The Slow Clock is the Master

Abstract We investigate the synchronization and stability of mutually coupled oscillators through small impacts and explore both master-master and master-slave synchrony. The synchronization behaviour between two oscillators, one operating at a frequency ω and the other at a frequency near a multiple of the first one, nω where n is an integer greater than 1, is investigated. It is observed that such systems tend to exhibit a consistent pattern of master-slave relationship. This phenomenon is geometrically elucidated by considering the interaction dynamics: while the multiple bursts of the faster oscillator tend to cancel each other out upon impacting the slower oscillator, the single burst of the slower oscillator on the faster has a secular effect. We propose that this synchronization pattern, arising from perturbative impacts, is prevalent across various physical systems. Further experimental validation of these findings could contribute to a deeper understanding of synchronization phenomena and their implications in natural and technological domains. 2020 Mathematics Subject Classification. Primary 34D06, Secondary 37E30.