Linear Stability

Consider the general system of two first-order ordinary differential equations

Let and denote fixed points with , so

Then expand about so

To first-order, this gives

where the matrix is called the stability matrix.

In general, given an -dimensional map , let be a fixed point, so that

Expand about the fixed point,

so

The map can be transformed into the principal axis frame by finding the eigenvectors and eigenvalues of the matrix

so the determinant

The mapping is

When iterated a large number of times, only if for all , but if any . Analysis of the eigenvalues (and eigenvectors) of therefore characterizes the type of fixed point.