# Deterministic Chaos

## Logistic Map Fixed Points

First, we find the fixed points of the logistic map. If xf stands for a fixed point of the logistic map, we know it must satisfy the fixed point equation
xf = L(xf)
Using the logistic map definition L(x) = s⋅x⋅(1 - x), the fixed point equation becomes
 xf = s⋅xf⋅(1 - xf) so s⋅xf2 + (1 - s)⋅xf = 0 so xf⋅(s⋅xf + (1 - s)) = 0
and we obtain two fixed points
xf = 0 and xf = (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 s-values 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 xf = 0 is stable for 0 ≤ s < 1. The fixed point xf = (s - 1)/s is stable for 1 < s < 3.