Delay embedding theorem — reconstruct attractors from a single time series
8
3
2000
45°
Takens theorem (1981): generically, a delay-coordinate map Φ: x(t) → (x(t), x(t−τ), …, x(t−(m−1)τ)) is an embedding when m ≥ 2d+1 (d = attractor dimension). The reconstructed attractor is diffeomorphic to the original — same topology, Lyapunov exponents, fractal dimension. Only one observable needed to recover the full attractor.