Parallel Dynamical Systems over Graphs and Related Topics: A Survey

Juan A Aledo,Silvia Martinez,Jose C Valverde

doi:10.1155/2015/594294

Juan A Aledo, Silvia Martinez + Show 1 more

Open Access

https://doi.org/10.1155/2015/594294

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Abstract

In discrete processes, as computational or genetic ones, there are many entities and each entity has a state at a given time. The update of states of the entities constitutes an evolution in time of the system, that is, a discrete dynamical system. The relations among entities are usually represented by a graph. The update of the states is determined by the relations of the entities and some local functions which together constitute (global) evolution operator of the dynamical system. If the states of the entities are updated in a synchronous manner, the system is called aparallel dynamical system. This paper is devoted to review the main results on the dynamical behavior of parallel dynamical systems over graphs which constitute a generic tool for modeling discrete processes.

Highlights

Modeling discrete processes is one of the most important tasks in modern mathematics
In [9], it is proved that, for a parallel dynamical system OR-PDS associated with the maxterm OR, all the orbits of the system are fixed points or eventually fixed points
In [42], it is demonstrated that for the general case of PDDS associated with maxterms and minterms as global evolution operators any period can appear, breaking the pattern found for the undirected case where only fixed points or 2-periodic orbits can exist [41]

Summary

Introduction

Modeling discrete processes is one of the most important tasks in modern mathematics. According to [6], one of the main goals in the study of a dynamical system is to give a complete characterization of its orbit structure This is the main purpose of the review paper in relation with parallel dynamical systems over graphs: to show as much information about the orbit structure as possible, based on the properties of the dependency graph and the local Boolean functions which constitute the evolution law of the system. In this particular case, as the state space of the system is finite, every orbit is periodic or eventually periodic. The paper finishes by setting out several interesting future research directions for the development of these kinds of models

Theoretical Foundations

Analytical Results on the Orbital Structure of PDS

Extensions of the Concept of PDS

Algorithms for the Computation of Orbits

Conclusions and Future Research