Journal of Statistical Physics, Vol. 65, Nos. 3/4, 1991 Embedology Tim Sauer, 1 James A. Yorke, 2 and Martin Casdagli 3 Received May 10, 1991 Mathematical formulations of the embedding methods commonly used for the reconstruction of attractors from data series are discussed. Embedding theorems, based on previous work by H. Whitney and F. Takens, are estab- lished for compact subsets A of Euclidean space R k. If n is an integer larger than twice the box-counting dimension of A, then almost every map from R k to R ", in the sense of prevalence, is one-to-one on A, and moreover is an embedding on smooth manifolds contained within A. If A is a chaotic attractor of a typical dynamical system, then the same is true for almost every delay-coordinate map from R ~ to R n. These results are extended in two other directions. Similar results are proved in the more general case of reconstructions which use moving averages of delay coordinates. Second, information is given on the self-intersec- tion set that exists when n is less than or equal to twice the box-counting dimension of A. KEY WORDS: Embedding; chaotic attractor; attractor reconstruction; probability one; prevalence; box-counting dimension; delay coordinates. 1. INTRODUCTION In this, work we give theoretical justification of data embedding techniques used by experimentalists to reconstruct dynamical information from time series. We focus on cases in which trajectories of the system under study are asymptotic to a compact attractor. We state conditions that ensure that the map from the attractor into reconstruction space is an embedding, meaning that it is one-to-one and preserves differential information. Our approach integrates and expands on previous results on embedding by Whitney (29) and Takens. (27) 1 Department of Mathematical Sciences, George Mason University, Fairfax, Virginia 22030. 2 Institute of Physical Science and Technology, University of Maryland, College Park, Maryland 20742. 3 Santa Fe Institute, Santa Fe, New Mexico 87501. Current address: Tech Partners, 4 Stamford Forum, 8th Floor, Stamford, Connecticut 06901. 579 0022-4715/91/1100-0579506.50/0 9 1991 Plenum Publishing Corporation580 Sauer et al. Whitney showed that a generic smooth map F from a d-dimensional smooth compact manifold M to R 2d+ 1 is actually a diffeomorphism on M. That is, M and F(M) are diffeomorphic. We generalize this in two ways: first, by replacing "generic" with "probability-one" (in a prescribed sense), and second, by replacing the manifold M by a compact invariant set A contained in R k that may have noninteger box-counting dimension (boxdim). In that case, we show that almost every smooth map from a neighborhood of A to R" is one-to-one as long as n > 2- boxdim(A) We also show that almost every smooth map is an embedding on compact subsets of smooth manifolds within A. This suggests that embedding techniques can be used to compute positive Lyapunov exponents (but not necessarily negative Lyapunov exponents). The positive Lyapunov exponents are usually carried by smooth unstable manifolds on attractors. We give precise definitions of one-to-one, embedding, and almost every in the next section. Takens dealt with a restricted class of maps called delay-coordinate maps. A delay-coordinate map is constructed from a time series of a single observed quantity from an experiment. Because of this, a typical delay- coordinate map is much more likely to be accessible to an experimentalist than a typical map. Takens (27) showed that if the dynamical system and the observed quantity are generic, then the delay-coordinate map from a d-dimensional smooth compact manifold M to R 2d+1 is a diffeomorphism on M. Our results generalize those of Takens ~27) in the same two ways as for Whitney's theorem. Namely, we replace generic with probability-one and the manifold M by a possibly fractal set. Thus, for a compact invariant subset A of R k, under mild conditions on the dynamical system, almost every delay-coordinate map F from R k to R n is one-to-one on A provided that n>2.boxdim(A). Also, any manifold structure within A will be preserved in F(A). These results hold for lower box-counting dimension (see Section 4) if boxdim does not exist. The ambient space R k can be replaced by a k-dimensional smooth manifold in the general case. In addition, we have made explicit the hypotheses on the dynamical system (discrete or continuous) that are needed to ensure that the delay-coor- dinate map gives an embedding. In particular, only C 1 smoothness is needed. For flows, the delay must be chosen so that there are no periodic orbits whose period is exactly equal to the time delay used or twice the delay. (A finite number of periodic orbits of a flow whose periods are p times the delay are allowed for p >~ 3.) Further, we explain what happensEmbedology 581 in the case that n ~<2.boxdim(A). In that case we put bounds on the dimension of the self-intersection set, which is the set on which the one- to-one property fails. Finally, we give more general versions of the delay- coordinate theorem which deals with filtered delay coordinates, which allow more versatile and useful applications of embedding methods. There are no analogues of these results where the box-counting dimension is replaced by Hausdorff dimension (see Theorem 4.7 and the discussion that follows). In an Appendix to this work written by I. Kan, examples are described of compact subsets of R k, for any positive integer k, which have Hausdorff dimension d= 0, and which are difficult to project in a one-to-one way. The requirement n > 2d discussed above translates in this case to n > 0. However, every projection of such a set to R n, n < k, fails to be one-to-one. In Section 2 we explain the new version of the Whitney and Takens embedding theorems. In Section 3 we discuss filtered delay coordinates. Section 4 contains proofs of the results. 2. HOW TO EMBED