Phase Space Reconstruction

Takens' embedding theorem · delay coordinates · attractor recovery
Time series x(t)
Reconstructed attractor [x(t), x(t+τ)]
3D reconstruction [x(t), x(t+τ), x(t+2τ)]
Autocorrelation (choose τ at first zero)
System
Delay ττ=10
Points1000
Takens' theorem (1981): given a 1D time series from a dynamical system, delay-embedding x(t), x(t+τ), x(t+2τ),... reconstructs the attractor up to diffeomorphism. The time delay τ should be chosen at the first zero of the autocorrelation function. This lets us recover the full attractor structure from a single observable — the basis of modern time-series analysis.