  In this paper we perform a thorough dynamical systems analysis of the cubic
galileon model non-minimally coupled to the dark matter. Three well-known
classes of interacting models are considered where the energy exchange between
the dark components is a function of the dark matter density and of the dark
energy density: $Q_1=3\alpha H\rho_m$, $Q_2=3\beta\rho_m\dot{\phi}$ and $Q_3=3
\epsilon H \rho_\phi$, respectively. We are able to show the global asymptotic
dynamics of the model for the exponential potential in a homogeneous and
isotropic background. The cosmological implications of the proposed scenarios
are explored and it is found that, in addition to the appearance of new
equilibrium configurations that do not appear neither in the non-interacting
cubic galileon model nor in the interacting quintessence model, there is a
significant impact of the non-minimal coupling through modification of the
stability properties of the critical points. The resulting cosmological
scenario provides a bigbang origin of the cosmic expansion, an early transient
stage of inflationary expansion, as well as matter-scaling late time stable
state of the universe, among other solutions of lesser cosmological interest.
This work extends previous studies of coupled dark energy to a broader class of
gravitational theories.
