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Stability Matrix


Given a system of two ordinary differential equations

x^.=f(x,y)
(1)
y^.=g(x,y),
(2)

let x_0 and y_0 denote fixed points with x^.=y^.=0, so

f(x_0,y_0)=0
(3)
g(x_0,y_0)=0.
(4)

Then expand about (x_0,y_0) so

deltax^.=f_x(x_0,y_0)deltax+f_y(x_0,y_0)deltay+f_(xy)(x_0,y_0)deltaxdeltay+...
(5)
deltay^.=g_x(x_0,y_0)deltax+g_y(x_0,y_0)deltay+g_(xy)(x_0,y_0)deltaxdeltay+....
(6)

To first-order, this gives

 d/(dt)[deltax; deltay]=[f_x(x_0,y_0) f_y(x_0,y_0); g_x(x_0,y_0) g_y(x_0,y_0)][deltax; deltay],
(7)

where the 2×2 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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