The Return Map for a Planar Vector Field with Nilpotent Linear Part: A Direct and Explicit Derivation

Rodica D Costin

doi:10.1155/2009/590856

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.

Highlights

The study of planar vector fields has been the subject of intense research, in connection to Hilbert’s 16th Problem
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

Summary

Introduction

The study of planar vector fields has been the subject of intense research, in connection to Hilbert’s 16th Problem. The Poincarefirst return maps have been studied in view of their relevance for establishing the existence of closed orbits, and due to their large number of applications see e.g., 3 and references therein , and in connection to o-minimality 4. A fundamental result concerns the asymptotic form of return maps states that if the singular points of a C∞ vector field are algebraically isolated, there exists a semitransversal arc such that the return map admits an asymptotic expansion is positive powers of x and logs with the first term linear , or has its principal part a finite composition of powers and exponentials 6, 7. In the case when the linear part of the vector field has nonzero eigenvalues there are important results containing the return map 8–14. The main goal is to establish techniques that allow to deduce the return map as a suitable series which can be calculated algorithmically and can be used in numerical calculations

Main Result

Normalization

Solutions in Other Quadrants

Matching at the Negative y-Axis

The First Return Map