Delay Embedding — Takens' Theorem

Reconstruct an attractor from a scalar time series using time-delay coordinates. One observable can reveal the full attractor topology.

Takens 1981: For generic τ, the delay map φ: x(t) → (x(t), x(t+τ), x(t+2τ)) is an embedding — it preserves the topology of the attractor.

Time Series x(t)

Raw scalar observable

Delay Embedding (τ)

x(t) vs x(t+τ)

Autocorrelation

First zero-crossing suggests optimal τ

Mutual Information

First minimum → optimal delay τ