Define the stable and unstable invariant subspaces of the linearisation of a dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ at a saddle point located at the origin in $\mathbb{R}^{n}$. How, according to the Stable Manifold Theorem, are the stable and unstable manifolds related to the invariant subspaces?

Calculate the stable and unstable manifolds, correct to cubic order, for the system

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