First, we find the fixed points of the logistic map. If
x_{f} stands for a fixed point of the logistic map, we know it
must satisfy the fixed point equation 
x_{f} = L(x_{f}) 
Using the logistic map
definition
L(x) = s⋅x⋅(1  x), the fixed point equation becomes 
x_{f} = s⋅x_{f}⋅(1  x_{f}) 
so 
s⋅x_{f}^{2} + (1  s)⋅x_{f} = 0 
so 
x_{f}⋅(s⋅x_{f} + (1  s)) = 0 

and we obtain two fixed points 
x_{f} = 0 and x_{f} = (s  1)/s 
Note the second fixed point is positive only when s > 1.
So for s ≤ 1 the logistic map has only one fixed point between 0 and 1.
Click on the picture to see how the fixed points of the logistic map change as s increases
to 4. 
For which svalues are these fixed points stable? Recall when we
studied graphical iteration we asserted fixed points are
stable if the graph crosses the diagonal inside the "45°
blue bowtie." 


Click the picture to see how the fixed points of the logistic map
change as s increases to 4. 
Click the picture to see how the logistic map and the blue bowties
interact as s increases to 4. 

In fact, with a little calculus we can prove 
The fixed point x_{f} = 0 is stable
for 0 ≤ s < 1. 
The fixed point x_{f} = (s  1)/s is stable
for 1 < s < 3. 
