  This paper is concerned with string cosmology and the dynamics of multiple
scalar fields in potentials that can become negative, and their features as
(Early) Dark Energy models. Our point of departure is the "String Axiverse", a
scenario that motivates the existence of cosmologically light axion fields as a
generic consequence of string theory. We couple such an axion to its
corresponding modulus. We give a detailed presentation of the rich cosmology of
such a model, ranging from the setting of initial conditions on the fields
during inflation, to the asymptotic future. We present some simplifying
assumptions based on the fixing of the axion decay constant $f_a$, and on the
effective field theory when the modulus trajectory is adiabatic, and find the
conditions under which these assumptions break down. As a by-product of our
analysis, we find that relaxing the assumption of fixed $f_a$ leads to the
appearance of a new meta-stable de-Sitter region for the modulus without the
need for uplifting by an additional constant. A dynamical systems analysis
reveals the existence of many fixed point attractors, repellers and saddle
points, which we analyse in detail. We also provide geometric interpretations
of the phase space. The fixed points can be used to bound the couplings in the
model. A systematic scan of certain regions of parameter space reveals that the
future evolution of the universe in this model can be rich, containing multiple
epochs of accelerated expansion.
