AI 中文总结
本文研究线性化矩阵为非零二阶幂零矩阵的平面向量场的特殊一维限制，明确其性质、递推规律及抵消机制，并通过平面模型验证其与幂零正规形理论的互补性。
AI 中文摘要
我们研究与线性化矩阵为非零二阶幂零矩阵的平面向量场相关的一类特殊一维限制。该限制在幂零特征方向上具有内禀解释，其在约旦坐标下的标量表示在约旦链的容许变换下满足简单变换律，其二阶泰勒系数与内禀二次Bogdanov–Takens系数一致。对于有理限制，我们证明其泰勒系数序列满足由最小分母决定的有限线性递推关系，其阶定义了一个在约旦链容许变换下不变的递推阶，且给出了最终齐次递推的最小阶。因此，在执行非线性正规形变换前，可直接从该限制中检测到精确抵消现象。将其应用于四个平面模型，展现出不同的递推阶与抵消机制，说明了该特殊限制与光滑幂零正规形理论之间的互补性。
英文摘要
We study a distinguished one dimensional restriction associated with planar vector fields whose linearization is a nonzero nilpotent matrix of index two. This restriction admits an intrinsic interpretation on the nilpotent eigendirection, while its scalar representation in Jordan coordinates satisfies a simple transformation law under admissible changes of Jordan chain. Its quadratic Taylor coefficient coincides with the intrinsic quadratic Bogdanov--Takens coefficient. % For rational restrictions, we show that the Taylor coefficient sequence satisfies a finite linear recurrence determined by the minimal denominator. Its degree defines a recurrence degree that is invariant under admissible changes of Jordan chain and gives the minimal order of an eventual homogeneous recurrence. Exact cancellations may therefore be detected directly from the restriction before nonlinear normal form transformations are performed. Applications to four planar models exhibit different recurrence degree and cancellation mechanisms, illustrating the complementarity between the distinguished restriction and smooth nilpotent normal form theory.
