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

TL;DR

This paper proves the uniqueness of the limit cycle in a specific quasi-homogeneous dynamical system by analyzing the behavior of heteroclinic separatrices at infinity.

Contribution

It provides a rigorous proof confirming the unique existence of the limit cycle for a particular quasi-homogeneous system, addressing an open problem.

Findings

Confirmed the limit cycle's uniqueness in the given system.

Analyzed the heteroclinic separatrix at infinity to establish the proof.

Resolved part of an open problem in low-dimensional dynamical systems.

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^2y+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.