Takens Embedding: Attractor Reconstruction

Takens' theorem (1981): a scalar time series x(t) reconstructs the full attractor as (x(t), x(t+τ), x(t+2τ)). The right delay τ reveals the Lorenz butterfly from a single variable.

Parameters

True Lorenz attractor (top-left)
Projected onto x-z plane.

Reconstructed attractor (right)
Using only x(t) and delay τ.

Optimal τ ≈ first minimum of
mutual information (≈6–10 steps).

Embedding dim ≥ 2d+1 by
Whitney; Takens shows d+1 works
generically.