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

Eyal Weiss, Michael Margaliot*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

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 languageEnglish
Article number109358
JournalAutomatica
Volume123
DOIs
StatePublished - Jan 2021

Keywords

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

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