  In experiments, an individual chemically active liquid crystal (LC) droplet
submerged in the bulk of a surfactant solution may self-propel along a
straight, helical, or random trajectory. In this paper, we develop a minimal
model capturing all three types of self-propulsion trajectories of a drop in
the case of a nematic LC with homeotropic anchoring at LC-fluid interface. We
emulate the director field within the drop by a single preferred polarization
vector that is subject of two reorientation mechanisms, namely, the internal
flow-induced displacement of the hedgehog defect and the droplet's rotation.
Within this reduced-order model, the coupling between the nematic ordering of
the drop and the surfactant transport is represented by variations of the
droplet's interfacial properties with nematic polarization. Our analysis
reveals that a novel mode of orientational instability emerges from the
competition of the two reorientation mechanisms and is characterized by a
spontaneous rotation of the self-propelling drop responsible for helical
self-propulsion trajectories. In turn, we also show that random trajectories in
isotropic and nematic drops alike stem from the advection-driven transition to
chaos. The succession of the different propulsion modes is consistent with
experimentally-reported transitions in the shape of droplet trajectories as the
drop size is varied.
