  The motion of a deformable active particle in linear shear flow is explored
theoretically. Based on symmetry considerations, in two spatial dimensions, we
propose coupled nonlinear dynamical equations for the particle position,
velocity, deformation, and rotation. In our model, both, passive rotations
induced by the shear flow as well as active spinning motions, are taken into
account. Our equations reduce to known models in the two limits of vanishing
shear flow and vanishing particle deformability. For varied shear rate and
particle propulsion speed, we solve the equations numerically and obtain a
manifold of different dynamical modes including active straight motion,
periodic motions, motions on undulated cycloids, winding motions, as well as
quasi-periodic and chaotic motions induced at high shear rates. The types of
motion are distinguished by different characteristics in the real-space
trajectories and in the dynamical behavior of the particle orientation and its
deformation. Our predictions can be verified in experiments on self-propelled
droplets exposed to a linear shear flow.
