  We perform the dynamical system analysis of a cosmological model in the
energy-momentum squared gravity (EMSG) of the form
$f(T_{\mu\nu}T^{\mu\nu})=\alpha\ln({\lambda}T_{\mu\nu} T^{\mu\nu})$, which is
known as energy-momentum log gravity (EMLG). In particular, we show that the
analytical cosmological solution of EMLG presented by Akarsu {\it et al.} (Eur.
Phys. J. C 79:846, 2019) is a future attractor. It includes new terms in the
right-hand side of the Einstein field equations, which yield constant inertial
mass density and provide a dynamical dark energy with a density passing below
zero at large redshifts, accommodating a mechanism for screening $\Lambda$ in
the past, suggested for alleviating some cosmological tensions. We show that
the model gives rise to an entire class of new stable late-time solutions with
$H\rightarrow\sqrt{(\Lambda+2\alpha)/3}$ as $a\rightarrow\infty$, where the new
term is due to the constant effective inertial mass density that arises from
EMLG contribution of dust, whereas $H\rightarrow\sqrt{\Lambda/3}$ as
$a\rightarrow\infty$ in the $\Lambda$CDM model. We also show existence of new
interesting features and trajectories that are absent in the $\Lambda$CDM
model.
