Updating method for the computation of orbits in parallel and sequential dynamical systems

Abstract

In this article, we provide a matrix method in order to compute orbits of parallel and sequential dynamical systems on Boolean functions. In this sense, we develop algorithms for systems defined over directed (and undirected) graphs when the evolution operator is a general minterm or maxterm and, likewise, when it is constituted by independent local Boolean functions, so providing a new tool for the study of orbits of these dynamical systems.

Predecessors Existence Problems and Gardens of Eden in Sequential Dynamical Systems

“…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.…”

“…Given a SDS, the dynamics of the rest of the states can be obtained computationally by brute-force, using algorithms like those in [34,40]. But, unfortunately, the results in such computations are only valid for such a SDS.…”

Parallel Dynamical Systems over Graphs and Related Topics: A Survey

“…However, it is not so easy to determine a priori the different coexistent periods of its orbits, because it depends fundamentally on the structure of the directed dependency graph. In this sense, in [90], algorithms for computing the (eventually) period of any orbit are given.…”

“…Since the coexistence of periodic orbits depends on the number of entities, their connections, and the Boolean operator, several matrix algorithms for the computation of orbits in PDDS over directed dependency graph are developed in [90]. These algorithms constitute a new tool for the study of orbits of these dynamical systems.…”

Linear sequential dynamical systems, incidence algebras, and Möbius functions

“…In [1,2,3,4,6,7,8,9,16] the reader can find a plethora of results on dynamical systems over particular graphs and updating functions. Linear systems together with a parallel update schedule, i.e.…”