Consider the dynamical system

$$\dot{x}=\mu x+x^{3}-a x y, \quad \dot{y}=\mu-x^{2}-y,$$

where $a$ is a constant.

(a) Show that there is a bifurcation from the fixed point $(0, \mu)$ at $\mu=0$.

(b) Find the extended centre manifold at leading non-trivial order in $x$. Hence find the type of bifurcation, paying particular attention to the special values $a=1$ and $a=-1$. [Hint. At leading order, the extended centre manifold is of the form $y=\mu+\alpha x^{2}+\beta \mu x^{2}+\gamma x^{4}$, where $\alpha, \beta, \gamma$ are constants to be determined.]