Kahan-Hirota-Kimura (KHK) maps are a specific discretization scheme for quadratic vector fields. They preserve certain algebraic structures — rational invariant fibrations — that make them candidates for integrable discrete dynamics. But “integrable” doesn't mean “simple.”
The authors (arXiv:2603.23382) show that KHK maps can exhibit surprisingly complex dynamics even when the original continuous vector field is trivial. A quadratic planar vector field with an isochronous center — all orbits periodic with the same period — discretizes to a KHK map that preserves the original first integral and admits all possible global periods except 2. For a dense set of timestep values, every orbit is periodic, but the period varies with the timestep.
The exclusion of period 2 is structural, not accidental. The KHK discretization scheme, by construction, maps each fiber to itself in a way that makes period-2 orbits impossible while allowing all other periods.
When the same technique is applied to other quadratic vector fields with isochronous centers, the KHK map is no longer integrable. The authors introduce “pseudo-KHK maps” as alternative discretizations that are specifically constructed to preserve integrability.
The through-claim: discretization is creative, not neutral. The KHK scheme applied to a trivial continuous system produces a discrete system with all periods except one — dynamics that the original system doesn't have. The discrete map doesn't approximate the continuous flow; it generates new behavior constrained by the algebraic structure of the discretization.