A proof of the uniqueness of the limit cycle of a quasi-homogeneous system

Abstract

A. Gasull shared a list of 33 open problems in low dimensional dynamical systems in his work in 2021. The second part of Problem 3 is about whether the limit cycle of a quasi-homogeneous system $\dot{x}=y,\dot{y}=-x^3+\alpha x^{2}y+y^3$ is unique. In this paper, we give a positive answer to this question by analysing the uniqueness of the heteroclinic separatrix at infinity.