Abstract
We study a collection of phase-coupled oscillators possessing a hierarchical coupling structure. We establish a necessary condition for the existence of a phase transition to collective synchrony for finite values of the coupling strength in terms of an inequality involving the connectivity between clusters of oscillators, the rate at which coupling strengths decrease with ultrametric distance, and the dispersion of intrinsic frequencies. When the inequality is not satisfied, there is a cascade of discrete transitions to intracluster synchrony as the coupling strength is increased, but no global synchronization is possible in the infinite size limit.
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