(i) Define a hyperbolic fixed point $x_{0}$ of a flow determined by a differential equation $\dot{x}=f(x)$ where $x \in R^{n}$ and $f$ is $C^{1}$ (i.e. differentiable). State the Hartman-Grobman Theorem for flow near a hyperbolic fixed point. For nonlinear flows in $R^{2}$ with a hyperbolic fixed point $x_{0}$, does the theorem necessarily allow us to distinguish, on the basis of the linearized flow near $x_{0}$ between (a) a stable focus and a stable node; and (b) a saddle and a stable node? Justify your answers briefly.

(ii) Show that the system:

$$\begin{aligned} &\dot{x}=-(\mu+1)+(\mu-3) x-y+6 x^{2}+12 x y+5 y^{2}-2 x^{3}-6 x^{2} y-5 x y^{2} \\ &\dot{y}=2-2 x+(\mu-5) y+4 x y+6 y^{2}-2 x^{2} y-6 x y^{2}-5 y^{3} \end{aligned}$$

has a fixed point $\left(x_{0}, 0\right)$ on the $x$-axis. Show that there is a bifurcation at $\mu=0$ and determine the stability of the fixed point for $\mu>0$ and for $\mu<0$.

Make a linear change of variables of the form $u=x-x_{0}+\alpha y, v=x-x_{0}+\beta y$, where $\alpha$ and $\beta$ are constants to be determined, to bring the system into the form:

$$\begin{aligned} &\dot{u}=v+u\left[\mu-\left(u^{2}+v^{2}\right)\right] \\ &\dot{v}=-u+v\left[\mu-\left(u^{2}+v^{2}\right)\right] \end{aligned}$$

and hence determine whether the periodic orbit produced in the bifurcation is stable or unstable, and whether it exists in $\mu<0$ or $\mu>0$.