Find the fixed points of the dynamical system (with $\mu \neq 0$ )

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

and determine their type as a function of $\mu$.

Find the stable and unstable manifolds of the origin correct to order $4 .$