Invariant manifolds of parabolic fixed points (II). Approximations by sums of homogeneous functions

We study the computation of local approximations of invariant manifolds of parabolic fixed points and parabolic periodic orbits of periodic vector fields. If the dimension of these manifolds is two or greater, in general, it is not possible to obtain polynomial approximations. Here we develop an algorithm to obtain them as sums of homogeneous functions by solving suitable cohomological equations. We deal with both the differentiable and analytic cases. We also study the dependence on parameters. In the companion paper [BFM] these approximations are used to obtain the existence of true invariant manifolds close by. Examples are provided.