which eigenvalue is the stable one

Question

I'm reading a textbook that says something along the lines of, given a function $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$. The Jacobian evaluated at $x_0$ has one eigenvalue smaller than one one eigenvalue greater than one. Then it says $[1 v]$ be the eigenvector that corresponds to the stable eignvalue. I'm confused now as to shich one is the stable one. Apparently the textbook assumes this is clear, but I'm not too familiar with dynamic systems so I'm not sure.

Answer 1

In general, the unstable manifold $E^u$ of a linear operator corresponds to all eigenvalues of modulus greater than 1, while the stable manifold $E^s$ corresponds to all eigenvalues smaller than 1 in modulus. Perturbations in the $E^s$ direction will decrease (as the eigenvalues are smaller than 1), while perturbations in the $E^u$ direction will increase--hence the names "stable" and "unstable."