Describe the different types of bifurcation from steady states of a one-dimensional map of the form $x_{n+1}=f\left(x_{n}\right)$, and give examples of simple equations exhibiting each type.

Consider the map $x_{n+1}=\alpha x_{n}^{2}\left(1-x_{n}\right), 0<x_{n}<1$. What is the maximum value of $\alpha$ for which the interval is mapped into itself?

Show that as $\alpha$ increases from zero to its maximum value there is a saddle-node bifurcation and a period-doubling bifurcation, and determine the values of $\alpha$ for which they occur.