\ShortSection {15}{CHAOS, CHANCE AND PREDICTABILITY}

\noindent The {\em Jacobian} $\vJ$ of a dynamical system 
$\dot \vx = \vf(\vx)$ is 
defined as $J_{ij} = \frac{\pa f_i}{\pa x_j}$, 
where $f_i$ and $x_j$ are the $i^{\mbox{\scriptsize th}}$ and 
$j^{\mbox{\scriptsize th}}$ components of $\vf(\vx)$ and 
$\vx$ respectively. The trace $\tau$ of the Jacobian is equal
to the sum of its eigenvalues, while the determinant $\Delta$ 
is equal to their product.

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