Algebraic limit cycles for quadratic polynomial differential systems

,Jaume Llibre,Claudia Valls

doi:10.3934/dcdsb.2018070

Abstract

We prove that for a quadratic polynomial differential system having three pairs of diametrally opposite equilibrium points at infinity that are positively rationally independent, has at most one algebraic limit cycle. Our result provides a partial positive answer to the following conjecture: Quadratic polynomial differential systems have at most one algebraic limit cycle.

Highlights

Let R[x, y] be the ring of all real polynomials in the variables x and y
For a quadratic polynomial differential system with three pairs of equilibrium points at infinity having an invariant algebraic curve with at most one pair of diametrally opposite singular points at infinity, being the endpoints of the y-axis has at most one algebraic limit cycle
In view of Theorem 13 in order to prove Theorem 3 we can restrict ourselves to the case in which the quadratic polynomial differential system has three pairs of equilibrium points at infinity

Summary

Introduction

Let R[x, y] be the ring of all real polynomials in the variables x and y. For other related papers concering this problem, see [18] and [23] It remains to know if when the invariant algebraic curves of a quadratic polynomial differential systems do not satisfy these generic conditions have at most one algebraic limit cycle. For a quadratic polynomial differential system with three pairs of equilibrium points at infinity having an invariant algebraic curve with at most no singular points at infinity, has at most one algebraic limit cycle. For a quadratic polynomial differential system with three pairs of equilibrium points at infinity having an invariant algebraic curve with at most one pair of diametrally opposite singular points at infinity, being the endpoints of the y-axis has at most one algebraic limit cycle.

Quadratic systems

Proof of Theorem 3