International Journal of Mathematics and Mathematical Sciences
Volume 2009 (2009), Article ID 590856, 11 pages
doi:10.1155/2009/590856

The return map for a planar vector field with nilpotent linear part: a direct and explicit derivation

Abstract

Using a direct approach the return map near a focus of a planar vector field with nilpotent linear part is found as a convergent power series which is a perturbation of the identity and whose terms can be calculated iteratively. The first nontrivial coefficient is the value of an Abelian integral, and the following ones are explicitly given as iterated integrals.