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We study the evolution of homogeneous and isotropic, flat cosmological modelswithin the general scalar-tensor theory of gravity with arbitrary couplingfunction and potential. After introducing the limit of general relativity wedescribe the details of the phase space geometry. Using the methods ofdynamical systems for the decoupled equation of the Jordan frame scalar fieldwe find the fixed points of flows in two cases: potential domination and matterdomination. We present the conditions on the mathematical form of the couplingfunction and potential which determine the nature of the fixed points(attractor or other). There are two types of fixed points, both arecharacterized by cosmological evolution mimicking general relativity, but onlyone of the types is compatible with the Solar System PPN constraints. The phasespace structure should also carry over to the Einstein frame as long as thetransformation between the frames is regular which however is not the case forthe latter (PPN compatible) fixed point.

Scalar-tensor cosmologies: fixed points of the Jordan frame scalar field