The Lorenz Butterfly

In the 1960's the American meteorologist Edward Lorenz (NOT Lorentz!) discovered, more or less by chance, that the mathematical models for the weather system were extremely sensitive to the initial conditions.
Small changes in these initial conditions result in big effects. Of course this had long been known as "popular wisdom"...:-), but now it was shown that this deterministic chaos was a result of the non-linearity of the underlying differential equations. As a result long-term weather predictions are impossible.
In 1972 he presented a talk with the title:

Predictability: Does the Flap of a Butterfly’s Wings in Brazil set off a Tornado in Texas?
This butterfly image has become a very popular one and you can find it in almost any of the hundreds of Internet sites about Chaos Theory. It is amusing to note that the webmasters often have been very creative in their geographical preferences, as can be seen from the list below.
My personal favourite is number 16, where the novelty of the sneezing butterfly is introduced.

1 With reference to Lorentz's strange attractor the famous statement on chaos says that the flapping of a butterfly's wings in Tokyo can cause a tornado in Texas.
2 Thus, a butterfly flapping its wings somewhere in Chicago may cause a tornado in Tokyo.
3 In other words, the flutter of a butterfly's wings in China could, in fact, actually effect weather patterns in New York City, thousands of miles away.
4 This is the cute little idea that a butterfly fluttering in Peking can cause storms to erupt in New York.
5 Can a butterfly stirring the air of Naupli today can transform storms in New York the next day or three months later?
6 The idea is that a butterfly flapping its wings in China could produce a tornado in Kansas.
7 This means that a butterfly in Japan can effect the wheather in the world.
8 The standard joke among chaologists is that a butterfly flapping its wings in the Amazon can influence a typhoon in the Pacific Ocean.
9 A butterfly flapping its wings in Turkey could change tornado patterns in Texas
10 This leads to the cliche that ``the flutter of a butterfly wing in Lima, Peru today can affect the weather in Toronto a month later.''
11 This effect became known as the Butterfly Effect - a butterfly in the USA may cause changes in air turbulance that could significantly effect weather in Asia
12 Dynamical instability is sometimes called "the butterfly effect": a butterfly flapping its wings in Brazil can cause a hurricane in Bengal.
13 A butterfly in Seattle could disrupt the weather in Cincinnati.
14 Edward Lorentz called this property the butterfly's effect, meaning that the butterfly moving its colourful wings over Peking in June can indirectly cause the disastrous hurricane over Argentina in September
15 This is sometimes referred to as the butterfly effect, i.e. a butterfly flapping its wings in South America can affect the weather in Central Park
16 The canonical example is the possibility of a butterfly's sneeze affecting the weather enough to cause a hurricane weeks later

In the applet below you can view the Lorenz attractor.

Click with the mouse inside the applet to start. What you see is a two-dimensional projection of the actual process (see below). Double-clicking will start two processes with a very small difference in the initial conditions. Watch how the two tracks diverge after some time. The tracks are circling around the two "eyes", but will never end there. The "eyes" are called Strange Attractors