Properties of Isostables and Basins of Attraction of Monotone Systems

TL;DR

This paper explores the geometric properties of monotone systems using the Koopman operator, extending Perron-Frobenius theory to nonlinear systems, and provides methods to estimate basins of attraction under uncertainty.

Contribution

It generalizes spectral properties of the Koopman operator for monotone systems and characterizes basins of attraction with parametric uncertainty.

Findings

Spectral properties of Koopman operator are extended to monotone systems.

Geometric bounds on basins of attraction are derived under uncertainty.

Computational methods for estimating isostables and basins are demonstrated.

Abstract

In this paper, we investigate geometric properties of monotone systems by studying their isostables and basins of attraction. Isostables are boundaries of specific forward-invariant sets defined by the so-called Koopman operator, which provides a linear infinite-dimensional description of a nonlinear system. First, we study the spectral properties of the Koopman operator and the associated semigroup in the context of monotone systems. Our results generalize the celebrated Perron-Frobenius theorem to the nonlinear case and allow us to derive geometric properties of isostables and basins of attraction. Additionally, we show that under certain conditions we can characterize the bounds on the basins of attraction under parametric uncertainty in the vector field. We discuss computational approaches to estimate isostables and basins of attraction and illustrate the results on two and four state monotone systems.