Properties of Immersions for Systems with Multiple Limit Sets with Implications to Learning Koopman Embeddings

TL;DR

This paper investigates the limitations of continuous linear immersions, including Koopman eigenfunctions, for nonlinear systems with multiple limit sets, showing they cannot distinguish all limit sets under mild conditions.

Contribution

It proves that continuous immersions for certain systems necessarily merge multiple limit sets, highlighting fundamental limitations in learning Koopman embeddings from data.

Findings

Continuous immersions collapse multiple limit sets

Approximate immersions from data share this limitation

Examples illustrate the theoretical results

Abstract

Linear immersions (such as Koopman eigenfunctions) of a nonlinear system have wide applications in prediction and control. In this work, we study the properties of linear immersions for nonlinear systems with multiple omega-limit sets. While previous research has indicated the possibility of discontinuous one-to-one linear immersions for such systems, it has been unclear whether continuous one-to-one linear immersions are attainable. Under mild conditions, we prove that any continuous immersion to a class of systems including finite-dimensional linear systems collapses all the omega-limit sets, and thus cannot be one-to-one. Furthermore, we show that this property is also shared by approximate linear immersions learned from data as sample size increases and sampling interval decreases. Multiple examples are studied to illustrate our results.