  We study a single, freely--floating, inextensible, elastic filament in a
linear shear flow: $\mathbf{U}_{0}(x,y) = \dot{\gamma} y \hat{x}$. In our
model: the elastic energy depends only on bending; the rate-of-strain,
$\dot{\gamma} = S \sin(\omega t)$ is a periodic function of time, $t$; and the
interaction between the filament and the flow is approximated by a local
isotropic drag force. Based on the shape of the filament we find five different
dynamical phases: straight, buckled, periodic (with period two, period three,
period four, etc), chaotic, and one with chaotic transients. In the chaotic
phase, we show that the iterative map for the angle, which the end-to-end
vector of the filament makes with the tangent its one end, has period three
solutions; hence it is chaotic. Furthermore, in the chaotic phase the flow is
an efficient mixer.
