With , the Logistic Equation becomes

(1) 
Now let

(2) 

(3) 

(4) 

(5) 
Manipulating (2) gives
so

(7) 

(8) 
But . Taking , then and

(9) 
For , and

(10) 
Combining

(11) 
which can be written

(12) 
the Tent Map with , so the Natural Invariant in is

(13) 
Transforming back to gives
This can also be derived from

(15) 
where is the Delta Function.
See also Logistic Equation
© 19969 Eric W. Weisstein
19990525