Kuramoto-like synchronization for droplets bouncing at a distance

Couder and Fort (PRL‘06) discovered that a fluid droplet bouncing on the surface of a vertically vibrating bath, forms a wave-particle system referred to as a hydrodynamic pilot-wave system. Such an object was only imagined in the quantum realm. Much research has been done since this discovery. Many mathematical problems arose, as will be outlined, where uncertainty related issues are recurrent. The main result of this talk regards a recent publication (N., Chaos, Sept.’18) where we show that two oscillating droplets, confined to separate wells, exhibit correlated features even when separated by a large distance. The particles’ phase space dynamics is described in a holistic fashion and may not be decomposed into separate subsystems. We detect “coherence” when the bouncing droplets behave as nonlinearly-coupled oscillators which synchronize spontaneously, as in the celebrated Kuramoto model for phase oscillators. The droplet coupling is dynamic and implicit, being wave-mediated as opposed to the Kuramoto model where phase-coupling is explicit and pre-defined.  We also discover a regime where “coherence” is defined in a statistical manner.