Canard points in planar mass-action networks: a flux criterion, a duality, and a complete classification
Abstract
We determine which mass-action reaction networks on two species, one fast and one slow, admit a canard point: a singular Hopf bifurcation at a fold of the critical manifold, followed by a canard explosion. In flux coordinates, existence reduces to three linear equations, one linear inequation and a sign condition on the product of two linear forms, and the criticality of the explosion is the sign of an explicit cubic form. For four reactions with a balance matrix of full rank, both become integer polynomial conditions on the stoichiometry. A vacancy duality, which exchanges the activator--inhibitor and substrate-depletion mechanisms, preserves canard points together with their criticality. The central structural result is a mode criterion: a network has a canard point if and only if its elementary flux modes contain a sign-meeting pair and a mode carrying a fold. Hence every canard network contains a minimal one carried by at most three modes, and so by at most twelve reactions, whatever the molecularity. For complexes of molecularity at most three we enumerate all 138\,754 minimal canard networks and label each exactly by the criticalities it admits. None has more than six reactions; 32 have canard points of codimension zero and no explosion; and most six-reaction networks admit both criticalities, separated by degenerate canard points. All labels rest on certificates checked in exact rational arithmetic. Resolving the cubic steps into bimolecular ones through fast dimers keeps the canard points and changes the criticality only by an explicit term.
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BibTeX
@misc{beldiman2026canard,
title = {{Canard points in planar mass-action networks: a flux criterion, a duality, and a complete classification}},
author = {Miruna-Mihaela Beldiman and Andrei-Dan Halanay},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.27188},
eprint = {2610.27188},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.27188},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - Canard points in planar mass-action networks: a flux criterion, a duality, and a complete classification
AU - Miruna-Mihaela Beldiman
AU - Andrei-Dan Halanay
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.27188
UR - https://arxiv.org/abs/2610.27188
AB - We determine which mass-action reaction networks on two species, one fast and one slow, admit a canard point: a singular Hopf bifurcation at a fold of the critical manifold, followed by a canard explosion. In flux coordinates, existence reduces to three linear equations, one linear inequation and a sign condition on the product of two linear forms, and the criticality of the explosion is the sign of an explicit cubic form. For four reactions with a balance matrix of full rank, both become integer polynomial conditions on the stoichiometry. A vacancy duality, which exchanges the activator--inhibitor and substrate-depletion mechanisms, preserves canard points together with their criticality. The central structural result is a mode criterion: a network has a canard point if and only if its elementary flux modes contain a sign-meeting pair and a mode carrying a fold. Hence every canard network contains a minimal one carried by at most three modes, and so by at most twelve reactions, whatever the molecularity. For complexes of molecularity at most three we enumerate all 138\,754 minimal canard networks and label each exactly by the criticalities it admits. None has more than six reactions; 32 have canard points of codimension zero and no explosion; and most six-reaction networks admit both criticalities, separated by degenerate canard points. All labels rest on certificates checked in exact rational arithmetic. Resolving the cubic steps into bimolecular ones through fast dimers keeps the canard points and changes the criticality only by an explicit term.
ER -