Linear Stability

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

			(1)
			(2)

Let and denote fixed points with , so

			(3)
			(4)

Then expand about so

			(5)
			(6)

To first-order, this gives

(7)

where the matrix is called the stability matrix.

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

(8)

Expand about the fixed point,

			(9)
			(10)

so

(11)

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

(12)

so the determinant

(13)

The mapping is

(14)

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.