Uniqueness of Limit Cycles for Quadratic Vector Fields

TL;DR

This paper investigates the maximum number of limit cycles around a critical point in quadratic planar vector fields, using algebraic geometry tools to analyze parameter spaces and classical criteria for Abel equations.

Contribution

It introduces a novel approach combining classical methods with computational algebraic geometry to study limit cycle uniqueness in quadratic vector fields.

Findings

Identifies parameter conditions for the existence of limit cycles.

Provides criteria for limit cycle uniqueness in quadratic systems.

Uses computational algebraic geometry to analyze semi-varieties.

Abstract

This article deals with the study of the number of limit cycles surrounding a critical point of a quadratic planar vector field, which, in normal form, can be written as $x'= a_1 x-y-a_3x^2+(2 a_2+a_5)xy + a_6 y^2$, $y'= x+a_1 y + a_2x^2+(2 a_3+a_4)xy -a_2y^2$. In particular, we study the semi-varieties defined in terms of the parameters $a_1,a_2,\ldots,a_6$ where some classical criteria for the associated Abel equation apply. The proofs will combine classical ideas with tools from computational algebraic geometry.