Consider the eventual fixed points of the logistic map $Lµ : [0, 1] → [0, 1], Lµ(x) = µx(1 − x)$, for $0 < µ < 4$.
1) Show that for $1 < µ ≤ 2$, the only eventual fixed point associated with the fixed point $x = 1 − 1/µ$ is $x = 1/µ$.
2) Show there are more eventually fixed points with $x = 1-1/µ$ when $2 < µ < 3$
I found that if I let $µ = 2$
The fixed points were found by $L2(x) = 2x(1 − x)$. To find fixed points $f(x) = x$ so $2x(1 − x) = x$ gives $x(1-2x) = 0$ ie. our fixed points are $ x=0$ and $ x=1/2$
How do you prove there is only one eventually fixed point?
When I worked it out, I thought that any point between 0 and 1/2 can be eventually fixed because they'll reach the fixed point after some iterations.
When I did this graphically, $f(1/2) = 1/2$ because the $x= 1/2$ automatically reaches the fixed point (which happens to be 1/2) without doing iterations. If I did $f(1/4) = 3/8...$ it will reach 1/2 eventually.