Almost sure convergence of differentially positive systems on a globally orderable Riemannian manifold

TL;DR

This paper proves that almost all trajectories of differentially positive systems on a globally orderable Riemannian manifold converge to equilibrium, addressing a measure-theoretic aspect of a conjecture related to nonlinear system stability.

Contribution

It establishes almost sure convergence of trajectories in differentially positive systems on globally orderable Riemannian manifolds, solving a measure-theoretic version of a 2016 conjecture.

Findings

Almost every orbit converges to equilibrium.

The result applies to systems originating from general relativity.

Addresses a measure-theoretic version of Forni-Sepulchre's conjecture.

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. Out of a set of measure zero, we show that almost every orbits will converge to equilibrium. This solved a reduced version (from a measure-theoretic perspective) of Forni-Sepulchre's conjecture in 2016 for globally orderable manifolds.