3D Poincaré Plot of the Logistic Map

The logistic map is given by , where is a parameter. The 3D Poincaré plot of this map is the set of points in 3D, which lie on the curve , where . The plot shows the curve and the points (, , ). The green ball shows the current iterate; past iterates are marked on the curve by smaller points. Observe that as approaches 4, the plot points fill nearly the entire curve, and the behavior becomes chaotic. For smaller values of , the iterates can either converge to a single point or oscillate between several limit points.

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