Consider the general system of two first-order Ordinary Differential Equations
Let and denote Fixed Points with
Then expand about so
To first-order, this gives
where the Matrix is called the Stability Matrix.
In general, given an -D Map
, let be a Fixed Point, so that
Expand about the fixed point,
The map can be transformed into the principal axis frame by finding the Eigenvectors and
Eigenvalues of the Matrix A
so the Determinant
The mapping is
When iterated a large number of times,
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 , ..., .
See also Fixed Point, Stability Matrix
Tabor, M. ``Linear Stability Analysis.'' §1.4 in
Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 20-31, 1989.
© 1996-9 Eric W. Weisstein