Consider the system

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

where $\mu$ and $\beta$ are constants with $\beta>0$.

(a) Find the fixed points, and classify those on $y=0$. State how the number of fixed points depends on $\mu$ and $\beta$. Hence, or otherwise, deduce the values of $\mu$ at which stationary bifurcations occur for fixed $\beta>0$.

(b) Sketch bifurcation diagrams in the $(\mu, x)$-plane for the cases $0<\beta<1, \beta=1$ and $\beta>1$, indicating the stability of the fixed points and the type of the bifurcations in each case. [You are not required to prove that the stabilities or bifurcation types are as you indicate.]

(c) For the case $\beta=1$, analyse the bifurcation at $\mu=-1$ using extended centre manifold theory and verify that the evolution equation on the centre manifold matches the behaviour you deduced from the bifurcation diagram in part (b).

(d) For $0<\mu+1 \ll 1$, sketch the phase plane in the immediate neighbourhood of where the bifurcation of part (c) occurs.