# [FOM] cycles in Boolean Networks

abuchan@mail.unomaha.edu abuchan at mail.unomaha.edu
Mon Feb 10 14:48:03 EST 2003

```Consider the following  Boolean network.

A(t+1) = B(t)D(t)
B(t+1) = 1+ A(t)
C(t+1) = B(t)

If we are interested in the cycle structure of the network we can iterated
the function discretely to get scalar equations.  For example, the scalar
equation of node A of the above system is

A(t+4) = (1+ A(t+t))(1+A(t))

Where the state of A at t+4 depends on the state of A at t and t+2.  So the
state if A at even time points is dependent only on the previous two even
time points.  So, the state of A at the first two even and odd time point
must be given and then the system can move along the trajectory mandated by
the logic.  In this example, independent of how node A starts, it move to a
period 3 orbit (0,0,1) on even time points and similarly of odd time
points.  So there are only two different possible orbits of node A, (0,0,1)
or (0,0,0,0,1,1), which give us a either a period 3 cycle or a period 6
cycle.  This method is not effective for large Boolean networks when we
wish to find the possible cycle structure, nor does it give any insight for
a general n-node network.   Further, it does not seem to give us much
information about the transient states.   What I would like to know is if
there is an effective algorithm for a general synchronous Boolean network
that will give information about lengths of transience and cycles?
References would be nice.

```