Discrete Synchronization under the Hierarchical viewpoint

Camille Poignard (1)

1 JAD - Laboratoire Jean Alexandre Dieudonné

We study the synchronization problem of dynamical systems in case of a hierarchical structure among them, of which interest comes from the growing necessity of understanding properties of complex systems, that often exhibit such an organization. Starting with a set of $2^n$ systems, we define a hierarchical structure inside it by a matrix representing all the steps of a matching process in groups of size $2$. This leads us naturally to the synchronization of a Cantor set of systems, indexed by $\{0,1\}^{\En}$: we obtain a global synchronization result generalizing the finite case. In the same context, we deal with this question when some defects appear in the hierarchy, that is to say when some links between certain systems are broken. We prove a local synchronization result under the condition that these defects are present at the $N$ smallest scales of the hierarchy (for a fixed integer $N$) and their number be in $\sqrt{n}$ when $n$ goes to infinity.

Mots clés: dynamical systems; synchronization; hierarchy

Domaines: Systèmes dynamiques [math.DS]

Camille Poignard. Discrete Synchronization under the Hierarchical viewpoint. 2012. ⟨hal-00755688v1⟩