WOLFRAM|DEMONSTRATIONS PROJECT

A Special Parameter Value for the Logistic Map

​
parameter a
3.67857
starting point
x
0
0.2
iterations
100
show orbit
zoom into
J
4
interval
display region
none
h
a
​​
J
5
​
h
a
​​
J
6
​
current iteration region
This Demonstration considers the iterations of the logistic map
h
a
(x)=ax(1-x)
for
3≤a≤4
. The Demonstration "An Interval Eventually Bounding Trajectories of the Logistic Map" showed how every trajectory with a starting point in (0, 1) is eventually contained in
J
4
=[
h
a
(a/4),a/4]
.
This Demonstration shows that there cannot be any odd periodic orbit if
a≤
a
d
=
3
2
1+
1/3
(19+3
33
)
+
1/3
(19-3
33
)
≈3.67857
. In fact, in that case the interval
J
5
=(
h
a
(a/4),(a-1)/a)
is mapped into the interval
J
6
=((a-1)/a,a/4)
and
J
6
is mapped into
J
5
. It is also easy to show that the points
h
a
(a/4),(a-1)/a,
and
a/4
do not lead to an odd periodic orbit. The previous statements are not true if
a
d
<a≤4
.
It is possible to limit
a
by
3≤a≤4
, as it is well known that for
a<3
there is no periodic orbit.