Phase-locking of hybrid oscillators

The synchronization and phase-locking behavior of oscillators with smooth dynamics is well captured by continuous phase-models \`a la Kuramoto. However, these models do not apply seamlessly for hybrid oscillators, where discrete events occur along their otherwise smooth dynamics. Consequently, the synchronization and phase-locking mechanisms of hybrid oscillators remain overlooked. Here, we combine experiments, theory, and simulations, to investigate and rationalize the coherent motion of pairs of hybrid oscillator. Using contact-charge electrophoretic (CCEP) oscillators as an experimental realization, we show that in-phase oscillations can occur, despite oscillators repelling each other. We rationalize this behavior by introducing a discrete-time framework that explicitly accounts for discrete events and elucidates the origin of phase-locking. Our model highlights the roles played by inertia and the decay of interaction strength along the oscillation cycle in promoting phase-locking of CCEP oscillators. More generally, it provides a phase-locking criterion that holds for a broader class of coupled hybrid oscillators, and reveals that various mechanisms can lead to their coherence.