  We study the phase space asymptotics of the so called Veneziano ghost dark
energy models. Models where the ghost field's energy density: i)
$\rho_{ghost}\propto H$, and ii) $\rho_{ghost}\propto H+H^2$, are investigated.
Both, cases with and without additional non-gravitational interaction between
cold dark matter and ghost dark energy, are subject to scrutiny. We pay special
attention to the choice of phase space variables leading to bounded and compact
phase space so that no critical point of physical interest is missing. A rich
asymptotic structure is revealed: depending on the kind of non-minimal coupling
critical points associated with radiation dominance, matter dominance, cold
dark matter/ghost dark energy scaling, and ghost dark energy dominance, are
found. Past and future attractors, as well as saddle equilibrium points, are
identified in the corresponding phase spaces.
