一类六次系统的中心焦点问题
The Center-Focus Problem for a Class of Sixth-Degree Systems
DOI: 10.12677/aam.2026.155207, PDF,    科研立项经费支持
作者: 江炫杰, 张文迪, 杨博彬, 付云龙, 樊智辉*：河南科技大学数学与统计学院，河南 洛阳
关键词: 六次系统；Lyapunov常数；中心；焦点；Sixth-Degree System； Lyapunov Constant； Center； Focus
摘要: 针对一类平面光滑六次自治微分系统，本文聚焦于其奇点处的中心焦点判定这一微分方程定性理论中的经典问题。研究中首先对系统进行极坐标变换，在此基础上开展幂级数展开，通过逐次积分递推构造得到系统的各阶Lyapunov常数；整个推导过程借助符号计算软件Maple完成严格的符号运算与分析，得到了8个中心条件和11个5阶弱焦点条件，为该类高次系统的局部动力学行为分析提供了可行的理论支撑。
Abstract: For a class of planar smooth sixth-degree autonomous differential systems, this paper focuses on the center-focus determination at singular points, a classical problem in the qualitative theory of differential equations. We first transform the system into polar coordinates and then perform a power series expansion. By successive integration, the Lyapunov constants are constructed recursively. The entire derivation is carried out with the aid of the symbolic computation software Maple, enabling rigorous symbolic computation and analysis. As a result, eight center conditions and eleven conditions for a weak focus of order five are obtained, providing a feasible theoretical basis for analyzing the local dynamical behavior of such high-degree systems.
文章引用：江炫杰, 张文迪, 杨博彬, 付云龙, 樊智辉. 一类六次系统的中心焦点问题[J]. 应用数学进展, 2026, 15(5): 44-60. https://doi.org/10.12677/aam.2026.155207

参考文献

[1] Poincaré, H. (1886) Sur les courbes définies par une équation différentielle. Journal de Mathématiques Pures et Appliquées, 2, 151-217.
[2] Lyapunov, A.M. (1992) The General Problem of the Stability of Motion. International Journal of Control, 55, 531-534. [Google Scholar] [CrossRef] 
[3] Bautin, N.N. (1939) On the Number of Limit Cycles Which Appear with the Variation of Coefficients from an Equilibrium Position of Focus or Center Type. Matematicheskii Sbornik, 30, 181-196.
[4] Dumortier, F., Llibre, J. and Artés, J.C. (2006) Qualitative Theory of Planar Differential Systems. Springer.
[5] Romanovski, V.G. and Shafer, D.S. (2009) The Center and Cyclicity Problems: A Computational Algebra Approach. Birkhäuser.
[6] Christopher, C. and Li, C. (2007) Limit Cycles of Differential Equations. Springer.
[7] Rondón, G. and Sadri, N. (2026) Global Dynamics of Generalized Duffing Oscillators with Global Centers. Applied Mathematics and Computation, 522, 130007. [Google Scholar] [CrossRef] 
[8] Llibre, J. and Rondón, G. (2023) Global Centers of a Class of Cubic Polynomial Differential Systems.

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