  We apply the Nyquist method to the Hamiltonian Mean Field (HMF) model in
order to settle the linear dynamical stability of a spatially homogeneous
distribution function with respect to the Vlasov equation. We consider the case
of Maxwell (isothermal) and Tsallis (polytropic) distributions and show that
the system is stable above a critical kinetic temperature T_c and unstable
below it. Then, we consider a symmetric double-humped distribution, made of the
superposition of two decentered Maxwellians, and show the existence of a
re-entrant phase in the stability diagram. When we consider an asymmetric
double-humped distribution, the re-entrant phase disappears above a critical
value of the asymmetry factor Delta>1.09. We also consider the HMF model with a
repulsive interaction. In that case, single-humped distributions are always
stable. For asymmetric double-humped distributions, there is a re-entrant phase
for 1<Delta<25.6, a double re-entrant phase for 25.6<Delta<43.9 and no
re-entrant phase for Delta>43.9. Finally, we extend our results to arbitrary
potentials of interaction and mention the connexion between the HMF model,
Coulombian plasmas and gravitational systems. We discuss the relation between
linear dynamical stability and formal nonlinear dynamical stability and show
their equivalence for spatially homogeneous distributions. We also provide a
criterion of dynamical stability for inhomogeneous systems.
