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3D Poincaré Plot of the Logistic Map

​
parameter r
3.7
number of iterations n
2
initial value
x
0
0.65
The logistic map is given by
x
n+1
=r
x
n
(1-
x
n
)
, where
r
is a parameter. The 3D Poincaré plot of this map is the set of points
(
x
n
,
x
n+1
,
x
n+2
)
in 3D, which lie on the curve
(x,f(x),f(f(x)))
, where
f(x)=rx(1-x)
. The plot shows the curve and the points (
x
n
,
x
n+1
,
x
n+2
). The green ball shows the current iterate; past iterates are marked on the curve by smaller points. Observe that as
r
approaches 4, the plot points fill nearly the entire curve, and the behavior becomes chaotic. For smaller values of
r
, the iterates can either converge to a single point or oscillate between several limit points.
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