Discrete synchronization of massively connected systems using hierarchical couplings

•We study the synchronization of networks having an uncountable number of systems.•We assume the couplings between the nodes arise from a two-by-two hierarchical organization.•Global synchronization takes place when such a network is fully coupled, if the parameters are close to 1/2.•We study the possibility of synchronization in case of an infinity of broken links inside the hierarchical structure.•We show local synchronization occurs if the broken links are well placed and the parameters are very close to 1/2.

We study the synchronization of massively connected dynamical systems for which the interactions come from the succession of couplings forming a global hierarchical coupling process. Motivations of this work come from the growing necessity of understanding properties of complex systems that often exhibit a hierarchical structure. Starting with a set of 2n2n systems, the couplings we consider represent a two-by-two matching process that gather them in larger and larger groups of systems, providing to the whole set a structure in nn stages, corresponding to nn scales of hierarchy. This leads us naturally to the synchronization of a Cantor set of systems, indexed by {0,1}N{0,1}N, using the closed-open sets defined by nn-tuples of 00 and 11 that permit us to make the link with the finite previous situation of 2n2n systems: we obtain a global synchronization result generalizing this case. In the same context, we deal with this question when some defects appear in the hierarchy, that is to say when some couplings among certain systems do not happen at a given stage of the hierarchy. We prove we can accept an infinite number of broken links inside the hierarchy while keeping a local synchronization, under the condition that these defects are present at the NN smallest scales of the hierarchy (for a fixed integer NN) and they be enough spaced out in those scales.