^{*}

Conceived and designed the experiments: ERD GS. Performed the experiments: ERD GS. Analyzed the data: ERD GS. Contributed reagents/materials/analysis tools: ERD GS. Wrote the paper: ERD GS.

The authors have declared that no competing interests exist.

Takens' theorem (1981) shows how lagged variables of a single time series can be used as proxy variables to reconstruct an attractor for an underlying dynamic process. State space reconstruction (SSR) from single time series has been a powerful approach for the analysis of the complex, non-linear systems that appear ubiquitous in the natural and human world. The main shortcoming of these methods is the phenomenological nature of attractor reconstructions. Moreover, applied studies show that these single time series reconstructions can often be improved

A growing realization in many natural sciences is that simple idealized notions of linearly decomposable, fixed equilibrium systems often do not accord with reality. Rather, empirical measurements on ecosystems, metabolic systems, financial networks, and the like suggest a more complex, but potentially more information-rich paradigm at work

In this paper, we present two general theorems that addresses the problem of characterizing the coupled dynamics of nonlinear systems using time series observations on a manifold

Here we prove the more general case of multivariate embeddings (embeddings using multiple time series and lags thereof), and show how time series information can be leveraged if multiple time series and their lags are used to construct embeddings of

The possibility of extending Takens' theorem to allow lags of multiple observation functions was mentioned in Remark 2.9 from

Consider the classic Lorenz attractor

The Lorenz attractor

An

The Lorenz attractor,

Let

Takens explicitly refers only to the unlagged

For a mapping

The question arises whether general multivariate mappings

It follows from Whitney

Recall that, for a compact manifold, a mapping that is an immersion and injective is also necessarily an embedding. Thus, Takens' general approach was to first show that (i) immersions are dense in the set of mappings

To demonstrate both (i) and (ii), Takens argues that even when the property of interest (e.g. the

Consider an arbitrary set of

For any point

Since immersions are local embeddings, we can find a

Next, we show that we can find a globally

We now consider the mapping

At each

Thus, we have shown that for any arbitrary set of

When mappings are confined to fixed lag relationships, Takens showed it is valid to independently perturb each component of

Before starting the proof, however, we must clarify exactly what the “subsets of interest” are. We define these sets as follows. First, we say

The proof of this theorem closely follows the logic of the previous proof and the original argument of Takens

Note that periodic points

We must also satisfy

We first perturb the

We now show that we can find a

Now we need to show that there is a

To perturb the manifold

The

For each

For all choices of

For

Take a partition of unity

Just as Takens extends the original result for discrete time to dynamical systems in continuous time, we can extend our result as follows:

In this case,

We now give an explicit proof of Remark 2.9 from

To apply these lemmas, it is necessary to restrict the dimension of the sets of periodic orbits—that is, the sets

Without loss of generality, assume we have ordered the components of

Let

To sensibly apply Lemma 5, we adopt the following convention: think of

We now check that the rank of the matrix

The dimension of the set

The dimension of the set

Now we want show that almost every

Theorem 7 can be extended to continuous dynamical systems (smooth vector fields on a manifold) by letting the flow

Theorem 1 and the more general result presented in Theorem 2 (and its corollary) were given proofs intended to follow those presented by Takens. The original “transversality” argument, however, has been replaced with what we reckon is a simpler and more direct argument. These clarify how perturbations to the observation functions can be constructed and highlight why

This work also develops a language to describe a wider family of cases for reconstructing state space manifolds from multiple observational time series to encourage wider applicability of SSR in the natural sciences. For example, these results can be extended to another special case of interest for reconstructions using time derivatives

More importantly, in terms of future applications, Theorems 2 and 7 set the stage for practical reconstruction of state space manifolds from multiple observation functions. This is significant in answering objections to single variable state space reconstruction (SSR) concerning the excessive phenomenology of lagged-coordinate embeddings

We wish to thank Hao Ye, James Crutchfield, John Melack, Donald DeAngelis, Simon Levin, J. Doyne Farmer, Martin Casdagli, Tim Sauer, Sarah Glaser, Chih-hao Hsieh, Stephen Munch and Charles Peretti, Michael Fogarty, Alec MacCall, Andrew Rosenberg, Les Kaufman, and Irit Altman for helpful comments and editorial advice.