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.

REFERENCES:

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