# 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