Consider the one-dimensional map $F: \mathbb{R} \rightarrow \mathbb{R}$ defined by

$$x_{i+1}=F\left(x_{i}\right)=x_{i}\left(a x_{i}^{2}+b x_{i}+\mu\right)$$

where $a$ and $b$ are constants, $\mu$ is a parameter and $a \neq 0$.

(i) Find the fixed points of $F$ and determine the linear stability of $x=0$. Hence show that there are bifurcations at $\mu=1$, at $\mu=-1$ and, if $b \neq 0$, at $\mu=1+b^{2} / 4 a$.

Sketch the bifurcation diagram for each of the cases:

$$\text { (1) } a>b=0, \quad \text { (2) } a, b>0 \text { and (3) } a, b<0$$

In each case show the locus and stability of the fixed points in the $(\mu, x)$-plane, and state the type of each bifurcation. [Assume that there are no further bifurcations in the region sketched.]

(ii) For the case $F(x)=x\left(\mu-x^{2}\right)$ (i.e. $\left.a=-1, b=0\right)$, you may assume that

$$F^{2}(x)=x+x\left(\mu-1-x^{2}\right)\left(\mu+1-x^{2}\right)\left(1-\mu x^{2}+x^{4}\right)$$

Show that there are at most three 2-cycles and determine when they exist. By considering $F^{\prime}\left(x_{i}\right) F^{\prime}\left(x_{i+1}\right)$, or otherwise, show further that one 2-cycle is always unstable when it exists and that the others are unstable when $\mu>\sqrt{5}$. Sketch the bifurcation diagram showing the locus and stability of the fixed points and 2 -cycles. State briefly what you would expect to occur in the region $\mu>\sqrt{5}$.