Lorenz attractor with three shadow manifolds.
The Lorenz attractor [37] is shown with three shadow manifolds created from lag-coordinate transformations. The typical parameters were used: 
, 
, and 
, giving the three coupled equations as 
, 
, and 
. The solution was computed using a fourth order Runge-Kutta method with a time step of 
, and the time lag used to create the shadow manifolds was 
. (A) The trajectory shown in the 
, 
, and 
coordinates of the original system reveals a two-lobed manifold. (B) A univariate transformation using time lags of the 
-coordinate, 
, preserves this two-lobed structure (and other topological properties), verifying Takens' theorem. (C) A univariate transformation using time lags of the 
-coordinate, 
, does not preserve the two-lobed structure. Local neighborhoods of the original attractor are, however, preserved. Thus, though this mapping violates a genericity assumption of the original theorem and is not an embedding, it is an immersion of the original manifold. (D) A multivariate transformation using both the 
- and 
-coordinates, 
, fulfills the assumptions of Theorems 2 and 7. As predicted, it also preserves the two-lobed structure of the Lorenz and is a valid embedding.
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