# Discrete synchronization of hierarchically organized dynamical systems.

Abstract : 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 2n 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 we can allow an infinite number of broken links inside the hierarchy while keeping a local synchronization, under the condition that these defects are present at the N smallest scales of the hierarchy (for a fixed integer N) and they be enough spaced out in those scales.
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https://hal.archives-ouvertes.fr/hal-00755688
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• HAL Id : hal-00755688, version 2

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Camille Poignard. Discrete synchronization of hierarchically organized dynamical systems.. 2014. ⟨hal-00755688v2⟩

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