Dynamical properties of hierarchical networks of Van Der Pol oscillators

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.