Takens's theorem

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.

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.

[1] Delay embedding theorems are simpler to state for discrete-time dynamical systems.

with box counting dimension dA.

Using ideas from Whitney's embedding theorem, A can be embedded in k-dimensional Euclidean space with That is, there is a diffeomorphism φ that maps A into

must be twice-differentiable and associate a real number to any point of the attractor A.

It must also be typical, so its derivative is of full rank and has no special symmetries in its components.

evolves according to an unknown but continuous and (crucially) deterministic dynamic.

, but also at observations made at times removed from us by multiples of some lag

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 (precisely, they are related by a smooth, invertible change of coordinates, or diffeomorphism).

In fact, the theorem says that determinism appears once you reach dimension

[2][3] Takens' theorem is usually used to reconstruct strange attractors out of experimental data, for which there is contamination by noise.

Whereas for data without noise, any choice of delay is valid, for noisy data, the attractor would be destroyed by noise for delays chosen badly.

The optimal delay is typically around one-tenth to one-half the mean orbital period around the attractor.