AI 中文总结
该研究针对连续两域平面流的高阶中性掠食接触问题，结合Hermite零定理推导了周期轨道数量的尖锐界，明确了循环性与接触成本的控制因素，实现了相关几何与构型的显式多项式构造。
AI 中文摘要
连续两域平面流的中性周期轨道可接触切换接缝而不穿越它，一侧的邻近轨道会进入第二个向量场并在短暂主动偏移后返回。若接缝接触阶数为偶数ν，符号穿透量q>0的轨道在主动区域的停留时间为O(q^{1/ν})。连续性要求X⁺-X⁻=hW，因此累积变化比不连续系统小一阶，为O(q^{1+1/ν})。我们将该机制转化为参数一致的动力学到标量定理，并结合跨接缝Hermite零定理。当光滑参考回归首次与恒等式的阶数n≥2不同时，固定阶ν层上的邻近周期轨道的尖锐数量为n或n+1，取决于光滑修正与主动修正的符号耦合；秩为n的物理展开达到适用界。因此n控制循环性，ν控制接触成本与通过尺度。在给定的接触射流与受限回归射流横截性假设下，阶ν接触层的环境余维为ν-2，同时中性掠食的余维为n+ν-2。横截接触扰动将中心幂替换为加权正部帽，从1+1/ν到通用二次3/2定律的起始交叉。尖锐界对精确帽和导数控制的物理单阱持续存在；仅含值的反例将阻碍定位于一般多阱通道。显式闭多项式族实现了几何、秩与周期轨道构型。
英文摘要
We study a neutral periodic orbit of a continuous two-field flow that is tangent to a switching seam while nearby orbits enter the second field. We separate the roles of two governing integers: the neutral-return order $n\geq2$ controls orbit count, whereas the even contact order $ν$ controls grazing codimension, passage scale, and transverse crossover. Continuity factors the branch mismatch as $X^+-X^-=hW$; an active excursion of duration $O(q^{1/ν})$ therefore produces a correction of order $q_+^{1+1/ν}$. A parameter-uniform passage theorem and a cross-seam Hermite zero theorem then give the sharp fixed-stratum cyclicity: $n$ or $n+1$ periodic orbits, according to the sign coupling of smooth and active terms. A rank-$n$ physical unfolding attains the applicable bound. Contact-jet incidence gives ambient codimension $ν-2$ for order-$ν$ grazing and $n+ν-2$ for simultaneous neutral grazing. Transverse contact parameters replace the central power by a weighted positive-part cap whose onset exponent crosses from $1+1/ν$ to the generic quadratic-contact value $3/2$. The same sharp bound holds for both the exact cap and a prepared physical multiwell return, including births, mergers, and multiple entry--exit pairs. A value-only $C^1$ counterexample shows that finite cyclicity requires event-derivative information. For every even $ν\geq4$, a closed polynomial family realizes the prescribed contact and rank conditions and the sharp periodic-orbit configurations through actual finite-time first-return maps.
Comments61 pages, 4 figures; strengthened prepared physical multiwell theorem; revised literature positioning and exposition; clarified Figure 2 event labels; 33-page Technical Companion supplied as ancillary material
