Upper bounds for the number of zeroes for some Abelian integrals

TL;DR

This paper establishes upper bounds on the number of limit cycles bifurcating from a specific class of planar vector fields with line-critical sets, using Abelian integrals and zeroes bounding techniques, improving previous results for certain cases.

Contribution

It introduces a new approach to bound the zeroes of Abelian integrals for vector fields with line-critical sets, providing improved bounds for cases with up to four lines.

Findings

Derived explicit bounds for the number of limit cycles.

Reproduced or improved previous results for up to four lines.

Developed a new method for bounding zeroes of certain real functions.

Abstract

Consider the vector field $x'= -yG(x, y), y'=xG(x, y),$ where the set of critical points $\{G(x, y) = 0\}$ is formed by $K$ straight lines, not passing through the origin and parallel to one or two orthogonal directions. We perturb it with a general polynomial perturbation of degree $n$ and study which is the maximum number of limit cycles that can bifurcate from the period annulus of the origin in terms of $K$ and $n.$ Our approach is based on the explicit computation of the Abelian integral that controls the bifurcation and in a new result for bounding the number of zeroes of a certain family of real functions. When we apply our results for $K\le4$ we recover or improve some results obtained in several previous works.