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

Jianfeng Huang; Haihua Liang

doi:10.3934/dcdsb.2020145

Article Contents

2021, Volume 26, Issue 2: 861-873. Doi: 10.3934/dcdsb.2020145

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

Jianfeng Huang^1, , and
Haihua Liang^2,

1.
Department of Mathematics, Jinan University, Guangzhou 510632, P.R. China
2.
School of Mathematics and Systems Science, Guangdong Polytechnic Normal University, Guangzhou 510665, P.R. China

^* Corresponding author: Jianfeng Huang
^* Corresponding author: Jianfeng Huang

Received: December 31, 2018

Revised: December 31, 2019

Early access: May 2020

Published: February 2021

The first author is supported by the NSF of China (No.11401255, No.11771101) and the China Scholarship Council (No.201606785007). The second author is supported by the NSF of China (No.11771101), the major research program of colleges and universities in Guangdong Province (No.2017KZDXM054), and the Science and Technology Program of Guangzhou, China (201805010001)

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Abstract

In this paper we consider the limit cycles of the planar system

$ \begin{align*} \frac{d}{dt}(x,y) = \boldsymbol X_n+\boldsymbol X_m, \end{align*} $

where $ \boldsymbol X_n $ and $ \boldsymbol 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 introduce 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.

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Mathematics Subject Classification: Primary: 34C07; Secondary: 37C05; Tertiary: 35F60.

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Figure 1. The dot curve represents the curve $ b_n(\theta)+b_m(\theta)r = 0 $ in Cartesian coordinates, which is the inner boundary of the region $ U $

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