WOLFRAM|DEMONSTRATIONS PROJECT

An Interval Eventually Bounding Trajectories of the Logistic Map

​
a
3.44
starting value
x
0
0.138
number of points
81
This Demonstration illustrates the iteration of the logistic map
h
a
(x)=ax(1-x)
for
2<a≤4
. All the points of the interval
J
2
=(a/4,1)
are mapped into
J
1
=(0,a/4)
. Also, the interval
J
4
=[
h
a
(a/4),a/4]
is mapped into itself. If an iteration belongs to
J
3
it can be proved that after a finite number of steps the trajectory is contained in
J
4
. Therefore every trajectory whose starting point belongs to (0, 1) is eventually contained in
J
4
.