Discrete, sequential dynamical systems Original Research Article

We study a class of discrete dynamical systems that consists of the following data: (a) a finite loop-free graph Y with vertex set {1, …, n} where each vertex has a binary state, (b) a vertex labeled multi-set of functions (Fi, Y : F2n→F2n)i and (c) a permutation π∈Sn. The function Fi, Y updates the state of vertex i as a function of the states of vertex i and its Y-neighbors and leaves the states of all other vertices fixed. The permutation π represents a Y-vertex ordering according to which the functions Fi, Y are applied. By composing the functions Fi, Y in the order given by π we obtain the dynamical system [FY, π]=∏i=1n Fπ(i),Y : F2n→F2n, which we refer to as a sequential dynamical system (SDS). Among various basic results on SDS we will study their invertibility and analyze the set |{[FY, π] | π∈Sn}| for fixed Y and (Fi, Y)i. Finally, we give an estimate for the number of non-isomorphic digraphs Γ[FY, π] (having vertex sets F2n and directed edges {(x, [FY, π](x)) | x∈F2n}) for a fixed graph Y and a fixed multi-set (Fi, Y)i.