Time flies and memory lingers: altering the stability of synchronized states in hydrodynamically coupled oscillators.

Minimal models of coupled oscillators have been used to shed light on the emergence of synchronized beating of nearby flagella or metachronal waves of arrays of cilia.  Here we focus on oscillators whose motion is driven by two alternating traps, one active and one not.  The activation state of the trap is governed by a geometric switch that depends upon the oscillator's distance to it.  In Stokes flow, it has been shown both analytically and experimentally that two in-tandem spherical colloids, each with their its own pair of traps (all co linear), will tend towards anti-phase oscillations, with in-phase oscillations unstable.    Here we show that adding some memory to the system can stabilize the in-phase oscillation. This memory may come from fluid inertia (unsteady Stokes or Navier-Stokes), weak elastic coupling of oscillators, or an altered switch that checks for a stall-velocity in addition to geometric information.   This stabilization of in-phase synchrony is robust – we have considered rigid spherical oscillators  in 3D Stokes and 3D unsteady Stokes, elastic filament oscillators in 3D Stokes, and flexible droplets in 2D Stokes and 2D Navier Stokes – and all demonstrate this phenomenon