Full characterization of attractors for two intersected asynchronous Boolean automata cycles

Nowadays, the understanding of Boolean automata networks dynamics takes an important place in various domains of computer science such as computability, complexity and discrete dynamical systems. Basing ourselves on these specific mathematical objects, in this paper, we make a step further in this understanding by focusing on their cycles, whose necessity in networks is known as the brick of their complexity. We indeed present new results that provide a characterisation of the transient and asymptotic dynamics, i.e. of the computational abilities, of asynchronous Boolean automata networks composed of two cycles that intersect by sharing one automaton, the so-called double-cycle. Moreover, we introduce here an efficient formalism inspired by algorithms to define long sequences of updates. Also, the analysis of these Boolean automata networks relies on technics developed for cellular automata. These technics allow us to be more precise for describing the asymptotic dynamics than previous works in this area.