  In this paper, we have investigated the phase space analysis in modified
Galileon cosmology, where the Galileon term is considered a coupled scalar
field, $F(\phi)$. We focus on the exponential type function of $F(\phi)$ and
the three well-motivated potential functions $V(\phi)$. We obtain the critical
points of the autonomous system, along with their stability conditions and
cosmological properties. The critical points of the autonomous system describe
different phases of the Universe. In the results of our analysis, we have found
the scaling solution for critical points, which determine different
evolutionary eras for the Universe. The dark-energy-dominated critical points
show stable behavior and indicate the Universe's late-time cosmic acceleration
phase. Further, the results are examined with the cosmological data sets of the
Hubble rate $H(z)$ and the Supernovae Ia.
