Juan Luis García Guirao

Modeling the dynamics of concurrent computing systems

This paper presents a very first step on the road to applying the theory of discrete dynamical systems (DDSs) in the analysis of concurrent computing systems. In order to proceed, Petri nets (PNs) are appropriately modeled as DDSs, so defining the corresponding phase space with its metric structure and the evolution operator of the system. We conclude this study by showing results for some identifiable problems.

On skew-product maps with the base having a closed set of periodic points

Sharkovskii˘ and Kolyada reported the problem of the characterization of skew-product maps having zero topological entropy. It is known that, even under additional assumptions, this aim has not been attained. Guirao and Chudziak partially solved this problem for the class of skew-product maps with the base map having a closed set of periodic points. The present paper has two aims for this class of map. On the one hand, to improve that solution showing the equivalence between the property ‘of having zero topological entropy’ and the fact of ‘not being Li–Yorke chaotic in the union of the ω-limit sets of recurrent points’. On the other hand, we show that the properties ‘of having a closed set of periodic points’ and ‘all non-wandering points are periodic’ are not mutually equivalent properties. In doing this we disprove a result of Efremova.

Petri Nets and Discrete Events Systems

This paper provides an overview over the relationship between Petri Nets and Discrete Event Systems as they have been proved as key factors in the cognitive processes of perception and memorization. In this sense, different aspects of encoding Petri Nets as Discrete Dynamical Systems that try to advance not only in the problem of reachability but also in the one of describing the periodicity of markings and their similarity, are revised. It is also provided a metric for the case of Non-bounded Petri Nets.

On solenoidal distribution of infinite ω-limit sets

The aim of this paper is to present a structure result on the infinite ω-limit sets of continuous triangular mappings defined on the unit square, i.e. maps of the form (x, y)→(f(x), g(x, y)) having zero topological entropy in the base map f and on the fibres defined over the periodic points of f. We show for this class of systems that the infinite ω-limit sets of the points of the form (x, y), where x is a periodic point of f, have a solenoidal distribution. It extends the results by Smítal [J. Smítal, Chaotic functions with zero topological entropy, Trans. Amer. Math. Soc. 297 (1986), pp. 269–282] on zero topological entropy of continuous interval maps. An application is presented.