Explain the difference between a hyperbolic and a nonhyperbolic fixed point $\mathbf{x}_{0}$ for a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{n}$.

Consider the system in $\mathbb{R}^{2}$, where $\mu$ is a real parameter,

$$\begin{aligned} &\dot{x}=x\left(\mu-x+y^{2}\right) \\ &\dot{y}=y\left(1-x-y^{2}\right) \end{aligned}$$

Show that the fixed point $(\mu, 0)$ has a bifurcation when $\mu=1$, while the fixed points $(0, \pm 1)$ have a bifurcation when $\mu=-1$.

[The fixed point at $(0,-1)$ should not be considered further.]

Analyse each of the bifurcations at $(\mu, 0)$ and $(0,1)$ in turn as follows. Make a change of variable of the form $\mathbf{X}=\mathbf{x}-\mathbf{x}_{0}(\mu), \nu=\mu-\mu_{0}$. Identify the (non-extended) stable and centre linear subspaces at the bifurcation in terms of $X$ and $Y$. By finding the leading-order approximation to the extended centre manifold, construct the evolution equation on the extended centre manifold, and determine the type of bifurcation. Sketch the local bifurcation diagram, showing which fixed points are stable.

[Hint: the leading-order approximation to the extended centre manifold of the bifurcation at $(0,1)$ is $Y=a X$ for some coefficient a.]

Show that there is another fixed point in $y>0$ for $\mu<1$, and that this fixed point connects the two bifurcations.