  We study the non-linear stability of fixed-point solutions to the
$\alpha'$-exact equations from O$(d,d)$ invariant cosmology, with and without
matter perturbations. Previous non-linear analysis in the literature is
revisited, and its compatibility with known linear perturbation results is
shown. Some formal aspects of cosmological perturbations in duality invariant
cosmology are discussed, and we show the existence of time-reparameterization
invariant variables for perturbations.
