Visible and hidden observables in super-linearization

TL;DR

This paper investigates the concept of super-linearization, introducing visible and hidden observables, and establishes a tight lower bound on the number of visible observables needed for such embeddings.

Contribution

It defines visible and hidden observables in super-linearizations and derives a tight lower bound for the number of visible observables in all embeddings.

Findings

Introduces the notions of visible and hidden observables.

Derives a tight lower bound for visible observables.

Provides insights into the structure of super-linearizations.

Abstract

We call a system super-linearizable if it admits finite-dimensional embedding as a linear system -- known as a finite-dimensional Koopman embedding; said otherwise, if its dynamics can be linearized by adding a finite set of observables. We introduce the notions of visible and hidden observables for such embeddings which, roughly speaking, are the observables that explicitly appear in the original system and the ones that do not, but yet are necessary for its embedding. Distinct embeddings can have different numbers of hidden and visible observables. In this paper, we derive a tight lower bound for the number of visible observables of a system among all its super-linearizations.