Consider the system

$$\dot{x}=y, \quad \dot{y}=\mu_{1} x+\mu_{2} y-(x+y)^{3}$$

where $\mu_{1}$ and $\mu_{2}$ are parameters.

By considering a function of the form $V(x, y)=f(x+y)+\frac{1}{2} y^{2}$, show that when $\mu_{1}=\mu_{2}=0$ the origin is globally asymptotically stable. Sketch the phase plane for this case.

Find the fixed points for the general case. Find the values of $\mu_{1}$ and $\mu_{2}$ for which the fixed points have (i) a stationary bifurcation and (ii) oscillatory (Hopf) bifurcations. Sketch these bifurcation values in the $\left(\mu_{1}, \mu_{2}\right)$-plane.

For the case $\mu_{2}=-1$, find the leading-order approximation to the extended centre manifold of the bifurcation as $\mu_{1}$ varies, assuming that $\mu_{1}=O\left(x^{2}\right)$. Find also the evolution equation on the extended centre manifold to leading order. Deduce the type of bifurcation, and sketch the bifurcation diagram in the $\left(\mu_{1}, x\right)$-plane.