Consider the dynamical system

$$\begin{aligned} &\dot{x}=x+y^{2}-a \\ &\dot{y}=y\left(4 x-x^{2}-a\right) \end{aligned}$$

for $(x, y) \in \mathbb{R}^{2}, a \in \mathbb{R}$.

Find all fixed points of this system. Find the three different values of $a$ at which bifurcations appear. For each such value give the location $(x, y)$ of all bifurcations. For each of these, what types of bifurcation are suggested from this analysis?

Use centre manifold theory to analyse these bifurcations. In particular, for each bifurcation derive an equation for the dynamics on the extended centre manifold and hence classify the bifurcation.