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

Abstract

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.