Given a system of two ordinary differential equations
let and denote fixed points with , so
Then expand about so
To first-order, this gives
|
(7) |
where the matrix, or its generalization to higher dimension, is called the stability matrix. Analysis of the eigenvalues (and eigenvectors) of the stability matrix characterizes the type of fixed point.
See also
Elliptic Fixed Point, Fixed Point, Hyperbolic Fixed Point, Linear Stability, Stable Improper Node, Stable Node, Stable Spiral Point, Stable Star, Unstable Improper Node, Unstable Node, Unstable Spiral Point, Unstable Star
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References
Tabor, M. "Linear Stability Analysis." ยง1.4 in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 20-31, 1989.
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Cite this as:
Weisstein, Eric W. "Stability Matrix." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/StabilityMatrix.html