Consider the dynamical system

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

where $a \geqslant 0$ and $b>0$.

(i) Find and classify the fixed points. Show that a bifurcation occurs when $4 b=a^{2}>0$.

(ii) After shifting coordinates to move the relevant fixed point to the origin, and setting $a=2 \sqrt{b}-\mu$, carry out an extended centre manifold calculation to reduce the two-dimensional system to one of the canonical forms, and hence determine the type of bifurcation that occurs when $4 b=a^{2}>0$. Sketch phase portraits in the cases $0<a^{2}<4 b$ and $0<4 b<a^{2}$.

(iii) Sketch the phase portrait in the case $a=0$. Prove that periodic orbits exist if and only if $a=0$.