Koopman Embedding and Super-Linearization Counterexamples with Isolated Equilibria

TL;DR

This paper disproves common claims in applied Koopman operator theory by constructing smooth dynamical systems with multiple isolated equilibria that are linearizable, challenging previous assumptions about system linearization limitations.

Contribution

The authors construct counterexamples of smooth dynamical systems with multiple isolated equilibria that are (super-)linearizable, refuting prior claims about linearization impossibility.

Findings

Counterexamples of linearizable systems with multiple equilibria.

Disproof of claims that systems with isolated equilibria cannot be linearized.

Construction of systems with any countable number of equilibria in .

Abstract

A frequently repeated claim in the "applied Koopman operator theory'' literature is that a dynamical system with multiple isolated equilibria cannot be linearized in the sense of admitting a smooth embedding as an invariant submanifold of a linear dynamical system. This claim is sometimes made only for the class of super-linearizations, which additionally require that the embedding "contain the state''. We show that both versions of this claim are false by constructing (super-)linearizable smooth dynamical systems on $\mathbb{R}^k$ having any countable (finite) number of isolated equilibria for each $k>1$.