Existence, coexistence and uniqueness of fixed points in parallel and sequential dynamical systems over directed graphs

Abstract

In this work, we solve the fixed-point existence, coexistence and uniqueness problems in the context of homogeneous parallel and sequential dynamical systems on maxterm and minterm Boolean functions over directed dependency graphs. More specifically, we give characterizations for the existence of fixed points. Likewise, we provide a sufficient condition and a necessary one for the non-coexistence of other periods in the dynamics of the system. In addition, we state a necessary and sufficient condition for the uniqueness of a fixed point, what allows us to establish a Fixed-Point Theorem in the sense of Banach for this kind of systems. Finally, we give an advance in the analysis of the number of fixed points of such systems, finding a lower bound for this number in the case of the most simplest maxterm and minterm. Numerical examples are given to show the feasibility of the results. Thus, this work completes the study of the fixed-point problems for homogeneous Boolean finite dynamical systems on maxterm and minterm functions.