On the maximal saddle order of $ p:-q $ resonant saddle

Guangfeng Dong; Changjian Liu; Jiazhong Yang

doi:10.3934/dcds.2019251

Article Contents

2019, Volume 39, Issue 10: 5729-5742. Doi: 10.3934/dcds.2019251

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On the maximal saddle order of $ p:-q $ resonant saddle

1.
Department of Mathematics, Jinan University, Guangzhou 510632, China
2.
School of Mathematics(Zhuhai), Sun Yat-Sen University, Zhuhai 519082, China
3.
School of Mathematical Sciences, Peking University, Beijing 100871, China

^* Corresponding author: donggf@jnu.edu.cn
^* Corresponding author: donggf@jnu.edu.cn

Received: June 30, 2018

Revised: February 28, 2019

Published: July 2019

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Abstract

In this paper, we obtain some estimations of the saddle order which is the sole topological invariant of the non-integrable resonant saddles of planar polynomial vector fields of arbitrary degree $n$. Firstly, we prove that, for any given resonance $p:-q$, $(p, q)=1$, and sufficiently big integer $n$, the maximal saddle order can grow at least as rapidly as $n^2$. Secondly, we show that there exists an integer $k_0$, which grows at least as rapidly as $3n^2/2$, such that $L_{k_0}$ does not belong to the ideal generated by the first $k_0-1$ saddle values $L_1, L_2, \cdots, L_{k_0-1}$, where $L_{k}$ represents the $k$-th saddle value of the given system. In particular, if $p=1$ (or $q=1$), we obtain a sharper result that $k_0$ can grow at least as rapidly as $2 n^2$. These results are valid for both cases of real and complex resonant saddles.

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Mathematics Subject Classification: Primary: 34C07; Secondary: 37G05.

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