  The dynamics of fluid vesicles in oscillatory shear flow was studied using
differential equations of two variables: the Taylor deformation parameter and
inclination angle $\theta$. In a steady shear flow with a low viscosity
$\eta_{\rm {in}}$ of internal fluid, the vesicles exhibit steady tank-treading
motion with a constant inclination angle $\theta_0$. In the oscillatory flow
with a low shear frequency, $\theta$ oscillates between $\pm \theta_0$ or
around $\theta_0$ for zero or finite mean shear rate $\dot\gamma_{\rm m}$,
respectively. As shear frequency $f_{\gamma}$ increases, the vesicle
oscillation becomes delayed with respect to the shear oscillation, and the
oscillation amplitude decreases. At high $f_{\gamma}$ with $\dot\gamma_{\rm
m}=0$, another limit-cycle oscillation between $\theta_0-\pi$ and $-\theta_0$
is found to appear. In the steady flow, $\theta$ periodically rotates
(tumbling) at high $\eta_{\rm {in}}$, and $\theta$ and the vesicle shape
oscillate (swinging) at middle $\eta_{\rm {in}}$ and high shear rate. In the
oscillatory flow, the coexistence of two or more limit-cycle oscillations can
occur for low $f_{\gamma}$ in these phases. For the vesicle with a fixed shape,
the angle $\theta$ rotates back to the original position after an oscillation
period. However, it is found that a preferred angle can be induced by small
thermal fluctuations.
