Consider the dynamical system $\dot{\mathbf{x}}=\mathbf{f}(\mathbf{x})$ in $\mathbb{R}^{n}$ which has a hyperbolic fixed point at the origin.

Define the stable and unstable invariant subspaces of the system linearised about the origin. Give a constraint on the dimensions of these two subspaces.

Define the local stable and unstable manifolds of the origin for the system. How are these related to the invariant subspaces of the linearised system?

For the system

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

calculate the stable and unstable manifolds of the origin, each correct up to and including cubic order.