Consider the system

$$\dot{x}=-2 a x+2 x y, \quad \dot{y}=1-x^{2}-y^{2}$$

where $a$ is a constant.

(a) Find and classify the fixed points of the system. For $a=0$ show that the linear classification of the non-hyperbolic fixed points is nonlinearly correct. For $a \neq 0$ show that there are no periodic orbits. [Standard results for periodic orbits may be quoted without proof.]

(b) Sketch the phase plane for the cases (i) $a=0$, (ii) $a=\frac{1}{2}$, and (iii) $a=\frac{3}{2}$, showing any separatrices clearly.

(c) For what values of a do stationary bifurcations occur? Consider the bifurcation with $a>0$. Let $y_{0}, a_{0}$ be the values of $y, a$ at which the bifurcation occurs, and define $Y=y-y_{0}, \mu=a-a_{0}$. Assuming that $\mu=O\left(x^{2}\right)$, find the extended centre manifold $Y=Y(x, \mu)$ to leading order. Further, determine the evolution equation on the centre manifold to leading order. Hence identify the type of bifurcation.