  In this paper we deal with analytic nonautonomous vector fields with a
periodic time-dependancy, that we study near an equilibrium point. In a first
part, we assume that the linearized system is split in two invariant subspaces
E0 and E1. Under light diophantine conditions on the eigenvalues of the linear
part, we prove that there is a polynomial change of coordinates in E1 allowing
to eliminate up to a finite polynomial order all terms depending only on the
coordinate u0 of E0 in the E1 component of the vector field. We moreover show
that, optimizing the choice of the degree of the polynomial change of
coordinates, we get an exponentially small remainder. In the second part, we
prove a normal form theorem with exponentially small remainder. Similar
theorems have been proved before in the autonomous case : this paper
generalizes those results to the nonautonomous periodic case.
