  We study the behavior of perturbations of small nonlinear Dirac standing
waves. We assume that the linear Dirac operator of reference $H=D_m+V$ has only
two double eigenvalues and that degeneracies are due to a symmetry of $H$
(theorem of Kramers). In this case, we can build a small 4-dimensional manifold
of stationary solutions tangent to the first eigenspace of $H$. Then we assume
that a resonance condition holds and we build a center manifold of real
codimension 8 around each stationary solution. Inside this center manifold any
$H^{s}$ perturbation of stationary solutions, with $s>2$, stabilizes towards a
standing wave. We also build center-stable and center-unstable manifolds each
one of real codimension 4. Inside each of these manifolds, we obtain
stabilization towards the center manifold in one direction of time, while in
the other, we have instability. Eventually, outside all these manifolds, we
have instability in the two directions of time. For localized perturbations
inside the center manifold, we obtain a nonlinear scattering result.
