  We consider the potential dominated era of
Friedmann-Lemaitre-Robertson-Walker flat cosmological models in the framework
of general Jordan frame scalar-tensor theories of gravity with arbitrary
coupling functions, and focus upon the phase space of the scalar field. To
study the regime suggested by the local weak field tests (i.e. close to the
so-called limit of general relativity) we propose a nonlinear approximation
scheme, solve for the phase trajectories, and provide a complete classification
of possible phase portraits. We argue that the topology of trajectories in the
nonlinear approximation is representative of those of the full system, and thus
can tell for which scalar-tensor models general relativity functions as an
attractor.
