Chaos, Differential Equations and the Weather.

The reason that weather prediction is so difficult and forecasts are so erratic is no longer thought to be the complexity of the system but the nature of the differential equations modelling it. These DEs belong to a class reffered to as chaotic. Such equations will produce wildly different results when their initial conditions are changed infinitesimally. In other words, accurate weather prediction depends crucially on the accuracey of the measurements of the initial conditions.
Edward Lorenz, a research meteoroligist, discovered the phenemenon in 1961. Although his original equations are far to complex to be considered here, the following much simpler system has the same essential chaotic features:

dx/dt = 10(y - x)
dy/dt = -xz + 28x - y
dz/dt = xy - 8z/3.