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Article

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

1
Department of Mathematics, University of Castilla-La Mancha, 02071 Albacete, Spain
2
Department of Mathematics and Computer Sciences, Hakim Sabzevari University, 9617976487 Sabzevar, Iran
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(7), 1088; https://doi.org/10.3390/math8071088
Received: 3 June 2020 / Revised: 15 June 2020 / Accepted: 16 June 2020 / Published: 3 July 2020
(This article belongs to the Special Issue Boolean Networks Models in Science and Engineering)
In this paper, based on previous results on AND-OR parallel dynamical systems over directed graphs, we give a more general pattern of local functions that also provides fixed point systems. Moreover, by considering independent sets, this pattern is also generalized to get systems in which periodic orbits are only fixed points or 2-periodic orbits. The results obtained are also applicable to homogeneous systems. On the other hand, we study the periodic structure of parallel dynamical systems given by the composition of two parallel systems, which are conjugate under an invertible map in which the inverse is equal to the original map. This allows us to prove that the composition of any parallel system on a maxterm (or minterm) Boolean function and its conjugate one by means of the complement map is a fixed point system, when the associated graph is undirected. However, when the associated graph is directed, we demonstrate that the corresponding composition may have points of any period, even if we restrict ourselves to the simplest maxterm OR and the simplest minterm AND. In spite of this general situation, we prove that, when the associated digraph is acyclic, the composition of OR and AND is a fixed point system. View Full-Text
Keywords: boolean networks; fixed points; periodic points; independent sets; conjugate and equivalent systems; boolean functions boolean networks; fixed points; periodic points; independent sets; conjugate and equivalent systems; boolean functions
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MDPI and ACS Style

Aledo, J.A.; Barzanouni, A.; Malekbala, G.; Sharifan, L.; Valverde, J.C. On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions. Mathematics 2020, 8, 1088. https://doi.org/10.3390/math8071088

AMA Style

Aledo JA, Barzanouni A, Malekbala G, Sharifan L, Valverde JC. On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions. Mathematics. 2020; 8(7):1088. https://doi.org/10.3390/math8071088

Chicago/Turabian Style

Aledo, Juan A.; Barzanouni, Ali; Malekbala, Ghazaleh; Sharifan, Leila; Valverde, Jose C. 2020. "On the Periodic Structure of Parallel Dynamical Systems on Generalized Independent Boolean Functions" Mathematics 8, no. 7: 1088. https://doi.org/10.3390/math8071088

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