Differentially positive systems

TL;DR

This paper introduces differentially positive systems, extending Perron Frobenius theory to analyze how local order constraints influence the long-term behavior of nonlinear systems.

Contribution

It generalizes positivity concepts to nonlinear systems via differential analysis, revealing new order-based constraints on system dynamics.

Findings

Differential positivity induces a conal order constraining asymptotic behavior.

Extends Perron Frobenius theory to nonlinear, trajectory-dependent positivity.

Behavioral constraints go beyond linear positive and monotone systems.

Abstract

The paper introduces and studies differentially positive systems, that is, systems whose linearization along an arbitrary trajectory is positive. A generalization of Perron Frobenius theory is developed in this differential framework to show that the property induces a (conal) order that strongly constrains the asymptotic behavior of solutions. The results illustrate that behaviors constrained by local order properties extend beyond the well-studied class of linear positive systems and monotone systems, which both require a constant cone field and a linear state space.