Résumé : 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, this degeneracy is due to a theorem of Kramers. In this case, we can build a small $8$-dimensional manifold of stationary solutions tangent to the first eigenspace of $H$. Then we assume that a resonance condition holds for the first eigenvalue. We build a center manifold of real codimension $16$ 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 $8$. Inside each manifold, 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.