Abstract
In this work, we solve the problem of the coexistence of periodic orbits in homogeneous Boolean graph dynamical systems that are induced by a maxterm or a minterm (Boolean) function, with a direct underlying dependency graph. Specifically, we show that periodic orbits of any period can coexist in both kinds of update schedules, parallel and sequential. This result contrasts with the properties of their counterparts over undirected graphs with the same evolution operators, where fixed points cannot coexist with periodic orbits of other different periods. These results complete the study of the periodic structure of homogeneous Boolean graph dynamical systems on maxterm and minterm functions.
Highlights
From the theoretical point of view, deterministic Boolean networks can be seen as discrete dynamical systems of the form
In the literature, they are known as Boolean dynamical systems [20] or, more properly, as Boolean finite dynamical systems [21,22] to emphasize that they are considered on a finite set of elements, without mentioning the term network
The main contribution of this paper is to show that periodic orbits of any period can exist and coexist together in MAX − PDDS, MIN − PDDS, MAX − SDDS and MIN − SDDS, which means a breaking of the pattern found in the case of their counterparts with undirected dependency graphs
Summary
Either deterministic or probabilistic, have demonstrated to be a very useful tool to formalize several phenomena coming from computer sciences [1,2], and from other sciences such as biology [3,4,5,6,7,8,9,10], chemistry [11,12], physics [13,14,15,16], mathematics [17,18]. In [37], the authors proved that, for the sequential case of Boolean graph dynamical systems whose evolution operators are given by threshold component functions, the only ω-limit sets are fixed points, in contrast with the results for homogeneous sequential systems induced by maxterms and minterms shown in [32]. The main contribution of this paper is to show that periodic orbits of any period can exist and coexist together in MAX − PDDS, MIN − PDDS, MAX − SDDS and MIN − SDDS, which means a breaking of the pattern found in the case of their counterparts with undirected dependency graphs These results complete the study of the periodic structure of homogeneous (Boolean) graph dynamical systems on maxterm and minterm Boolean functions. We finish the paper by showing the most important conclusions and future research directions which can be derived from these results
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