Periods in XOR parallel dynamical systems over directed dependency graphs

Abstract

We analyze the periodic structure of parallel dynamical systems over directed dependency graphs, whose evolution operator is the Boolean function XOR. We prove that such systems can present periodic orbits of any period. Moreover, we demonstrate that any kinds of periods can coexist at the same time. In view of these results, we study the periodic structure of these dynamical systems over complete digraphs, complete bipartite digraphs, acyclic digraphs and out-trees.