On the dynamics of semilattice networks

Abstract

Finite dynamical systems are used to model problems from different branches of sciences such as biological networks, computer process, physics, and engineered control systems. An important combinatorial tool for studying limit cycles of a finite dynamical system is dependency graph. In Veliz-Cuba and Laubenbacher (2019), cycle structure of semilattice networks has completely been characterized when the dependency graph is strongly connected. This characterization is in terms of the loop number of the dependency graph.In this paper, we use the methods developed in Jarrah et al. (2010); Veliz-Cuba and Laubenbacher (2019) and we study cycle structure of a semilattice network with an arbitrary dependency graph. We show that the period of any limit cycle divides the loop number. Next, we prove that all periodic points of a semilattice network are fixed points if and only if the loop number of dependency graph is 1. We also find some sufficient conditions under which system has periodic points of any period that divides the loop number. Then, using the same idea as Chen et al. (2020), we give a reduction process in studying cycle structure of a semilattice networks. Finally, we completely characterized the set of fixed points of a semilattice networks in terms of the set of isotone maps between two certain partially ordered sets. Using this characterization, we give a sharp lower bound for the number of fixed points. The results in this paper apply to certain types of Boolean networks and diffusion models, and in particular extends and recovers some of the results of Chen et al. (2020); Jarrah et al. (2010).