# Geometric Properties of Isostables and Basins of Attraction of Monotone Systems

## TL;DR

This paper explores the geometric structure of basins of attraction in monotone systems using spectral operator theory, revealing properties of isostables and how they relate to system stability and multistability.

## Contribution

It introduces a novel geometric analysis of basins of attraction in monotone systems using Koopman operator theory, highlighting properties of eigenfunctions and isostables.

## Key findings

- Dominant eigenfunction is increasing in all variables for monotone systems
- Isostables are nested forward-invariant sets simplifying stability analysis
- Bounds on basin variations under parametric uncertainty

## Abstract

In this paper, we study geometric properties of basins of attraction of monotone systems. Our results are based on a combination of monotone systems theory and spectral operator theory. We exploit the framework of the Koopman operator, which provides a linear infinite-dimensional description of nonlinear dynamical systems and spectral operator-theoretic notions such as eigenvalues and eigenfunctions. The sublevel sets of the dominant eigenfunction form a family of nested forward-invariant sets and the basin of attraction is the largest of these sets. The boundaries of these sets, called isostables, allow studying temporal properties of the system. Our first observation is that the dominant eigenfunction is increasing in every variable in the case of monotone systems. This is a strong geometric property which simplifies the computation of isostables. We also show how variations in basins of attraction can be bounded under parametric uncertainty in the vector field of monotone systems. Finally, we study the properties of the parameter set for which a monotone system is multistable. Our results are illustrated on several systems of two to four dimensions.