Takens's theorem
In the study of dynamical systems, a delay embedding theorem gives the conditions under which a chaotic dynamical system can be reconstructed from a sequence of observations of the state of a dynamical system. The reconstruction preserves the properties of the dynamical system that do not change under smooth coordinate changes (i.e., diffeomorphisms), but it does not preserve the geometric shape of structures in phase space. Takens' theorem is the 1981 delay embedding theorem of Floris Takens. It provides the conditions under which a smooth attractor can be reconstructed from the observations made with a generic function . Later results replaced the smooth attractor with a set of arbitrary box counting dimension and the class of generic functions with other classes of functions. Delay embedding theorems are simpler to state for discrete-time dynamical systems. The state space of the dynamical system is a formula_1-dimensional manifold formula_2. The dynamics is given by a smooth map Assume that the dynamics formula_4 has a strange attractor formula_5 with box counting dimension formula_6. Using ideas from Whitney's embedding theorem, formula_7 can be embedded in formula_8-dimensional Euclidean space with That is, there is a diffeomorphism formula_10 that maps formula_7  into formula_12 such that the derivative of formula_10 has full rank. A delay embedding theorem uses an "observation function" to construct the embedding function. An observation function formula_14 must be twice-differentiable and associate a real number to any point of the attractor formula_7. It must also be typical, so its derivative is of full rank and has no special symmetries in its components. The delay embedding theorem states that the function is an embedding of the strange attractor formula_7 in formula_12. Simplified, slightly inaccurate version. Suppose the formula_19-dimensional state vector formula_20 evolves according to an unknown but continuous  and (crucially) deterministic dynamic. Suppose, too, that the one-dimensional observable formula_21 is a smooth function of formula_22, and “coupled” to all the components of formula_22. Now at any time we can look not just at the present measurement formula_24, but also at observations made at times removed from us by multiples of some lag formula_25, etc. If we use formula_8 lags, we have a formula_8-dimensional vector. One might expect that, as the number of lags is increased, the motion in the lagged space will become more and more predictable, and perhaps in the limit formula_28 would become deterministic . In fact, the dynamics of the lagged vectors become deterministic at a finite dimension; not only that, but the deterministic dynamics are completely equivalent to those of the original state space (More exactly, they are related by a smooth, invertible change of coordinates, or diffeomorphism.) The magic embedding dimension formula_8 is at most formula_30, and often less.