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

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 $\mathrm{\Gamma }$-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.