Abstract
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.
Highlights
Discrete-time dynamical systems on a finite state space have an important role in several different branches of science and engineering
We study the periodic structure of parallel dynamical systems given by the composition of two parallel systems that are conjugate by means of an invertible map whose inverse is equal to the original map
In the rest of this section, we complete the study by considering general independent local functions and prove that, such systems can present periodic orbits of any period greater than one, under particular conditions related to independent sets, they can only present fixed points or 2-periodic orbits, as in the known case of AND-OR-NAND-NOR-parallel dynamical system (PDS)
Summary
Discrete-time dynamical systems on a finite state space have an important role in several different branches of science and engineering. In the rest of this section, we complete the study by considering general independent local functions and prove that, such systems can present periodic orbits of any period greater than one, under particular conditions related to independent sets, they can only present fixed points or 2-periodic orbits, as in the known case of AND-OR-NAND-NOR-PDS. Both fixed and 2-periodic points can coexist, so breaking the pattern found for homogeneous PDS on any of these Boolean functions. If we consider directed dependency graphs, we prove that the corresponding composition may have points of any period
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