The first return map for planar vector fields with nilpotent linear part with a center or a focus

TL;DR

This paper derives a convergent power series for the return map of planar vector fields with nilpotent linear parts, explicitly calculating coefficients using Abelian and iterated integrals, advancing understanding of local dynamics near centers and foci.

Contribution

It provides a new method to explicitly compute the return map for nilpotent linear systems with centers or foci, including iterative calculation of coefficients.

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

The return map for planar vector fields with nilpotent linear part (having a center or a focus and under an assumption generically satisfied) is found as a convergent power series 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 built with algebraic functions.