一类四次系统的中心焦点问题

The Central Focal Issue of a Class IV System

摘要: 聚焦一类平面四次微分系统在奇点处的类型判定问题，所涉系统仅保留二次项与单一四次幂项。借助极坐标变换导出径向方程，通过幂级数展开依次计算Lyapunov常数，确定前三阶为零的四类参数分布。在此基础上，给出奇点成为四阶细焦点的四组参数关系，同时获得奇点为中心的四个充分条件，并由此构造首次积分函数完成充分性证明。上述结果系统总结了该类模型低阶情形下的中心与高阶焦点特征，为后续极限环分支分析奠定理论基础。

Abstract: This study focuses on the type determination of a class of planar fourth-order differential systems at singularities, where the systems retain only quadratic terms and a single fourth-power term. Using polar coordinate transformations, radial equations are derived, and Lyapunov constants are calculated sequentially via power series expansions to identify four parameter distributions with zero Lyapunov coefficients up to the third order. Building upon this, four sets of parameter relationships are established for singularities to become fourth-order fine focal points, along with four sufficient conditions centered around these singularities. The sufficiency is demonstrated through the construction of first-integral functions. These results systematically summarize the central and higher-order focal characteristics of such models in low-order cases, laying a theoretical foundation for subsequent analysis of limit cycle bifurcation.