一类具有十参数的复三次多项式微分系统的弱持续中心问题

doi:10.16088/j.issn.1001-6600.2025090101

摘要/Abstract

摘要： 本文研究一类含10个参数的复平面三次多项式系统在原点处的弱持续中心问题。首先,通过对由前7个焦点量生成理想的代数簇进行计算和分解,得到系统原点成为弱持续中心的必要条件;然后,运用Darboux可积理论构造Darboux首次积分或Darboux积分因子,验证系统满足可反性条件,用归纳法证明系统存在多项式首次积分等方法证明这些条件也是充分的;最后,当该系统的变量和系数满足共轭关系时,给出相应实系统原点成为弱持续中心的完整分类。

关键词: 三次多项式微分系统, 中心, 弱持续中心, 焦点量, Darboux可积理论

Abstract: The problem of weakly persistent centers for a ten-parameter family of complex planar cubic polynomial differential systems at the origin is studied. Firstly, the necessary conditions for the origin of the system to be a weakly persistent center are obtained by calculating and decomposing the algebraic variety of the ideal generated by the first seven focal quantities. Then, it is proved that these conditions are also sufficient by either using Darboux integrable theory to construct the Darboux first integral or Darboux integrating factor, or verifying the time reversibility of the system, or applying the induction to demonstrate the existence of polynomial first integral. Finally, the complete classifications for a weakly persistent center in a family of real planar cubic polynomial differential systems is derived, which is obtained by setting the variables and the coefficients of the complex systems are conjugation.

Key words: cubic polynomial differential systems, center, weakly persistent center, focal quantity, Darboux integrable theory

中图分类号: O175

罗珵, 黄文韬, 何东平, 张越. 一类具有十参数的复三次多项式微分系统的弱持续中心问题[J]. 广西师范大学学报（自然科学版）, 2026, 44(4): 121-129.

Luo Cheng, Huang Wentao, He Dongping, Zhang Yue. Weakly persistent centers for a ten-parameter family of complex planar cubic polynomial differential systems[J]. Journal of Guangxi Normal University(Natural Science Edition), 2026, 44(4): 121-129.

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