The applet to the right shows the "Lorenz attractor" being created by taking randomly distributed points and following their evolution under the Lorenz equations. Newly introduced points are red, they turn blue as they "age".

Lorenz considered the relatively harmless looking coupled non-linear differential equations.

One commonly used set of constants is s = 10, b = 8 / 3, r = 28; this produces the "Lorenz attractor" shown in the applet above.

If it were not for the two non-linear terms (xz in the second equation and xy in the third), the complete set of solutions would be expressible using at most one natural frequency. Lorenz found solutions that are nonperiodic; they cannot be represented using any finite number of frequencies. These solutions are also sensitively dependent on initial conditions, which means that for all practical purposes, prediction of the state of the system is limited to relatively short times. Moreover, regardless of the initial conditions, all solutions are attracted to the same strange attractor, which resembles a surface with two wings; it is, in fact, a fractal.

The sensitive dependence of the evolution of a system to the most infinitesimal changes of initial state is known as the butterfly effect, after the title of a talk by Lorenz:

Predictability: Does the flap of a Butterfly's Wings in Brazil set off a Tornado in Texas?

It raises the question: if Lorenz' equations do not allow long time prediction, why should more complicated dynamical models of the atmosphere do any better?

These pages are hosted by Mathematical Sciences Institute, ANU, and maintained by Brian Davies .

e-mail to Brian.Davies@anu.edu.au
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