State the normal-form equations for (i) a saddle-node bifurcation, (ii) a transcritical bifurcation and (iii) a pitchfork bifurcation, for a one-dimensional map $x_{n+1}=F\left(x_{n} ; \mu\right)$.

Consider a period-doubling bifurcation of the form

$$x_{n+1}=-x_{n}+\alpha+\beta x_{n}+\gamma x_{n}^{2}+\delta x_{n}^{3}+O\left(x_{n}^{4}\right),$$

where $x_{n}=O\left(\mu^{1 / 2}\right), \alpha, \beta=O(\mu)$, and $\gamma, \delta=O(1)$ as $\mu \rightarrow 0$. Show that

$$X_{n+2}=X_{n}+\hat{\mu} X_{n}-A X_{n}^{3}+O\left(X_{n}^{4}\right),$$

where $X_{n}=x_{n}-\frac{1}{2} \alpha$, and the parameters $\hat{\mu}$ and $A$ are to be identified in terms of $\alpha, \beta$, $\gamma$ and $\delta$. Deduce the condition for the bifurcation to be supercritical.