  In this work we study the phase-space analysis of generalized Rastall gravity
(GRG) which has recently been introduced as a modification to the original
version of Rastall gravity (RG). In GRG, the coupling parameter assumes a
dynamical feature and may play the role of dark energy (DE) which is
responsible for the present accelerating expansion of the Universe. Our
investigation shows that such a modification of General Relativity (GR) admits
a stable critical point corresponding to the late time accelerated expansion of
the Universe. Assuming both dark matter (DM) and an ultra-relativistic perfect
fluid (radiation) as the cosmic ingredients, we find that the underlying model
presents a viable sequence of cosmic evolution, beginning from radiation
dominated era passing then through DM domination and finally reaches the late
time DE dominated era. Moreover, taking into account the contribution due to
the spatial curvature within the total energy density leads to a growing mode
for it in such a way that its present values are consistent with those reported
in recent observations. We also present a numerical simulation of the model
which particularizes our dynamical system representation. These studies
demonstrate the ability of GRG to describe the present accelerated expansion of
the Universe.
