Consider the system

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

Show that when $\mu=0$ the fixed point at the origin has a stationary bifurcation.

Find the centre subspace of the extended system linearised about $(x, y, \mu)=(0,0,0)$.

Find an approximation to the centre manifold giving $y$ as a function of $x$ and $\mu$, including terms up to quadratic order.

Hence deduce an expression for $\dot{x}$ on the centre manifold, and identify the type of bifurcation at $\mu=0$.