Explain what is meant by a steady-state bifurcation of a fixed point $\mathbf{x}_{0}(\mu)$ of a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x}, \mu)$ in $\mathbb{R}^{n}$, where $\mu$ is a real parameter.

Consider the system in $x \geqslant 0, y \geqslant 0$, with $\mu>0$,

$$\begin{aligned} \dot{x} &=x\left(1-y^{2}-x^{2}\right), \\ \dot{y} &=y\left(\mu-y-x^{2}\right) . \end{aligned}$$

(i) Show that both the fixed point $(0, \mu)$ and the fixed point $(1,0)$ have a steady-state bifurcation when $\mu=1$.

(ii) By finding the first approximation to the extended centre manifold, construct the normal form near the bifurcation point $(1,0)$ when $\mu$ is close to unity, and show that there is a transcritical bifurcation there. Explain why the symmetries of the equations mean that the bifurcation at $(0,1)$ must be of pitchfork type.

(iii) Show that two fixed points with $x, y>0$ exist in the range $1<\mu<5 / 4$. Show that the solution with $y<1 / 2$ is stable. Identify the bifurcation that occurs at $\mu=5 / 4$.

(iv) Draw a sketch of the values of $y$ at the fixed points as functions of $\mu$, indicating the bifurcation points and the regions where each branch is stable. [Detailed calculations are not required.]