Linear Stability

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

$$\dot x = f(x,y)$$ (1)

$$\dot y = g(x,y).$$ (2)

Let $x_0$ and $y_0$ denote Fixed Points with $\dot x=\dot y=0$, so

$$f(x_0,y_0) = 0$$ (3)

$$g(x_0,y_0) = 0.$$ (4)

Then expand about $(x_0,y_0)$ so

$$\delta\dot x = f_x(x_0,y_0)\delta x+f_y(x_0,y_0)\delta y+f_{xy}(x_0,y_0)\delta x\delta y+\ldots$$ (5)

$$\delta\dot y = g_x(x_0,y_0)\delta x+g_y(x_0,y_0)\delta y+g_{xy}(x_0,y_0)\delta x\delta y+\ldots.$$ (6)

To first-order, this gives

$${d\over dt}\left[{\matrix{\delta x\cr \delta y}}\right] = \l... ...\cr}}\right] \left[{\matrix{\delta x\cr \delta y\cr}}\right],$$ (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

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

Expand about the fixed point,

$$T({\bf x}_0+\delta{\bf x}) = T({\bf x}_0)+{\partial T\over\partial{\bf x}}\delta{\bf x}+{\mathcal O}(\delta {\bf x})^2$$

$$\equiv T({\bf x}_0)+\delta T,$$ (9)

so

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

The map can be transformed into the principal axis frame by finding the Eigenvectors and Eigenvalues of the Matrix A

$$({\hbox{\sf A}}-\lambda{\hbox{\sf I}})\,\delta{\bf x} = {\bf0},$$ (11)

so the Determinant

$$\vert{\hbox{\sf A}}-\lambda{\hbox{\sf I}}\vert = 0.$$ (12)

The mapping is

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

When iterated a large number of times,

$$\delta T_{\rm princ}'\to 0$$ (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

References

Tabor, M. ``Linear Stability Analysis.'' §1.4 in Chaos and Integrability in Nonlinear Dynamics: An Introduction. New York: Wiley, pp. 20-31, 1989.