A one-dimensional map is defined by

$$x_{n+1}=F\left(x_{n}, \mu\right)$$

where $\mu$ is a parameter. What is the condition for a bifurcation of a fixed point $x_{*}$ of $F$ ?

Let $F(x, \mu)=x\left(x^{2}-2 x+\mu\right)$. Find the fixed points and show that bifurcations occur when $\mu=-1, \mu=1$ and $\mu=2$. Sketch the bifurcation diagram, showing the locus and stability of the fixed points in the $(x, \mu)$ plane and indicating the type of each bifurcation.