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

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