Advances in parallel and sequential dynamical systems over graphs

In this dissertation, the dynamics of homogeneous parallel and sequential dynamical systems on maxterm and minterm Boolean functions are analyzed. In particular, for parallel and sequential dynamical systems over undirected graphs, the dynamics are described completely, while some advances are provided for such systems over directed graphs. Specifically, for the case of homogeneous parallel dynamical systems on maxterm or minterm Boolean functions over undirected graphs, it is proved that they can present only two kinds of periodic orbits: fixed points and 2-periodic orbits. Furthermore, it is demonstrated that fixed points and 2-periodic orbits cannot coexist. In addition, uniqueness results of such periodic orbits are provided. Finally, the study of the periodic structure of such systems is completed by showing optimal upper bounds for the number of fixed points and 2-periodic orbits, and examples where these bounds are attained. The dynamics of non-periodic orbits are also studied for this kind of systems, by solving the classical predecessor problems (existence, uniqueness, coexistence and number of predecessors), obtaining a characterization of the Garden-of-Eden configurations and an optimal bound for the number of them. Additionally, it is provided a characterization of attractors and a method to obtain their basins of attraction. Finally, optimal upper bounds for the transient in such systems are shown. In the case of homogeneous sequential dynamical systems on maxterm or minterm Boolean functions over undirected graphs, it is demonstrated that they can present periodic orbits of any period. Besides, it is proved that periodic orbits with different periods greater than or equal to 2 can coexist, but when these systems have fixed points, periodic orbits of other periods cannot appear. Finally, as in the parallel update case, the study of the periodic structure of such systems is completed by showing optimal upper bounds for the number of fixed points and periodic orbits of period greater than 1, and examples where these bounds are attained. In this case, the dynamics of non-periodic orbits are also studied, by solving the same problems as in the case of parallel dynamical systems on maxterm or minterm Boolean functions over undirected graphs. Indeed, the classical predecessor problems (existence, uniqueness, coexistence and number of predecessors) are solved, providing a characterization of the Garden-of-Eden configurations and an optimal bound for the number of them. A characterization of attractors and a method to obtain their basins of attraction are shown, also providing optimal upper bounds for the transient in such systems. Finally, for homogeneous parallel and sequential dynamical systems on maxterm or minterm Boolean functions over directed graphs, it is proved that periodic orbits of any periods can appear and coexist, even fixed points and periodic orbits with greater periods. Also, a solution to the predecessor problems is provided, so extending the results given for systems over undirected graphs. Consequently, a characterization of the Garden-of-Eden states is achieved, providing the best bound for the number of them