  We investigate the evolution of the homogeneous and isotropic universe within
the framework of the effective string gravity with string-loop modifications of
dilaton couplings. In the case of barotropic perfect fluid as a
nongravitational source the set of cosmological equations is presented in the
form of the third-order autonomous dynamical system. The cases are considered
when this system is integrable in terms of integrals depending on dilaton
coupling functions. Without specifying these functions we describe generic
evolution of the universe, using dynamical systems methods. The critical points
and their stability are found for all regions of the parameters. The
qualitatively different phase space diagrams are presented for the
spatially-flat, closed and open universes. The case of the tree-level models is
considered separately. The issue of dilaton stabilization is discussed within
the framework of Damour-Polyakov mechanism.
