  The generation of droplets at low Reynolds numbers is driven by non-linear
dynamics that give rise to complex patterns concerning both the
droplet-to-droplet spacing and the individual droplet sizes. Here we
demonstrate an experimental system in which a time-varying energy landscape
provides a periodic magnetic force that generates an array of droplets from an
immiscible mixture of ferrofluid and silicone oil. The resulting droplet
patterns are periodic, owing to the nature of the magnetic force, yet the
droplet spacing and size can vary greatly by tuning a single bias pressure
applied on the ferrofluid phase; for a given cycle period of the magnetic
force, droplets can be generated either at integer multiples (1, 2, etc.), or
at rational fractions (3/2, 5/3, 5/2, etc.) of this period with mono- or
multidisperse droplet sizes. We develop a discrete-time dynamical systems model
not only to reproduce the phenotypes of the observed patterns but also provide
a framework for understanding systems driven by such periodic energy
landscapes.
