodic orbit. We characterize in this paper all
periodic orbits of a conjunctive Boolean network whose underlying graph is
strongly connected. In particular, we establish a bijection between the set of
periodic orbits and the set of binary necklaces of a certain length. We further
investigate the stability of a periodic orbit. Specifically, we perturb a state
in the periodic orbit by changing the value of a single entry of the state. The
trajectory, with the perturbed state being the initial condition, will enter
another (possibly the same) periodic orbit in finite time steps. We then
provide a complete characterization of all such transitions from one periodic
orbit to another. In particular, we construct a digraph, with the vertices
being the periodic orbits, and the (directed) edges representing the
transitions among the orbits. We call such a digraph the stability structure of
the conjunctive Boolean network.

Dynamical Systems;Combinatorics