  We study non oscillating bifurcations of non homogeneous steady states of the
Vlasov equation, a situation occurring in galactic models, or for
Bernstein-Greene-Kruskal modes in plasma physics. We show that resonances are
strongly suppressed, leading to very different phenomena with respect to the
homogeneous case. Through an unstable manifold expansion, we show that the
dynamics is very sensitive to the initial perturbation: the instability may
saturate at small amplitude -generalizing the "trapping scaling" of plasma
physics- or may grow to produce a large scale modification of the system. These
analytical findings are illustrated and extended by direct numerical
simulations with a cosine interaction potential.
