  We first make more precise a recent "Hamiltonian" reformulation of the
Hohm-Zwiebach approach to the tree-level, $O(d,d)$-invariant string cosmology
equations at all orders in the $\alpha'$ expansion, and recall how it allows to
give a simple characterization of a large class of cosmological scenarios
connecting, through a non-singular bounce, two duality-related perturbative
solutions at early and late times. We then discuss the effects of adding to the
action a non-perturbative, $O(d,d)$-breaking, dilaton potential $V(\phi)$. The
resulting cosmological solutions, assumed to approach at early times the
perturbative string vacuum (with vanishing curvature and string coupling), can
stabilize the dilaton at late times and simultaneously approach either a
matter-dominated FLRW cosmology or a de-Sitter-like inflationary phase,
depending on initial conditions and on the properties of $V(\phi)$ at
moderate-coupling. We also identify a general mechanism for generating
isotropic late-time attractors from a large basin of anisotropic initial
conditions.
