AI 中文总结
研究二次向量场的标记 \(H_{14}^{3}\) 半双曲半周期的局部一致有限循环性，通过构建有限地图集、相交论证等方法，对解析方程进行多方面处理，得出在特定邻域内孤立极限环数量一致有界且估计在五参数中保持一致的结论。
AI 中文摘要
我们证明了二次向量场的标记 \(H_{14}^{3}\) 半双曲半周期的局部一致有限循环性。具体而言，在紧致化图形的一个固定环形邻域内，对于全五参数源归一化商展开的所有足够小的值，孤立极限环的数量是一致有界的。这是相应二次半周期分析中未解决的情况，因为非紧致源、两个半双曲端点和上赤道退化同时出现。证明通过在形成任何全圈返回之前构建一个有限的停止首次命中地图集。一个相交论证通过恰好一个保留的行程来表示每个计数的循环。所得的解析方程通过匹配源估计、在精确混合面上的直接Liénard - Dulac论证以及在其余区域的双曲、中心、严格利普希茨、中间和根尺度零定理来处理。一个有限特化论证包括系数、边界、坍缩和恒等式值。独特之处在于所有估计在五个原始参数中保持一致。所得的界是存在性的。
英文摘要
We prove local uniform finite cyclicity of the labelled $H_{14}^{3}$ hemicycle in a full neighborhood of its base field in the twelve-dimensional space of planar quadratic vector fields. Thus there is one fixed two-sided annular neighborhood of the compactified graphic in which the number of isolated limit cycles is bounded uniformly for every sufficiently close quadratic field. A local analytic slice theorem is part of the result: the displayed five-parameter source-normalized family is transverse at the $B=0$ field to the seven-dimensional action of affine phase changes and positive constant time rescalings. This removes the normalization and transfers the same cyclicity bound to the full quadratic coefficient space. The normalized problem combines a noncompact period annulus, two semihyperbolic endpoints at infinity, and a degeneration at the upper equatorial point. We replace the unavailable global Poincare map by finitely many stopped transition maps and prove an exact multiplicity-preserving correspondence between collar cycles and displacement zeros. At the noncompact source a matched nonlinear return on a common physical domain, controlled through six derivatives, gives a two-zero bound by Rolle's theorem. At the two saddle-nodes an exhaustive physical incidence and scale decomposition reduces all limits to source, mixed, hyperbolic, central, and lips estimates. A finite specialization argument then extends these bounds across coefficient and identity strata. The resulting bound is existential and is not claimed to be sharp. In the terminology of the quadratic finite-cyclicity program, this completes the labelled $H_{14}^{3}$ open case.
Comments99 pages, 5 figures; unified manuscript including all proof appendices. This replacement proves a local analytic slice theorem for the full twelve-dimensional quadratic coefficient space and substantially reorganizes the complete proof
