Friday, August 06, 2004

Chaos theory

Chaos theory deals with the behaviour of certain nonlinear dynamical systems that, under the proper conditions, exhibit chaotic behavior -- that is, behavior which is characterised by long-term sensitivity to initial conditions. This means that the final outcome will vary tremendously, based on minor variations in the initial conditions. This has been observed in mathematics, economics, fluid dynamics, plate tectonics and a wide variety of other systems.

This is NOT the same as noise or modeling uncertainty. In a non-chaotic system ( e.g. a linear system), the variability in the outcome (if any) will depend on the degree of noise present. As a result, establishing bounds on the amount of noise in the initial conditions will likewise establish bounds on the final outcome variability. In a chaotic system, however, even the tiniest variation will be amplified to essentially unpredictable results. (See: Butterfly effect.)

One consequence of chaotic behavior is that even the most precise measuring instruments cannot be used to predict the final outcome with any degree of precision. Even with a perfect model of the system, finite measurement precision and computational round-off errors will combine to produce uncertain results.