It is a nonlinear system of three differential equations. With the most commonly used values of three parameters, there are two unstable critical points. The solutions remain bounded, but orbit chaotically around these two points. The program "lorenzgui" provides an app for investigating the Lorenz attractor.
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The Butterfly: Rotate, pan or zoom! Two butterflies starting at exactly the same position will have exactly the same path. There is nothing random in the system - it is deterministic.
Two butterflies that are arbitrarily close to each other but not at exactly the same position, will diverge after a number of times steps, making it impossible to predict the position of any butterfly after many time steps.
Any approximation, such as approximate measurements of real life data, will give rise to unpredictable motion. This behavior can be seen if the butterflies are placed at random positions inside a very small cube, and then watch how they spread out.
Press the "Small cube" button! It is certain that all butterflies will be on the attractor, but it is impossible to foresee where on the attractor. This is an example of deterministic chaos. The Lorenz attractor was first described in by the meteorologist Edward Lorenz. The thing that has first made the origin of the phrase a bit uncertain is a peculiarity of the first chaotic system I studied in detail. Here an abbreviated graphical representation of a special collection of states known as "strange attractor" was subsequently found to resemble a butterfly, and soon became known as the butterfly.
Before the Washington meeting I had sometimes used a sea gull as a symbol for sensitive dependence. The switch to a butterfly was actually made by the session convenor, the meteorologist Philip Merilees, who was unable to check with me when he submitted the program titles. Perhaps the butterfly, with its seemingly frailty and lack of power, is a natural choice for a symbol of the small that can produce the great.
Lorenz, , University of Washington Press, pp
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