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Linear Stability

Consider the general system of two first-order Ordinary Differential Equations

$\displaystyle \dot x$ $\textstyle =$ $\displaystyle f(x,y)$ (1)
$\displaystyle \dot y$ $\textstyle =$ $\displaystyle g(x,y).$ (2)

Let $x_0$ and $y_0$ denote Fixed Points with $\dot x=\dot y=0$, so
$\displaystyle f(x_0,y_0)$ $\textstyle =$ $\displaystyle 0$ (3)
$\displaystyle g(x_0,y_0)$ $\textstyle =$ $\displaystyle 0.$ (4)

Then expand about $(x_0,y_0)$ so
$\displaystyle \delta\dot x$ $\textstyle =$ $\displaystyle f_x(x_0,y_0)\delta x+f_y(x_0,y_0)\delta y+f_{xy}(x_0,y_0)\delta x\delta y+\ldots$  
$\displaystyle \delta\dot y$ $\textstyle =$ $\displaystyle g_x(x_0,y_0)\delta x+g_y(x_0,y_0)\delta y+g_{xy}(x_0,y_0)\delta x\delta y+\ldots.$  

To first-order, this gives
{d\over dt}\left[{\matrix{\delta x\cr \delta y}}\right] = \l...
\left[{\matrix{\delta x\cr \delta y\cr}}\right],
\end{displaymath} (7)

where the $2\times 2$ Matrix is called the Stability Matrix.

In general, given an $n$-D Map ${\bf x}'=T({\bf x})$, let ${\bf x}_0$ be a Fixed Point, so that

T({\bf x}_0) = {\bf x}_0.
\end{displaymath} (8)

Expand about the fixed point,
$\displaystyle T({\bf x}_0+\delta{\bf x})$ $\textstyle =$ $\displaystyle T({\bf x}_0)+{\partial T\over\partial{\bf x}}\delta{\bf x}+{\mathcal O}(\delta {\bf x})^2$  
  $\textstyle \equiv$ $\displaystyle T({\bf x}_0)+\delta T,$ (9)

\delta T={\partial T\over\partial{\bf x}}\delta{\bf x}\equiv{\hbox{\sf A}}\,\delta{\bf x}.
\end{displaymath} (10)

The map can be transformed into the principal axis frame by finding the Eigenvectors and Eigenvalues of the Matrix A
({\hbox{\sf A}}-\lambda{\hbox{\sf I}})\,\delta{\bf x} = {\bf0},
\end{displaymath} (11)

so the Determinant
\vert{\hbox{\sf A}}-\lambda{\hbox{\sf I}}\vert = 0.
\end{displaymath} (12)

The mapping is
\delta{{\bf x}_{\rm princ}}' = \left[{\matrix{\lambda_1 & \c...
...ots & \ddots & \vdots \cr 0 & \cdots & \lambda_n \cr}}\right].
\end{displaymath} (13)

When iterated a large number of times,
\delta T_{\rm princ}'\to 0
\end{displaymath} (14)

only if $\vert\Re(\lambda_i)\vert<1$ for $i=1$, ..., $n$ but $\to\infty$ if any $\vert\lambda_i\vert>1$. Analysis of the Eigenvalues (and Eigenvectors) of A therefore characterizes the type of Fixed Point. The condition for stability is $\vert\Re(\lambda_i)\vert<1$ for $i=1$, ..., $n$.

See also Fixed Point, Stability Matrix


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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© 1996-9 Eric W. Weisstein