Consider the general system of two first-order Ordinary Differential Equations
Let and denote Fixed Points with
, so
Then expand about so
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) |
Expand about the fixed point,
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 , ..., .
See also Fixed Point, Stability Matrix
References
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
1999-05-25