Consider the one-dimensional real map $x_{n+1}=F\left(x_{n}\right)=r x_{n}^{2}\left(1-x_{n}\right)$, where $r>0$. Locate the fixed points and explain for what ranges of the parameter $r$ each fixed point exists. For what range of $r$ does $F$ map the open interval $(0,1)$ into itself?

Determine the location and type of all the bifurcations from the fixed points which occur. Sketch the location of the fixed points in the $(r, x)$ plane, indicating stability.