  The dynamics of a two dimensional autophoretic disk is quantified as a
minimal model for the chaotic trajectories undertaken by active droplets. Via
direct numerical simulations, we show that the mean-square displacement of the
disk in a quiescent fluid is linear at long times. Surprisingly, however, this
apparently diffusive behavior is non-Brownian, owing to strong
cross-correlations in the displacement tensor. The effect of a shear flow field
on the chaotic motion of a 2D autophoretic disk is examined. Here, the
stresslet on the disk is chaotic for weak shear flows; a dilute suspension of
such disks would exhibit a chaotic shear rheology. This chaotic rheology is
quenched first into a periodic state and ultimately a steady state as the flow
strength is increased.
