Many of you know what the butterfly effect is and and have seen the movie with that same title, but its doubtful whether many people know the origins of the name. The butterfly effect is the idea that a single flap of a butterfly’s wings could have large scale effects across the world. To the extreme, could a butterfly in Brazil trigger a hurricane (or prevent one) in Texas with a single flap of its wings. This question was posed by Edward Lorenz and Philip Merilees, and the more scientific concepts they were addressing were: can tiny changes in the weather pattern cause drastic changes, and how does this relate to chaos.
Today we know that a key characteristic of chaotic systems is this high sensitivity to minute changes in the initial conditions. These changes can be seen in the the Lorenz Attractor. Try clicking various places in the window to start multiple attractors. You will notice that sometimes moving a few pixels wont have a significant affect on the created attractor, but in other instances moving a single pixel will result in completely different systems. The actually Lorenz attractor is a three dimension function:

and the sensitivity becomes more apparent when the medium better demonstrates this extra dimensionality.

The lorenz attractor is given by the following three differential equations:

Where the values of sigma, rho, and beta affect the how the systems behave.

The lorenz attractor was originally a mathematical model of the atmosphere. In this simplified model of the atmosphere the constants relate to various properties of the volume being modeled. y and z are related to temperature distributions throughout that volume where as x is related to convection.

The findings of Lorenz in the 60’s and 70’s are very relevant to modern scientific computing. They reaffirm what we already know, the tiniest change in a single value can perturb an entire system; every bit of data matters. Imagine how small the metaphorical butterfly is in comparison to the atmosphere surrounding earth. Its minute, and yet these tiny factors matter. This is one of the many issues that plague the field of forecasting. It seems likely that even as modeling and forecasting systems improve they will remain greatly handicapped for decades to come since tiny, near unmeasurable changes have the ability to be magnified enormously throughout the atmosphere.

For more information about Lorenz Attractors:
http://hypertextbook.com/chaos/21.shtml
http://en.wikipedia.org/wiki/Lorenz_attractor
http://mathworld.wolfram.com/LorenzAttractor.html