Simplified version
Suppose the d-dimensional state vector x_t evolves according to an unknown but continuous and (crucially) deterministic dynamic. Suppose, too, that the one-dimensional observable y is a smooth function of x, and “coupled” to all the components of x. Now at any time we can look not just at the present measurement y(t), but also at observations made at times removed from us by multiples of some lag \tau: y_{t+\tau}, y_{t+2\tau} , etc. If we use k lags, we have a k-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 k \to \infty 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 (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 2d+1, and the minimal embedding dimension is often less. == Choice of delay ==
Choice of delay
Takens' theorem is usually used to reconstruct strange attractors out of experimental data, for which there is contamination by noise. As such, the choice of delay time becomes important. 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. == See also ==