Some intuition for why cooperative systems "look 1-dimensional" and 2-cooperative systems "look 2-dimensional"

Eduardo D. Sontag

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arXiv · 2607.27176

Some intuition for why cooperative systems "look 1-dimensional" and 2-cooperative systems "look 2-dimensional"

Eduardo D. Sontag

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Abstract

It is known that cooperative systems, and more generally systems monotone with respect to cones, behave under appropriate irreducibility conditions like one-dimensional systems, as evidenced by results such as Hirsch's generic convergence theorem. It is also known, following work of Sanchez and others, that two-cooperative systems, those preserving a generally nonconvex 2-dimensional cone (tested through the diminishing of sign variations), behave like two-dimensional systems, as evidenced by Poincare--Bendixson-type theorems. In these notes I attempt to give some geometric intuition for these dimensionality reductions, based on Birkhoff--Hilbert contractions of the projective metric on a positive cone.

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Eduardo D. Sontag. 2026-07-29. Some intuition for why cooperative systems "look 1-dimensional" and 2-cooperative systems "look 2-dimensional". https://arxiv.org/abs/2607.27176

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