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) |
Matrix, or its generalization to higher dimension, is called the stability matrix. Analysisof the Eigenvalues (and Eigenvectors) of the stability matrix characterizes the type of Fixed Point.See also Elliptic Fixed Point (Differential Equations), Fixed Point, Hyperbolic Fixed Point(Differential Equations), Linear Stability,Stable Improper Node, Stable Node, Stable Spiral Point, Stable Star, Unstable ImproperNode, Unstable Node, Unstable Spiral Point, Unstable Star
References
Tabor, M. ``Linear Stability Analysis.'' §1.4 in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 20-31, 1989.
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