  We consider Friedmann-Lema\^{\i}tre-Robertson-Walker flat cosmological models
in the framework of general Jordan frame scalar-tensor theories of gravity with
arbitrary coupling function and potential. For the era when the cosmological
energy density of the scalar potential dominates over the energy density of
ordinary matter, we use a nonlinear approximation of the decoupled scalar field
equation for the regime close to the so-called limit of general relativity
where the local weak field constraints are satisfied. We give the solutions in
cosmological time with a particular attention to the classes of models
asymptotically approaching general relativity. The latter can be subsumed under
two types: (i) exponential convergence, and (ii) damped oscillations around
general relativity. As an illustration we present an example of oscillating
dark energy.
