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

arXiv: https://arxiv.org/abs/2312.17045 · HTML: https://mlsanity.com/paper/2312.17045/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Dec 2023.

## Built on by (most cited first, 2 of 2)

- [On the relationship between Koopman operator approximations and neural ordinary differential equations for data-driven time-evolution predictions](https://mlsanity.com/paper/2411.12940/) (2024)
- [Sparse Koopman Autoencoders Identify Local Dynamical Regimes in Multibasin Systems](https://mlsanity.com/paper/2608.29057/) (2026)

## Abstract (arXiv)

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.