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Local maximal-canard threshold shifts under Runge--Kutta discretization: an observable-specific order condition — arXiv:2608.04304

Source: arXiv (en)
Original: https://arxiv.org/abs/2608.04304
License (as archived): CC0-1.0
Archived: 2026-10-31T11:45:52.028Z
SHA-256 of the archived text: 9a3a393068b62437651914244cbf0f7f9a7ea70e86a74c03bf2df6d36c19a107
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Local maximal-canard threshold shifts under Runge--Kutta discretization: an observable-specific order condition — arXiv:2608.04304 — arXiv (en), https://arxiv.org/abs/2608.04304 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_bb014339bf13 (Metraipolis archive dc_bb014339bf13, sha256 9a3a393068b62437651914244cbf0f7f9a7ea70e86a74c03bf2df6d36c19a107)

BibTeX

@misc{dcbb014339bf13,
  title = {Local maximal-canard threshold shifts under Runge--Kutta discretization: an observable-specific order condition — arXiv:2608.04304},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2608.04304},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_bb014339bf13; sha256 9a3a393068b62437651914244cbf0f7f9a7ea70e86a74c03bf2df6d36c19a107},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2608.04304. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $(\alpha,\beta)\mapsto-3\beta\Xi(J)/8$. Here $\Xi(J)$ 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 $\lambda_{\mathrm{RK}}-\lambda_{\mathrm{flow}}=K_\theta(J)h^2\varepsilon^2+o(h^2\varepsilon^2)$. 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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-08-06 in math.NA).
Categorie arXiv: Dynamical Systems (math.DS) ; Numerical Analysis (math.NA).
Pagina del lavoro: https://arxiv.org/abs/2608.04304 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-31T11:34:54.272Z dalla pagina https://arxiv.org/catchup/math.NA/2026-08-06?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).