Prevalence of Delay Embeddings with a Fixed Observation Function

TL;DR

This paper proves that delay embeddings with a fixed observation function are prevalent in dynamical systems, showing that generic polynomial perturbations to the system ensure the delay map is an embedding with probability one.

Contribution

The paper introduces a novel technique using Lebesgue points to establish the prevalence of delay embeddings with a fixed observation function under polynomial perturbations.

Findings

Delay map is an embedding for generic observation functions.

Embedding property is prevalent for fixed observation functions.

Probability one result for delay embeddings under polynomial perturbations.

Abstract

Let $x_{j+1}=\phi(x_{j})$, $x_{j}\in\mathbb{R}^{d}$, be a dynamical system with $\phi$ being a diffeomorphism. Although the state vector $x_{j}$ is often unobservable, the dynamics can be recovered from the delay vector $\left(o(x_{1}),\ldots,o(x_{D})\right)$, where $o$ is the scalar-valued observation function and $D$ is the embedding dimension. The delay map is an embedding for generic $o$, and more strongly, the embedding property is prevalent. We consider the situation where the observation function is fixed at $o=\pi_{1}$, with $\pi_{1}$ being the projection to the first coordinate. However, we allow polynomial perturbations to be applied directly to the diffeomorphism $\phi$, thus mimicking the way dynamical systems are parametrized. We prove that the delay map is an embedding with probability one with respect to the perturbations. Our proof introduces a new technique for proving prevalence using the concept of Lebesgue points.