  Individual chemically active drops suspended in a surfactant solution were
observed to self-propel spontaneously with straight, helical, or chaotic
trajectories. To elucidate how these drops can exhibit such strikingly
different dynamics and `decide' what to do, we propose a minimal axisymmetric
model of a spherical active drop, and show that simple and linear interface
properties can lead to both steady self-propulsion of the droplet as well as
chaotic behavior. The model includes two different mobility mechanisms, namely,
diffusiophoresis and the Marangoni effect, that convert self-generated
gradients of surfactant concentration into the flow at the droplet surface. In
turn, surface-driven flow initiates surfactant advection that is the only
nonlinear mechanism and, thus, the only source of dynamical complexity in our
model. Numerical investigation of the fully-coupled hydrodynamic and advection
diffusion problems reveals that strong advection (e.g., large droplet size) may
destabilize a steadily self-propelling drop; once destabilized, the droplet
spontaneously stops and a symmetric extensile flow emerges. If advection is
strengthened even further in comparison with molecular diffusion, the droplet
may perform chaotic oscillations. Our results indicate that the thresholds of
these instabilities depend heavily on the balance between diffusiophoresis and
the Marangoni effect. Using linear stability analysis, we demonstrate that
diffusiophoresis promotes the onset of high-order modes of monotonic
instability of the motionless drop. We argue that diffusiophoresis has a
similar effect on the instabilities of a moving drop.
