Shear flows with curved streamlines lead to viscoelastic instabilities which occur even at zero Reynolds number. The viscoelastic Taylor problem, in particular, has attracted much attention in the past few years. The instabilities which occur have much in common with the Newtonian Taylor problem for counterrotating cylinders, in particular, there are instabilities with oscillatory onset, and there are specific parameter values where several modes become unstable simultaneously. In this paper, we use the center manifold reduction to derive amplitude equations governing these mode interactions, and we numerically compute the coefficients in these amplitude equations. For the particular cases analyzed, we find that no stable solutions of small amplitude exist, and a finite transition takes place.