  We study the autonomous system for a scalar-tensor model of dark energy with
Gauss-Bonnet and non-minimal couplings. 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 the exponential functions the existence of stable
quintessence, phantom or de Sitter solutions, allows an asymptotic behavior
where the effective Newtonian coupling becomes constant. The phantom solutions
could be realized without appealing to ghost degrees of freedom. Transient
inflationary and radiation dominated phases can also be described.
