  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
Poincar\'e-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.
