局部最大鸭阈值中的龙格-库塔偏差：链树条件
Local maximal-canard threshold shifts under Runge--Kutta discretization: an observable-specific order condition
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中文总结 AI 辅助
该研究分析了龙格-库塔离散化在局部最大鸭阈值中的偏差，揭示三阶B级数缺陷仅链树坐标被阈值泛函检测，证明了特定自伴龙格-库塔族的主导偏差抵消条件及相关数值验证结果。
中文摘要 AI 辅助
固定步长龙格-库塔离散化可精准近似普通轨迹，同时会偏移参数，使得选定的吸引和排斥慢流形形成最大鸭。对于极小多项式模型 $x'=x^2-y+xy$，$y'=\epsilon(x-\lambda)$，我们在吸引和排斥慢流形的所有允许选择上，比较流的实际截面定义阈值及其精确龙格-库塔映射的阈值。我们首先在对称、A-稳定的两级切片上揭示该机制，然后在完整自伴两级族的紧致双参数域及单独的自伴三级族上证明该机制。三阶B级数缺陷具有独立的浓密树和链树坐标，但阈值泛函仅检测链树缺陷。因此，全类主导偏差抵消等价于 $b^T A c=1/6$，而其他经典三阶条件可能仍未满足。对于尖锐模型，声明的两级和三级族的主导系数分别为 $(12\rho-1)/32$ 和 $(24\nu-1)/128$。当 $\epsilon=r^2\to0$ 且 $r^2\le h\le r^{3/4}$ 时，这些系数在附近实际映射根与流根的偏移中乘以 $h^2\epsilon^2$。证明通过指数选择屏蔽、精确共同目标递推及离散高斯梅尔尼科夫积分，将有限龙格-库塔缺陷传输到实际截面定义的根。右折范德波尔公式提供经典校准，有理罗森茨威格-麦克阿瑟特例给出原始源时钟中的实际局部选定阈值定律，且联合 $(r,h)$ 有限边界代理计算（以192位精度独立验证）说明了预测的符号反转和主导偏差抵消。
英文摘要
Near a planar fast--slow fold, a local maximal canard is selected by the parameter at which the attracting and repelling slow manifolds meet. We compare this threshold for a physical flow and a Runge--Kutta map, using actual invariant manifolds on a common fold section. An order-two Runge--Kutta method has two independent order-three rooted-tree defects. Both enter the pointwise one-step residual, but Gaussian fold transport acts on their leading contribution by $(α,β)\mapsto-3βΞ(J)/8$. Here $Ξ(J)$ is an explicit functional of the fold jet. Thus the singular passage filters the numerical defect space: it annihilates the bushy-tree direction and can retain only the chain-tree direction. For compact analytic classes of affinely normalizable folds and every fixed compact, uniformly finite-stage family of real Runge--Kutta methods of order at least two, the actual flow and map splittings obtained from independent continuations have unique roots whose displacement satisfies a uniform absolute estimate throughout the full small-step rectangle. Whenever the step-independent, exponentially small selection ambiguity is $o(h^2\varepsilon^2)$, the joint-fold law is $λ_{\mathrm{RK}}-λ_{\mathrm{flow}}=K_θ(J)h^2\varepsilon^2+o(h^2\varepsilon^2)$. Fold-matched continuations additionally give ordinary second-order convergence as $h\to0$ with $\varepsilon$ fixed. The leading joint-fold bias therefore vanishes under $b^T A c=1/6$, without classical third order. This is cancellation in one nonlinear observable, not an increase in trajectory order or, in general, in fixed-$\varepsilon$ threshold order. Affine covariance transfers the coefficient to physical fold germs, and a van der Pol invariant-graph computation illustrates the sign change, cancellation, and fixed-$\varepsilon$ convergence.
