Given a system of two ordinary differential equations
let fixed points with
Then expand about
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
Explore with Wolfram|Alpha
References Tabor, M. "Linear Stability Analysis." §1.4 in Chaos
and Integrability in Nonlinear Dynamics: An Introduction. Referenced on Wolfram|Alpha Stability Matrix
Cite this as:
Weisstein, Eric W. "Stability Matrix."
From MathWorld https://mathworld.wolfram.com/StabilityMatrix.html
Subject classifications
and 
denote fixed points with 
, so
so
![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],](/images/equations/StabilityMatrix/NumberedEquation1.svg)
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.
