EXPLICIT DERIVATION OF RETURN MAPS FOR PLANAR VECTOR FIELDS WITH NILPOTENT LINEAR PART WITH A CENTER OR A FOCUS

Abstract

For planar vector fields with nilpotent linear part with a center or a focus it is shown that, remarkably, there exist substitutions after which the equations become regular perturbations of Hamiltonian systems. The return map is then obtained as a convergent series by iterating a contractive map, yielding formulas suitable for numerical calculations and estimates for the errors.