Takens' theorem in action for the Lorenz chaotic attractor

Movies from the supplementary info of the paper "Detecting Causality in Complex Ecosystems," authored by George Sugihara, Robert May, Hao Ye, Chih-hao Hsieh, Ethan Deyle, Michael Fogarty, and Stephan Munch. Published Online September 20 2012 in Science Express: http://dx.doi.org/10.1126/science.122...

Movie S1
This movie demonstrates the relationship between time series and dynamic attractors (manifolds, M).

Movie S2
This movie illustrates how Takens' Theorem (19) can be used to reconstruct a shadow manifold MX from a single time series, and illustrates the 1:1 mapping between M and MX.

Movie S3
This movie explains how convergent cross mapping (CCM) estimates states across variables.