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.

The study of planar vector fields has been the subject of intense research, particularly in connection to Hilbert's 16th Problem. Significant progress has been made in the geometric theory of these fields, as well as in bifurcation theory, normal forms, foliations, and the study of Abelian integrals [

The Poincaré first return maps have been studied in view of their relevance for establishing the existence of closed orbits, and also due to their large number of applications (see e.g., [

The monodromy problem (determining when the singularity is a center or a focus) was solved by Andreev [

A fundamental result concerns the asymptotic form of return maps states that if the singular points of a

In the case when the linear part of the vector field has nonzero eigenvalues there are important results containing the return map [

The present paper studies an example of a field with nilpotent linear part, near a focus. 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.

The paper studies the return map for the system

The main result is the following.

Let

The solution of (

The coefficients

The proof of Proposition

It is convenient to normalize the variables

While the initial condition

System (

Let

Note the following Lyapunov function for (

Since the set

Consider the solution of (

Since

Denote by

Similar arguments show that the solution

Solutions

Note that following the path

Lemma

Substituting

There exists

Let

We will use the results of Lemma

Local analysis shows that solutions of (

Denote

Note that if

Equation (

Lemma

These exists

Moreover,

Let

Let

We have

Moreover, the operator

Therefore the operator

To obtain the power series (

Substitution of (

From (

In particular, we have (

The following gathers the conclusions of the present section.

There exists

Moreover, this solution has the form

We found an expression for the solution

Let

Let

Then:

the function

The function

The function

Let

The following lemma finds

Let

There exists a unique

Let

We have

We have

The expansion of

Let

Let

The following lemma finds

Let

There exists a unique

We need to find

Note that the function

Let

Combining (

To obtain the point

The first coefficient of the return map (

With the notation

For the present system we have

The author is grateful to Chris Miller for suggesting the problem.