## On the Enumeration of Cycle-equivalent Classes of
Sequential Dynamical Systems

### Andrew Dove

Sequential dynamical systems (SDSs) is a class of dynamical systems
that can
be used to model many complex systems. Two SDSs are cycle equivalent
if their periodic
orbits are isomorphic as directed graphs. This means that the two
systems have similar
long-term dynamics. This paper approaches the problem of applying the
vertex functions
sequentially to the states of the vertices in a given graph *Y*, using
an update order instead
of applying all the vertex functions to *Y* in parallel. Three forms of
equivalence for SDSs
are functional equivalence, dynamical equivalence, and cycle
equivalence. This paper
describes upper bounds for the number of equivalence classes for each
of these forms of
equivalence, focusing mainly on cycle equivalence and its
bound, κ. This paper gives new
results for the evaluation of κ for certain families of
graphs.