Universal Regular Autonomous Asynchronous Systems: Fixed Points, Equivalencies and Dynamic Bifurcations

The asynchronous systems are the non-deterministic models of the asynchronous circuits from the digital electrical engineering. In the autonomous version, such a system is a set of functions x : R → {0, 1} called states (R is the time set). If an asynchronous system is defined by making use of a so called generator function Φ : {0, 1} → {0, 1}, then it is called regular. The property of universality means the greatest in the sense of the inclusion. The purpose of the paper is that of defining and of characterizing the fixed points, the equivalencies and the dynamical bifurcations of the universal regular autonomous asynchronous systems. We use analogies with the dynamical systems theory. 1 Preliminaries Definition 1 We denote by B = {0, 1} the binary Boole algebra, endowed with the discrete topology and with the usual laws. Definition 2 Let be the Boolean function Φ : B → B,Φ = (Φ1, ...,Φn) and ν ∈ B, ν = (ν1, ..., νn). We define Φ ν : B → B by ∀μ ∈ B, Φ(μ) = (ν1 · μ1 ⊕ ν1 · Φ1(μ), ..., νn · μn ⊕ νn · Φn(μ)). Remark 3 Φ represents the function resulting from Φ when this one is not computed, in general, on all the coordinates Φi, i = 1, n : if νi = 0, then Φi is not computed, Φ ν i (μ) = μi and if νi = 1, then Φi is computed, Φi (μ) = Φi(μ). Definition 4 Let be the sequence α, α, ..., α, ... ∈ B. The functions Φ 0α1...αk : B → B are defined iteratively by ∀k ∈ N, ∀μ ∈ B, Φ 0α1...αkαk+1(μ) = Φ k+1 (Φ 0α1...αk(μ)).