Delay Embedding of Periodic Orbits Using a Fixed Observation Function

TL;DR

This paper proves that for periodic orbits, delay embedding using a fixed observation function generically results in an embedding, extending classical results to fixed observation scenarios.

Contribution

It establishes that delay coordinates produce embeddings for periodic orbits with a fixed observation function, a case not covered by previous generic embedding theorems.

Findings

Delay embeddings are valid for periodic orbits with fixed observations.

Embedding results hold generically over the space of flows in the $C^{r}$ topology.

Applicable to any nonzero linear combination of state coordinates.

Abstract

Delay coordinates are a widely used technique to pass from observations of a dynamical system to a representation of the dynamical system as an embedding in Euclidean space. Current proofs show that delay coordinates of a given dynamical system result in embeddings generically over a space of observations (Sauer, Yorke, Casdagli, J. Stat. Phys., vol. 65 (1991), p. 579-616). Motivated by applications of the embedding theory, we consider the situation where the observation function is fixed. For example, the observation function may simply be some fixed coordinate of the state vector. For a fixed observation function (any nonzero linear combination of coordinates) and for the special case of periodic solutions, we prove that delay coordinates result in an embedding generically over the space of flows in the $C^{r}$ topology with $r\geq2$.