On the basins of attraction of the regular autonomous asynchronous systems

The Boolean autonomous dynamical systems, also called regular autonomous asynchronous systems are systems whose ‘vector …eld’is a function : f0; 1gn ! f0; 1gn and time is discrete or continuous. While the synchronous systems have their coordinate functions 1; :::; n computed at the same time: ; ; ; ::: the asynchronous systems have 1; :::; n computed independently on each other. The purpose of the paper is that of studying the basins of attraction of the …xed points, of the orbits and of the !-limit sets of the regular autonomous asynchronous systems, by continuing the study started in [8]. The bibliography consists in analogies. MSC: 94C10 keywords: asynchronous system, ! limit set, invariance, basin of attraction 1 Introduction The R ! f0; 1g functions model the digital electrical signals and they are not studied in literature. An asynchronous circuit without input, considered as a collection of n signals, should be deterministically modelled by a function x : R ! f0; 1g called state. Several parameters related with the asynchronous circuit are either unknown, or perhaps variable or simply ignored in modeling: the temperature, the tension of the mains, the delays the occur in the computation of the Boolean functions etc. For this reason, instead of a function x we have in general a set X of functions x; called state space or autonomous system, where each x represents a possibility of modeling the circuit. When X is constructed by making use of a ’vector …eld’ : f0; 1g ! f0; 1g; the system X is