Linear Stability
المؤلف:
Tabor, M.
المصدر:
"Linear Stability Analysis." §1.4 in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley
الجزء والصفحة:
pp. 20-31
9-10-2021
1236
Linear Stability
Consider the general system of two first-order ordinary differential equations
Let 
and 
denote fixed points with 
, so
Then expand about 
so
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],](https://mathworld.wolfram.com/images/equations/LinearStability/NumberedEquation1.gif) |
(7)
|
where the 
matrix is called the stability matrix.
In general, given an 
-dimensional map 
, let 
be a fixed point, so that
 |
(8)
|
Expand about the fixed point,
so
 |
(11)
|
The map can be transformed into the principal axis frame by finding the eigenvectors and eigenvalues of the matrix 
 |
(12)
|
so the determinant
 |
(13)
|
The mapping is
 |
(14)
|
When iterated a large number of times, 
only if ![R[lambda_i]<0](https://mathworld.wolfram.com/images/equations/LinearStability/Inline35.gif)
for all 
, but 
if any ![R[lambda_i]>0](https://mathworld.wolfram.com/images/equations/LinearStability/Inline38.gif)
. Analysis of the eigenvalues (and eigenvectors) of 
therefore characterizes the type of fixed point.
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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