For ekpyrotic/cyclic scenarios we show that one can make predictions for density perturbations which are independent of the details of the bouncing phase. This is possible, as in inflationary cosmology, thanks to the existence of a dynamical attractor, which makes local observables equal to the unperturbed solution up to exponentially small terms. Assuming that physics at the bounce is not extremely sensitive to these corrections, perturbations can be evolved even at non-linear level. The resulting spectrum is not scale invariant and thus incompatible with experimental data.