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.

REFERENCES:

Tabor, M. "Linear Stability Analysis." §1.4 in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 20-31, 1989.