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

TL;DR

This paper derives an explicit, convergent power series for the return map near a focus in a planar vector field with a nilpotent linear part, using a direct iterative approach involving Abelian and iterated integrals.

Contribution

It provides a novel direct method to explicitly compute the return map as a convergent power series for such vector fields.

Findings

Return map expressed as a convergent power series

First nontrivial coefficient is an Abelian integral

Subsequent coefficients are explicitly given as iterated integrals

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.