Consider the dynamical system

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

where $a>0$.

Find the fixed points of the dynamical system. Show that for any fixed value of $a$ there exist three values $b_{1}>b_{2} \geqslant b_{3}$ of $b$ where a bifurcation occurs. Show that $b_{2}=b_{3}$ when $a=1 / 2$.

In the remainder of this question set $a=1 / 2$.

(i) Being careful to explain your reasoning, show that the extended centre manifold for the bifurcation at $b=b_{1}$ can be written in the form $X=\alpha Y+\beta \mu+p(Y, \mu)$, where $X$ and $Y$ denote the departures from the values of $x$ and $y$ at the fixed point, $b=b_{1}+\mu, \alpha$ and $\beta$ are suitable constants (to be determined) and $p$ is quadratic to leading order. Derive a suitable approximate form for $p$, and deduce the nature of the bifurcation and the stability of the different branches of the steady state solution near the bifurcation.

(ii) Repeat the calculations of part (i) for the bifurcation at $b=b_{2}$.

(iii) Sketch the $x$ values of the fixed points as functions of $b$, indicating the nature of the bifurcations and where each branch is stable.