  We study the autonomous system for a scalar-tensor model of dark energy with
non-minimal coupling to curvature and non-minimal kinetic coupling to the
Einstein tensor. The critical points describe important stable asymptotic
scenarios including quintessence, phantom and de Sitter attractor solutions.
Two functional forms for the coupling functions and the scalar potential were
considered: power-law and exponential functions of the scalar field. For
power-law couplings, the restrictions on stable quintessence and phantom
solutions lead to asymptotic freedom regime for the gravitational interaction.
The model with dimensionless kinetic coupling constant gives stable de Sitter
solutions. For the exponential functions the stable quintessence, phantom or de
Sitter solutions, allow asymptotic behaviors where the effective Newtonian
coupling can reach either the asymptotic freedom regime or constant value. The
phantom solutions could be realized without appealing to ghost degrees of
freedom. Transient inflationary and radiation dominated phases can also be
described.
