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." §1.4 in Chaos
and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley,
pp. 20-31, 1989. Referenced on Wolfram|Alpha Stability Matrix
Cite this as:
Weisstein, Eric W. "Stability Matrix."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/StabilityMatrix.html
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