Limit cycles of planar system defined by the sum of two quasi-homogeneous vecter fields

TL;DR

This paper investigates the limit cycles of a planar system formed by the sum of two quasi-homogeneous vector fields, establishing conditions under which at most one limit cycle exists, and introduces a novel divergence-based method for analysis.

Contribution

The paper provides a new hypothesis ensuring at most one limit cycle in such systems and develops a divergence formula linking Abel and generalized-polar equations for better analysis.

Findings

Maximal number of limit cycles is one under the new hypothesis.

The divergence formula facilitates auxiliary function construction for system analysis.

Application extends to systems where previous results do not apply.

Abstract

In this paper we consider the limit cycles of the planar system $$\frac{d}{dt}(x,y)=\mathbf X_n+\mathbf X_m, $$ where $\mathbf X_n$ and $\mathbf X_m$ are quasi-homogeneous vector fields of degree $n$ and $m$ respectively. We prove that under a new hypothesis, the maximal number of limit cycles of the system is $1$. We also show that our result can be applied to some systems when the previous results are invalid. The proof is based on the investigations for the Abel equation and the generalized-polar equation associated with the system, respectively. Usually these two kinds of equations need to be dealt with separately, and for both equations, an efficient approach to estimate the number of periodic solutions is constructing suitable auxiliary functions. In the present paper we develop a formula on the divergence, which allows us to construct an auxiliary function of one equation with the auxiliary function of the other equation, and vice versa.