Frequency and phase synchronization in large groups: Low dimensional description of synchronized clapping, firefly...

A common observation is that large groups of oscillatory biological units often have the ability to synchronize. A paradigmatic model of such behavior is provided by the Kuramoto model, which achieves synchronization through coupling of the phase dynamics of individual oscillators, while each oscillator maintains a different constant inherent natural frequency. Here we consider the biologically likely possibility that the oscillatory units may be capable of enhancing their synchronization ability by adaptive frequency dynamics. We propose a simple augmentation of the Kuramoto model which does this. We also show that, by the use of a previously developed technique [Ott and Antonsen, Chaos 18, 037113 (2008)], it is possible to reduce the resulting dynamics to a lower dimensional system for the macroscopic evolution of the oscillator ensemble. By employing this reduction, we investigate the dynamics of our system, finding a characteristic hysteretic behavior and enhancement of the quality of the achieved synchronization.