Delay-Coordinate Embedding — Takens' Theorem

System

Attractor type Delay τ = 10 steps Embedding dim m = 2 Series length N = 3000

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ModeEmbedding

Takens' Theorem

Given scalar x(t), form delay vectors:
y(t) = [x(t), x(t+τ), ..., x(t+(m-1)τ)]
For generic τ and m ≥ 2d_A+1
(d_A = attractor dimension),
the map is a diffeomorphic embedding.
False nearest neighbors → optimal m