Takens Embedding Theorem

Reconstructing strange attractors from scalar time series via delay embedding

Parameters

Takens (1981): For a generic smooth system with attractor dim d, delay embedding with m≥2d+1 yields a diffeomorphism between the attractor and its reconstruction.
State: x(t), x(t-τ), x(t-2τ), ...
Current delay: τ=— steps

Original Attractor (x-y projection)

Time Series x(t)

Delay Embedding Reconstruction