Linear Stability

Consider the general system of two first-order Ordinary Differential Equations

 (1) (2)

Let and denote Fixed Points with , so
 (3) (4)

 (5) (6)

To first-order, this gives
 (7)

where the Matrix is called the Stability Matrix.

In general, given an -D Map , let be a Fixed Point, so that

 (8)

 (9)

so
 (10)

The map can be transformed into the principal axis frame by finding the Eigenvectors and Eigenvalues of the Matrix A
 (11)

so the Determinant
 (12)

The mapping is
 (13)

When iterated a large number of times,
 (14)

only if for , ..., but if any . Analysis of the Eigenvalues (and Eigenvectors) of A therefore characterizes the type of Fixed Point. The condition for stability is for , ..., .

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