  We study the evolution of homogeneous and isotropic, flat cosmological models
within the general scalar-tensor theory of gravity with arbitrary coupling
function and potential and scrutinize its limit to general relativity. Using
the methods of dynamical systems for the decoupled equation of the Jordan frame
scalar field we find the fixed points of flows in two cases: potential
domination and matter domination. We present the conditions on the mathematical
form of the coupling function and potential which determine the nature of the
fixed points (attractor or other). There are two types of fixed points, both
are characterized by cosmological evolution mimicking general relativity, but
only one of the types is compatible with the Solar System PPN constraints.
