Describe informally the concepts of extended stable manifold theory. Illustrate your discussion by considering the 2-dimensional flow

$$\dot{x}=\mu x+x y-x^{3}, \quad \dot{y}=-y+y^{2}-x^{2},$$

where $\mu$ is a parameter with $|\mu| \ll 1$, in a neighbourhood of the origin. Determine the nature of the bifurcation.