State the normal-form equations for (a) a saddle-node bifurcation, (b) a transcritical bifurcation, and (c) a pitchfork bifurcation, for a dynamical system $\dot{x}=f(x, \mu)$.

Consider the system

$$\begin{aligned} &\dot{x}=\mu+y-x^{2}+2 x y+3 y^{2} \\ &\dot{y}=-y+2 x^{2}+3 x y \end{aligned}$$

Compute the extended centre manifold near $x=y=\mu=0$, and the evolution equation on the centre manifold, both correct to second order in $x$ and $\mu$. Deduce the type of bifurcation and show that the equation can be put in normal form, to the same order, by a change of variables of the form $T=\alpha t, X=x-\beta \mu, \tilde{\mu}=\gamma(\mu)$ for suitably chosen $\alpha, \beta$ and $\gamma(\mu)$.