A Complete Classification on the Center-Focus Problem of a Generalized Cubic Kukles System with a Nilpotent Singular Point

被引：0

作者：

Feng Li

Ting Chen

Yuanyuan Liu

Pei Yu

机构：

[1] Linyi University,School of Mathematics and Statistics

[2] Guangdong University of Finance and Economics,School of Statistics and Mathematics

[3] Western University,Department of Mathematics

来源：

Qualitative Theory of Dynamical Systems | 2024年 / 23卷

关键词：

Generalized cubic Kukles system; Center-focus problem; Nilpotent singular point; Center; Analytic center;

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摘要：

In this paper, we study the center-focus problem for a generalized cubic Kukles system with a nilpotent singular point, which consists of a cubic system with an extra 4th-order term. A complete classification is given on the center conditions which are explicitly expressed in term of the system parameters. A total of 15 cases are obtained, among them 4 for the generalized cubic Kukles system and 12 for the cubic Kukles system, with one common for both. One of the center conditions is analytic. Moreover, it is shown that 8 small-amplitude limit cycles can bifurcate in the neighborhood of the singular point for the generalized cubic Kukles system, while only 7 small-amplitude limit cycles can exist around the singular point for the cubic Kukles system. The center-focus problem for the generalized cubic Kukles system with a nilpotent origin is thoroughly solved.

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