  Oscillator networks found in social and biological systems are characterized
by the presence of wide ranges of coupling strengths and complex organization.
Yet robustness and synchronization of oscillations are found to emerge on
macro-scales that eventually become key to the functioning of these systems. In
order to model this kind of dynamics observed, for example, in systems of
circadian oscillators, we study networks of Van der Pol oscillators that are
connected with hierarchical couplings. For each isolated oscillator we assume
the same fundamental frequency. Using numerical simulations, we show that the
coupled system goes to a phase-locked state, with both phase and frequency
being the same for every oscillator at each level of the hierarchy. The
observed frequency at each level of the hierarchy changes, reaching an
asymptotic lowest value at the uppermost level. Notably, the asymptotic
frequency can be tuned to any value below the fundamental frequency of an
uncoupled Van der Pol oscillator. We compare the numerical results with those
of an approximate analytic solution and find them to be in qualitative
agreement.
