On the Periods of Parallel Dynamical Systems

Abstract

In this work, we provide conditions to obtain fixed point theorems for parallel dynamical systems over graphs with (Boolean) maxterms and minterms as global evolution operators. In order to do that, we previously prove that periodic orbits of different periods cannot coexist, which implies that Sharkovsky’s order is not valid for this kind of dynamical systems.

Predecessors Existence Problems and Gardens of Eden in Sequential Dynamical Systems

“…The periodic structure of SDS on maxterm and minterm Boolean functions is studied in [37], where it is proved that any period can appear in their phase portrait, although fixed points cannot coexist with other periods. This issue is someway a generalization of the results in PDS on the same Boolean functions [29,33], where only fixed points and 2-periodic orbits can appear but not coexist. This generalization on the periodic structure, obtained when passing from PDS to SDS, motivates the study of the behavior of the rest of the states for SDS that we perform in this work.…”

“…The interactions among elements of a phenomenon do typically not occur simultaneously. When it occurs, it is said that the model updates synchronously or parallelly (see [25,26,[28][29][30][31][32][33][34][35][36]); otherwise it is said that the model updates asynchronously or sequentially (see [20,27,[37][38][39][40][41]). In this last case, an update order is needed to specify the sequence in which the states of the elements evolve.…”

On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions

“…In particular, in [30], it is proved that any OR-PDS and AND-PDS are fixed point systems, while NAND-PDS and NOR-PDS are 2-periodic point systems, independently of the associated (simple) graph. Furthermore, these results are generalized in [25][26][27][28][29]33,34] where the authors study the periodic structure of MAX-PDS, MAX-PDDS, MIN-PDS, and MIN-PDDS. That is, parallel dynamical systems where the future state of each node is computed using the Boolean maxterm MAX or the Boolean minterm MIN.…”

“…Usually, the acronym PDS is used for parallel dynamical systems over undirected graphs, whereas if the associated graph is properly a digraph, F is said to be a parallel directed dynamical system (PDDS). As can be seen in the recent literature [25][26][27][28][29][30][31][32], the dynamics of PDDS are, in general, much more involved than the dynamics of PDS.…”

Predecessors and Gardens of Eden in sequential dynamical systems over directed graphs

“…As shown in [2,8,9], the dynamics of GDS over undirected graphs can be very different from the dynamics over directed ones. This motivates the study in both contexts separately to extract such differences, as done in [2,5,6,8,9,22,23,26,27]. Thus, in this work, once the situation in the undirected context is known [7], we deal with the predecessors existence problems for SDS over directed graphs (SDDS).…”