Generic behavior of differentially positive systems on a globally orderable Riemannian manifold

TL;DR

This paper demonstrates that in most cases, differentially positive systems on globally orderable Riemannian manifolds have trajectories that converge to a single equilibrium, extending the understanding of their long-term behavior.

Contribution

It proves that generic orbits in such systems tend to a single equilibrium, addressing a conjecture for globally orderable manifolds using cone fields and Perron-Frobenius vector fields.

Findings

Most orbits converge to a single equilibrium

Addresses a conjecture from 2016 for orderable manifolds

Utilizes cone fields and Perron-Frobenius vector fields

Abstract

Differentially positive systems are the nonlinear systems whose linearization along trajectories preserves a cone field on a smooth Riemannian manifold. One of the embryonic forms for cone fields in reality is originated from the general relativity. By utilizing the Perron-Frobenius vector fields and the $\Gamma$-invariance of cone fields, we show that generic (i.e.,``almost all" in the topological sense) orbits are convergent to certain single equilibrium. This solved a reduced version of Forni-Sepulchre's conjecture in 2016 for globally orderable manifolds.