On sequential dynamical systems and simulation
Chris Barrett, Henning Mortveit, Christian M. Reidys · University of North Texas Digital Library (University of North Texas) · 1999
The generic structure of computer simulations motivates a new class of discrete dynamical systems that captures this structure in a mathematically precise way. This class of systems consists of (1) a loopfree graph {Upsilon} with vertex set {l_brace}1,2,{hor_ellipsis},n{r_brace} where each vertex has a binary state, (2) a vertex labeled set of functions (F{sub i,{Upsilon}}:F{sub 2}{sup n} {r_arrow} F{sub 2}{sup n}){sub i} and (3) a permutation {pi} {element_of} S{sub n}. The function F{sub i,{Upsilon}} updates the state of vertex i as a function of the states of vertex i and its {Upsilon}-neighbors and leaves the states of all other vertices fixed. The permutation {pi} represents the update ordering, i.e., the order in which the functions F{sub i,{Upsilon}} are applied. By composing the functions F{sub i,{Upsilon}} in the order given by {pi} one obtains the dynamical system (equation given in paper) which the authors refer to as a sequential dynamical system, or SDS for short. The authors will present bounds for the number of functionally different systems and for the number of nonisomorphic digraphs {Gamma}[F{sub {Upsilon}},{pi}] that can be obtained by varying the update order and applications of these to specific graphs and graph classes. This will be done using both combinatorial/algebraic techniques and probabilistic techniques. Finally the authors give results on dynamical system properties for some special systems.