Consider the equations

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

as a function of the parameter $a$. Find the fixed points and plot their location in the $(a, x)$ plane. Hence, or otherwise, deduce that there are bifurcations at $a=0$ and $a=1$.

Investigate the bifurcation at $a=1$ by making the substitutions $u=x-1, v=y-x$ and $\mu=a-1$. Find the equation of the extended centre manifold to second order. Find the evolution equation on the centre manifold to second order, and determine the stability of its fixed points.

Show which branches of fixed points in the $(a, x)$ plane are stable and which are unstable, and state, without calculation, the type of bifurcation at $a=0$. Hence sketch the structure of the $(x, y)$ phase plane very near the origin for $|a| \ll 1$ in the cases (i) $a<0$ and (ii) $a>0$.

The system is perturbed to $\dot{x}=\left(a-x^{2}\right)\left(a^{2}-y\right)+\epsilon$, where $0<\epsilon \ll 1$, with $\dot{y}=x-y$ still. Sketch the possible changes to the bifurcation diagram near $a=0$ and $a=1$. [Calculation is not required.]