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

TL;DR

This paper establishes conditions under which singularly perturbed systems are strongly monotone with respect to a matrix cone of rank k, leading to specific asymptotic behaviors and extending to nonlinear systems.

Contribution

It provides a new sufficient condition for strong monotonicity in singularly perturbed systems with respect to matrix cones of rank k, including nonlinear extensions.

Findings

Systems are strongly monotone under the derived condition.

Asymptotic properties like convergence and Poincaré-Bendixson are inherited.

Numerical example demonstrates theoretical results.

Abstract

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.