  We investigate the dynamical stability of a fully-coupled system of $N$
inertial rotators, the so-called Hamiltonian Mean Field model. In the limit $N
\to \infty$, and after proper scaling of the interactions, the $\mu$-space
dynamics is governed by a Vlasov equation. We apply a nonlinear stability test
to (i) a selected set of spatially homogeneous solutions of Vlasov equation,
qualitatively similar to those observed in the quasi-stationary states arising
from fully magnetized initial conditions, and (ii) numerical coarse-grained
distributions of the finite-$N$ dynamics. Our results are consistent with
previous numerical evidence of the disappearance of the homogenous
quasi-stationary family below a certain energy.
