  Polymer solutions can develop chaotic flows, even at low inertia. This purely
elastic turbulence is well studied, but little is known about the transition to
chaos. In 2D channel flow and parallel shear flow, traveling wave solutions
involving coherent structures are present for sufficiently large fluid
elasticity. We numerically study 2D periodic parallel shear flow in
viscoelastic fluids and show that these traveling waves become oscillatory and
undergo a series of period-doubling bifurcations en-route to chaos.
