AI 中文总结
本文研究$\u211d^n$中具有二重零特征值的向量场的Bogdanov--Takens范式二次系数，通过特征不变量的方向导数给出显式公式，证明了系数的二分性，给出非退化检验与几何解释，并附Wolfram Language验证程序。
AI 中文摘要
设$X$是$\u211d^n$的一个开集上的向量场，满足$X(p)=0$，且其雅可比矩阵$J=DX(p)$的秩为$n-1$，并以0作为代数重数为2的特征值。设$q_0$张成$\u043a\u0435\u0440 J$，记$e_k(A)$为矩阵$A$的所有$k\u00d7k$阶主子式之和。在横截块的通常双曲性假设下，我们证明中心流形上Bogdanov--Takens范式的二次系数$a,b$满足$a=-\frac{1}{2}\frac{D_{q_0}e_n}{e_{n-2}}$，$b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}D_{q_0}e_n}{e_{n-2}^2}$。其背后的谱恒等式仅要求横截块可逆，在无�ite双曲性条件下依然成立。两个系数均作为$DX$沿$\u043a\u0435\u0440 J$的一阶谱射流的生成恒等式的最低阶项出现；所有更高阶系数都依赖于横截块。我们证明这种二分性是严格的。当$n=2$时，可恢复平面情形的公式$a=-\frac{1}{2}D_{q_0}\u0434\u0435\u0442$和$b=D_{q_0}\u0442\u0440$。我们还得到了一种不依赖坐标的非退化检验方法，以及基于核直线与两个不变超曲面横截性的几何解释。本文附带一份自包含的Wolfram Language笔记本，可符号化验证所有结果。
英文摘要
Let $X$ be a vector field on an open set of $\mathbb{R}^n$ with $X(p)=0$ and Jacobian $J=DX(p)$ of rank $n-1$ having $0$ as an eigenvalue of algebraic multiplicity two. Let $q_0$ span $\ker J$ and let $e_k(A)$ denote the sum of the principal $k\times k$ minors of $A$. Under the usual hyperbolicity assumption on the transverse block, we prove that the quadratic coefficients $a,b$ of the Bogdanov--Takens normal form on the centre manifold are $a=-\frac{1}{2}\frac{D_{q_0}e_n}{e_{n-2}}$ and $b=\frac{D_{q_0}e_{n-1}}{e_{n-2}}-\frac{e_{n-3}D_{q_0}e_n}{e_{n-2}^2}$. The underlying spectral identity requires only invertibility of the transverse block and remains valid without hyperbolicity. Both coefficients arise as the lowest-order terms of a generating identity for the first-order spectral jet of $DX$ along $\ker J$; all higher coefficients depend on the transverse block. We prove that this dichotomy is sharp. The planar formulas $a=-\frac{1}{2}D_{q_0}\det$ and $b=D_{q_0}\operatorname{tr}$ are recovered when $n=2$. We also obtain a coordinate-free nondegeneracy test and a geometric interpretation in terms of the transversality of the kernel line to two invariant hypersurfaces. A self-contained Wolfram Language notebook accompanies the paper and verifies the results symbolically.
Comments23 pages, 1 figure
