Consider the system

$$\begin{aligned} &\dot{x}=y+x y \\ &\dot{y}=x-\frac{3}{2} y+x^{2} \end{aligned}$$

Show that the origin is a hyperbolic fixed point and find the stable and unstable invariant subspaces of the linearised system.

Calculate the stable and unstable manifolds correct to quadratic order, expressing $y$ as a function of $x$ for each.