TY - JOUR

T1 - A generalization of linear positive systems with applications to nonlinear systems

T2 - Invariant sets and the Poincaré–Bendixson property

AU - Weiss, Eyal

AU - Margaliot, Michael

N1 - Publisher Copyright:
© 2020 Elsevier Ltd

PY - 2021/1

Y1 - 2021/1

N2 - 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.

AB - 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.

KW - Asymptotic stability

KW - Compound matrices

KW - Cyclic feedback systems

KW - Poincaré–Bendixson property

KW - Sign variation diminishing property

KW - Totally positive matrices

UR - http://www.scopus.com/inward/record.url?scp=85096191746&partnerID=8YFLogxK

U2 - 10.1016/j.automatica.2020.109358

DO - 10.1016/j.automatica.2020.109358

M3 - מאמר

AN - SCOPUS:85096191746

VL - 123

JO - Automatica

JF - Automatica

SN - 0005-1098

M1 - 109358

ER -