A Special Parameter Value for the Logistic Map
A Special Parameter Value for the Logistic Map
This Demonstration considers the iterations of the logistic map (x)=ax(1-x) for . 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 =[(a/4),a/4].
h
a
3≤a≤4
J
4
h
a
This Demonstration shows that there cannot be any odd periodic orbit if . In fact, in that case the interval =((a/4),(a-1)/a) is mapped into the interval =((a-1)/a,a/4) and is mapped into . It is also easy to show that the points (a/4),(a-1)/a, and do not lead to an odd periodic orbit. The previous statements are not true if <a≤4.
a≤=1++≈3.67857
a
d
3
2
1/3
(19+3
33
)1/3
(19-3
33
)J
5
h
a
J
6
J
6
J
5
h
a
a/4
a
d
It is possible to limit by , as it is well known that for there is no periodic orbit.
a
3≤a≤4
a<3