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

Eyal Weiss; Michael Margaliot

doi:10.1016/j.automatica.2020.109358

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

Eyal Weiss, Michael Margaliot

Department of Computer Science

Tel Aviv University

Research output: Contribution to journal › Article › peer-review

25 Scopus citations

Abstract

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.

Original language	English
Article number	109358
Journal	Automatica
Volume	123
DOIs	https://doi.org/10.1016/j.automatica.2020.109358
State	Published - Jan 2021

Bibliographical note

Publisher Copyright:
© 2020 Elsevier Ltd

Funding

This research was partially supported by research grants from the Israel Science Foundation and the Binational Science Foundation . The material in this paper was partially presented at the 27th Mediterranean Conference on Control and Automation July 1–4, 2019, Akko, Israel. This paper was recommended for publication in revised form by Associate Editor Björn S. Rüffer under the direction of Editor Daniel Liberzon.

Funders	Funder number
United States-Israel Binational Science Foundation
Israel Science Foundation

Keywords

Asymptotic stability
Compound matrices
Cyclic feedback systems
Poincaré–Bendixson property
Sign variation diminishing property
Totally positive matrices

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10.1016/j.automatica.2020.109358

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

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Keywords

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