A restricted version of the Hilbert's 16th problem for quadratic vector fields

TL;DR

This paper provides an upper estimate for the number of limit cycles in quadratic vector fields that are distant from centers and singular fields, considering their distance from singular points and infinity.

Contribution

It introduces a new upper bound for limit cycles in quadratic vector fields based on their distance from singularities and centers, addressing a restricted version of Hilbert's 16th problem.

Findings

Upper estimate for limit cycles in quadratic vector fields

Limit cycles are characterized by their distance from singular points and infinity

Results apply to fields distant from centers and singular quadratic fields

Abstract

The restricted version of the Hilbert 16th problem for quadratic vector fields requires an upper estimate of the number of limit cycles through a vector parameter that characterizes the vector fields considered and the limit cycles to be counted. In this paper we give an upper estimate of the number of limit cycles of quadratic vector fields $"\sigma $--distant from centers and $\ka $-distant from singular quadratic vector fields" provided that the limit cycles are $"\delta $--distant from singular points and infinity".