Abstract
This paper shows that the celebrated embedding theorem of Takens is a particular case of a much more general statement according to which, randomly generated linear state-space representations of generic observations of an invertible dynamical system carry in their wake an embedding of the phase space dynamics into the chosen Euclidean state space. This embedding coincides with a natural generalized synchronization that arises in this setup and that yields a topological conjugacy between the state-space dynamics driven by the generic observations of the dynamical system and the dynamical system itself. This result provides additional tools for the representation, learning, and analysis of chaotic attractors and sheds additional light on the reservoir computing phenomenon that appears in the context of recurrent neural networks.
| Original language | English |
|---|---|
| Pages (from-to) | 4674-4708 |
| Number of pages | 35 |
| Journal | Nonlinearity |
| Volume | 36 |
| Issue number | 9 |
| Early online date | 27 Jul 2023 |
| DOIs | |
| Publication status | Published - 1 Sept 2023 |
Bibliographical note
Funding Information:We thank Friedrich Philipp for kindly communicating to us the proof of proposition . We also thank Henrik Brautmeier for his assistance with some of the numerical illustrations in the paper. AH is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), Project EP/L015684/1. J P O acknowledges partial financial support coming from the Swiss National Science Foundation (Grant No. 200021_175801/1).
Funding Information:
We thank Friedrich Philipp for kindly communicating to us the proof of proposition 4.1. We also thank Henrik Brautmeier for his assistance with some of the numerical illustrations in the paper. AH is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), Project EP/L015684/1. J P O acknowledges partial financial support coming from the Swiss National Science Foundation (Grant No. 200021_175801/1).
Funding
We thank Friedrich Philipp for kindly communicating to us the proof of proposition . We also thank Henrik Brautmeier for his assistance with some of the numerical illustrations in the paper. AH is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), Project EP/L015684/1. J P O acknowledges partial financial support coming from the Swiss National Science Foundation (Grant No. 200021_175801/1). We thank Friedrich Philipp for kindly communicating to us the proof of proposition 4.1. We also thank Henrik Brautmeier for his assistance with some of the numerical illustrations in the paper. AH is supported by a scholarship from the EPSRC Centre for Doctoral Training in Statistical Applied Mathematics at Bath (SAMBa), Project EP/L015684/1. J P O acknowledges partial financial support coming from the Swiss National Science Foundation (Grant No. 200021_175801/1).
Keywords
- attractor
- dynamical systems
- echo state property
- fading memory property
- generalized synchronization
- reservoir computing
- Takens embedding
ASJC Scopus subject areas
- Statistical and Nonlinear Physics
- Mathematical Physics
- General Physics and Astronomy
- Applied Mathematics