Nonlinear left and right eigenvectors for max-preserving maps

TL;DR

This paper introduces a novel approach to defining nonlinear left and right eigenvectors for max-preserving maps, utilizing a semiring structure and closure operation to analyze stable dynamical systems and construct Lyapunov functions.

Contribution

It develops a new framework for eigenvector concepts in max-preserving maps using semiring theory and closure operations, with applications to stability analysis.

Findings

Defined nonlinear eigenvectors via inequalities for max-preserving maps

Established a semiring structure for these maps and their closure operation

Provided explicit examples and applications in Lyapunov function construction

Abstract

It is shown that max-preserving maps (or join-morphisms) on the positive orthant in Euclidean $n$-space endowed with the component-wise partial order give rise to a semiring. This semiring admits a closure operation for maps that generate stable dynamical systems. For these monotone maps, the closure is used to define suitable notions of left and right eigenvectors that are characterized by inequalities. Some explicit examples are given and applications in the construction of Lyapunov functions are described.