
 [No title] 
  If these data come from a logistic map or any onedimensional map, then the Nth vector will have the form: X[N] = (x[N], f(x[N])), and you should be able to see that all the 2vectors will collapse on to the quadratic curve (x,f(x)) rather than being sprinkled over the plane. 
  This idea generalizes to higherdimensional spaces although one needs to turn to a computer algorithm since it is difficult to visualize structures in 3d, 4d, etc spaces. 
  This may seem phony but there is a change of variables: x[i] = (1/2) (1  Cos(Pi y[i])), which converts the logistic map with r=4 into the tent map for y[i]. 
 www.phy.duke.edu /~hsg/213/lectures/9803.txt (847 words) 
