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Takens Embedding Theorem

Recover strange attractor topology from a 1D time series · delay embedding · mutual information

Time Series (observed)

System: Lorenz x(t)
τ = 8 | m = 2

Delay Embedding: x(t) vs x(t+τ)

Takens 1981: generically, delay embedding
reconstructs attractor up to diffeomorphism.
m ≥ 2d+1 sufficient (d = attractor dim)

Mutual Information & False Neighbors

Optimal τ: first minimum of I(τ)
Optimal m: FNN fraction drops to ~0