  The dynamics of red blood cells (RBCs) in oscillatory shear flow was studied
using differential equations of three variables: a shape parameter, the
inclination angle $\theta$, and phase angle $\phi$ of the membrane rotation. In
steady shear flow, three types of dynamics occur depending on the shear rate
and viscosity ratio. i) tank-treading (TT): $\phi$ rotates while the shape and
$\theta$ oscillate. ii) tumbling (TB): $\theta$ rotates while the shape and
$\phi$ oscillate. iii) intermediate motion: both $\phi$ and $\theta$ rotate
synchronously or intermittently. In oscillatory shear flow, RBCs show various
dynamics based on these three motions. For a low shear frequency with zero mean
shear rate, a limit-cycle oscillation occurs, based on the TT or TB rotation at
a high or low shear amplitude, respectively. This TT-based oscillation well
explains recent experiments. In the middle shear amplitude, RBCs show an
intermittent or synchronized oscillation. As shear frequency increases, the
vesicle oscillation becomes delayed with respect to the shear oscillation. At a
high frequency, multiple limit-cycle oscillations coexist. For a high mean
shear rate with small shear oscillation, the shape and $\theta$ oscillate in
the TT motion but only one attractor exists even at high shear frequencies. The
measurement of these oscillatory modes is a promising tool for quantifying the
viscoelasticity of RBCs and synthetic capsules.
