I'm going through a book on Chaotic Dynamical Systems and have just read that to determine the stability of a fixed point of a 2-dimensional map, one need simply check whether the eigenvalues of the map's Jacobian are less than or greater than 1.
Granted, I don't remember as much as I should about the properties of eigenvalues from Linear Algebra, but I can't quite figure out what eigenvalues have anything at all to do with a fixed point's stability. Isn't an eigenvalue dealing with a function's transformation from a field to itself?
Imagine a system of the form
$$ x_{n + 1} = f(x_n), $$
where $x\in R^k$, and let $x_0$ be a fixed point
$$ x_0 = f(x_0) $$
Note that
$$ x_{n + 1} = f(x_n) = f((x_n - x_0) + x_0) \approx f(x_0) + J(x_0)(x_n - x_0) = x_0 + J_0(x_n - x_0) $$
where $J_0 = J(x_0)$ is the Jacobian evaluated at the fixed point. Rearranging
$$ x_{n + 1} - x_0 = J_0(x_n - x_0) $$
Let $y = x - x_0$, so that
$$ y_{n + 1} = J_0 y_n $$
whose solution is
$$ y_n = J_0^n y_0 $$
If $J_0 = M\Lambda M^{-1}$ with $\Lambda = {\rm diag}(\lambda_1, \lambda_2, \cdots)$ being the eigenvalues of $J_0$ then
$$ y_n = M\Lambda^n M^{-1}y_0 $$
this thing will diverge if the eigenvalues of $J_0$ are larger than one.
note: You may argue that this approach will break when the distance between $x_n$ and $x_0$ becomes large, that is when $y_n$ becomes large, but remember that this is only a local stability argument