Stability Matrix

Given a system of two 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, or its generalization to higher dimension, is called the stability matrix. Analysis of the eigenvalues (and eigenvectors) of the stability matrix characterizes the type of fixed point.