A generalization of linear positive systems with applications to nonlinear systems: Invariant sets and the Poincaré–Bendixson property

• Published in: Automatica (Oxford) Vol. 123; p. 109358
• Main Authors: Weiss, Eyal, Margaliot, Michael
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.01.2021
• Subjects: Asymptotic stability; Compound matrices; Cyclic feedback systems; Poincaré–Bendixson property; Sign variation diminishing property; Totally positive matrices; Totally positive matrices; Asymptotic stability; Compound matrices; Poincaré–Bendixson property; Cyclic feedback systems; Sign variation diminishing property
• ISSN: 0005-1098; 1873-2836
• DOI: 10.1016/j.automatica.2020.109358

The dynamics of linear positive systems maps the positive orthant to itself. In other words, it maps a set of vectors with zero sign variations to itself. What linear systems map the set of vectors with k sign variations to itself? We address this question using tools from the theory of cooperative dynamical systems and the theory of totally positive matrices. This yields a generalization of positive linear systems called k-positive linear systems, that reduces to positive systems for k=1. We describe applications of this new type of systems to the analysis of nonlinear dynamical systems. In particular, we show that such systems admit certain explicit invariant sets, and for the case k=2 establish the Poincaré–Bendixson property for any bounded trajectory.