An operator-theoretic approach to differential positivity

TL;DR

This paper explores the relationship between differential positivity in dynamical systems and the spectral properties of the Koopman operator, providing new insights into system stability and limit cycle behavior.

Contribution

It establishes a connection between geometric properties of differentially positive systems and Koopman spectral characteristics, including converse results and cone field constructions.

Findings

Hyperbolic limit cycles are differentially positive within their basins.

Converse results linking spectral properties to differential positivity.

Construction of contracting cone fields for these systems.

Abstract

Differentially positive systems are systems whose linearization along trajectories is positive. Under mild assumptions, their solutions asymptotically converge to a one-dimensional attractor, which must be a limit cycle in the absence of fixed points in the limit set. In this paper, we investigate the general connections between the (geometric) properties of differentially positive systems and the (spectral) properties of the Koopman operator. In particular, we obtain converse results for differential positivity, showing for instance that any hyperbolic limit cycle is differentially positive in its basin of attraction. We also provide the construction of a contracting cone field.