Operator-Theoretic Characterization of Eventually Monotone Systems

TL;DR

This paper introduces a spectral characterization of nonlinear eventually monotone systems using the Koopman operator framework, extending the understanding of such systems beyond classical monotonicity conditions.

Contribution

It provides the first explicit spectral characterization of nonlinear eventually monotone systems, bridging a gap in the theory using Koopman operator techniques.

Findings

Spectral characterization of nonlinear eventually monotone systems established.

Koopman operator framework effectively analyzes eventual monotonicity.

Biologically inspired examples demonstrate practical relevance.

Abstract

Monotone systems are dynamical systems whose solutions preserve a partial order in the initial condition for all positive times. It stands to reason that some systems may preserve a partial order only after some initial transient. These systems are usually called eventually monotone. While monotone systems have a characterization in terms of their vector fields (i.e. Kamke-Muller condition), eventually monotone systems have not been characterized in such an explicit manner. In order to provide a characterization, we drew inspiration from the results for linear systems, where eventually monotone (positive) systems are studied using the spectral properties of the system (i.e. Perron-Frobenius property). In the case of nonlinear systems, this spectral characterization is not straightforward, a fact that explains why the class of eventually monotone systems has received little attention to date. In this paper, we show that a spectral characterization of nonlinear eventually monotone systems can be obtained through the Koopman operator framework. We consider a number of biologically inspired examples to illustrate the potential applicability of eventual monotonicity.