AI 中文总结
该研究分析平面快慢回路中，不变直线进出延迟与二次 grazing 共存时的极限环分岔数，推导参数一致的位移展开式，构造捕食模型验证结论。
AI 中文摘要
我们确定，当不变直线进出延迟与切换线的二次 grazing 发生在同一平面快慢返回回路中时，可分支出多少个极限环。我们考虑连续分段光滑系统，其中这些机制被规则流段分隔。对于正的 ε，当光滑参考曲率与主导 grazing 失配系数符号相同时，物理局部循环性至多为 2；当二者符号相反时，至多为 3。在秩 2 展开下，对于每个充分小的固定正奇异参数，每个符号类都达到适用界限。为证明这些结论，我们在平衡中性循环附近按任意指定有限阶推导了参数一致的 Poincaré 位移展开式，以及零点计数所需的单侧导数估计。其主导非光滑贡献是经典的 3/2 阶 grazing 项，该系数由接触、分支失配和全局传输数据显式给出。一个多项式族实现了两种符号情况。我们还构造了已发表的截断 Gause 捕食-食饵模型的定性可接受四次扩展，其决定性曲率和秩条件通过区间算术得到验证。我们的贡献是将经典 grazing 定律与单位乘数下的进出通道按符号精确、参数一致地组合。
英文摘要
We study planar continuous piecewise-smooth slow--fast return circuits in which an invariant-line entry--exit passage is followed by a separated quadratic grazing and the reference cycle has unit multiplier. We first prove that the physical entry--exit map for positive slow parameter extends, uniformly to any prescribed finite order, to the singular parameter, including nonvertical endpoint fibers. In the exact moving penetration coordinate, composition with the grazing passage gives the extended Poincare displacement $Δ(q,p)=S(q,p)+q_+^{3/2}K(\sqrt{q_+},q,p)$. Under a rank-two unfolding, we use the exact constant and linear coefficients of $Δ$ as parameters and construct, for every sufficiently small fixed positive slow parameter, a unique nonpenetrating fold, a unique penetrating fold, and the grazing-incidence stratum. We give the complete marked chamber decomposition by nonpenetrating, grazing, and penetrating cycles and determine their stability from the Poincare multiplier. The open chambers contain zero or two cycles when the smooth and grazing coefficients have the same sign, and one or three when their signs are opposite; the corresponding cyclicity bounds are sharp corollaries. A compact polynomial family realizes both sign classes. In a cutoff-Gause family, interval certificates isolate one singular balanced-neutral point in each of two parameter boxes and verify the required signs and rank; an analytic local pullback transfers the diagram to the response parameters. Thus one local classification records cycle number, itinerary, and stability across the grazing and fold boundaries.
Comments43 pages, 3 figures. Revised literature positioning and exposition; complete marked two-parameter bifurcation diagram, itinerary and stability classification, and cutoff-Gause parameter pullback. Ancillary files contain dual-backend interval certificates, a CAPD validated-ODE certificate, separately labelled positive-parameter numerical material, and a reproduction guide
