Discrete, sequential dynamical systems

Abstract

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 (F i, Y : F 2 n→ F 2 n) i and (c) a permutation π∈ S n . The function F i, 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 F i, Y are applied. By composing the functions F i, Y in the order given by π we obtain the dynamical system [ F Y, π]=∏ i=1 n F π(i),Y : F 2 n→ F 2 n, 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 |{[ F Y, π] | π∈S n}| for fixed Y and (F i, Y ) i . Finally, we give an estimate for the number of non-isomorphic digraphs Γ[ F Y, π] (having vertex sets F 2 n and directed edges {(x, [ F Y, π](x)) | x∈ F 2 n} ) for a fixed graph Y and a fixed multi-set (F i, Y ) i .