## The Characterization of Fixed Points in a Sequential Dynamical
System

### Jame Duvall

Graph dynamical systems are central to the modeling of a wide range
of different phenomena on networks.
As a particular type of graph dynamical systems, sequential dynamical
systems (SDS) have numerous applications
ranging from disease dynamics to the mapping of traffic
flows. These applications also extend
to modeling cellular automata (CA), which have a lot in common,
structurally, with SDS. The goal of this
paper is to thoroughly examine one of the most commonly studied graph
structures, the circle graph, which
is denoted by Circ_{n}. Using some function
*f*^{3} : {0,1}^{3} --> {0,1},
the process of characterizing fixed points
over this graph is delineated and the specific outcomes for each
result are then discussed. Specifically, this
paper will examine a special case, the majority function which is
denoted by maj_{3}, and the outcomes it
produces when applied to Circ_{n}. In general, computing
these fixed points for an arbitrary graph is hard and
computationally intractable. Therefore, the final portion of this
paper will discuss the wheel graph, denoted
Wheel_{n}, and the impending complications that arise when
attempting to compute its local fixed points.