On singularly perturbed systems that are monotone with respect to a matrix cone of rank k

We derive a sufficient condition guaranteeing that a singularly perturbed linear time-varying system is strongly monotone with respect to a matrix cone C of rank k. This implies that the singularly perturbed system inherits the asymptotic properties of systems that are strongly monotone with respect to C, which include convergence to the set of equilibria when $k=1$, and a Poincaré-Bendixson property when $k=2$. We extend this result to singularly perturbed nonlinear systems with a compact and convex state-space. We demonstrate our theoretical results using a simple numerical example.