# Embedding Information onto a Dynamical System

arXiv: https://arxiv.org/abs/2105.10766 · HTML: https://mlsanity.com/paper/2105.10766/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 34 references that are in the map.
- First posted on arXiv 22 May 2021.

## Builds on (most cited first, 3 of 3)

- [Echo state networks are universal](https://mlsanity.com/paper/1806.00797/) (2018)
- [Discrete-time signatures and randomness in reservoir computing](https://mlsanity.com/paper/2010.14615/) (2020)
- [Differentiable reservoir computing](https://mlsanity.com/paper/1902.06094/) (2019)

## Built on by (most cited first, 5 of 5)

- [Infinite-dimensional reservoir computing](https://mlsanity.com/paper/2304.00490/) (2023)
- [State-space systems as dynamic generative models](https://mlsanity.com/paper/2404.08717/) (2024)
- [Stochastic dynamics learning with state-space systems](https://mlsanity.com/paper/2508.07876/) (2025)
- [Echoes of the Past: A Unified Perspective on Fading memory and Echo States](https://mlsanity.com/paper/2508.19145/) (2025)
- [Data-Specific Hyper-Parameter Design: A Paradigm Shift in Reservoir Computing](https://mlsanity.com/paper/2605.25221/) (2026)

## Abstract (arXiv)

The celebrated Takens' embedding theorem concerns embedding an attractor of a dynamical system in a Euclidean space of appropriate dimension through a generic delay-observation map. The embedding also establishes a topological conjugacy. In this paper, we show how an arbitrary sequence can be mapped into another space as an attractive solution of a nonautonomous dynamical system. Such mapping also entails a topological conjugacy and an embedding between the sequence and the attractive solution spaces. This result is not a generalization of Takens embedding theorem but helps us understand what exactly is required by discrete-time state space models widely used in applications to embed an external stimulus onto its solution space. Our results settle another basic problem concerning the perturbation of an autonomous dynamical system. We describe what exactly happens to the dynamics when exogenous noise perturbs continuously a local irreducible attracting set (such as a stable fixed point) of a discrete-time autonomous dynamical system.