  We continue our consideration of a class of models describing the reversible
dynamics of $N$ Boolean variables, each with $K$ inputs. We investigate in
detail the behavior of the Hamming distance as well as of the distribution of
orbit lengths as $N$ and $K$ are varied. We present numerical evidence for a
phase transition in the behavior of the Hamming distance at a critical value
$K_c\approx 1.65$ and also an analytic theory that yields the exact bounds on
$1.5 \le K_c \le 2.$
  We also discuss the large oscillations that we observe in the Hamming
distance for $K<K_c$ as a function of time as well as in the distribution of
cycle lengths as a function of cycle length for moderate $K$ both greater than
and less than $K_c$. We propose that local structures, or subsets of spins
whose dynamics are not fully coupled to the other spins in the system, play a
crucial role in generating these oscillations. The simplest of these structures
are linear chains, called linkages, and rings, called circuits. We discuss the
properties of the linkages in some detail, and sketch the properties of
circuits. We argue that the observed oscillation phenomena can be largely
understood in terms of these local structures.
