Thursday, April 17, 2008

Edward Lorenz, 1917-2008

Edward Norton Lorenz was born in 1917, forecast the weather for the U.S. army during the war, and then he decided to construct accurate computer models to predict weather.

Instead, in 1960 or so, he tried to repeat a calculation and entered the same data as before. However, he got a completely different result. What was the reason? Was the computer broken? Not really. He found that the reason were tiny changes of the initial conditions that he didn't enter quite accurately.

A small change in the initial conditions may cause huge changes in the final results. I am actually unable to buy this story about the birth of chaos theory because it seems likely to me that people had to understand the instability and certainly non-integrability of most differential equations for quite some time. Euler presented his numerical method to solve differential equations back in 1768 and many numerical problems of this method were found soon.

Moreover, Henri Poincaré who was not completely unknown has dedicated a large portion of his career to something that could be called chaos theory, too. It is probably fair to say that Lorenz only rediscovered something that wasn't terribly popular at the time.

At any rate, Lorenz was able to generalize the lesson from his computer experiment and determined the cause of changing weather. No, it was not man. Instead, it was butterfly wings! :-) Here, I am talking about changing weather and not changing climate because the latter is effectively inconsequential. As Lorenz said in 1982,
"Climate is what you expect, weather is what you get".
From this point of view, climatology is a part of psychology because it studies people's expectations. For the same reason, catastrophic climate change science is a part of psychiatry. What you see is also weather because what you see is what you get. :-)

The history of the popular science of chaos is more mysterious than you might think. The discovery of the sensitivity of some systems of differential equations on tiny modifications of the initial conditions is called the butterfly effect. Why is it mysterious? One of the more technical things that Lorenz is famous for is the Lorenz attractor that you see on the picture. Recall that an attractor is a point or set of points into which a dynamical system evolves after a very long time.

What does the Lorenz attractor look like? Well, it looks like a butterly. ;-) The yellow set contains asymptotic points of a set of three first-order ordinary differential equations for x(t), y(t), z(t):
dx/dt = 10(y-x)
dy/dt = x(rho-z)-y
dz/dt = xy-8z/3.
Note that they are "minimally non-linear", employing bilinear functions of x,y,z on the right hand side. That's why they can be chosen as the simplest models of an unexpected behavior. Here, rho is a parameter - one parameter is sufficient to get qualitatively different types of behavior - while the conventional choices of other parameters have been explicitly substituted. For rho=28, the behavior gives the chaotic butterfly above but e.g. for rho=99.96, you obtain a knotted periodic orbit known as the T(3,2) torus knot, pictured on the left.

The Hausdorff dimension and the correlation dimension of the butterfly set are numbers close to 2.05. Because the dimension is not integer, the Lorenz attractor is an example of a strange attractor.

Evil apples and evil tigers in a 5D Lorentz attractor. Try also some "related videos".

Lorenz died yesterday in Cambridge, Massachusetts. I think I have met him during a dinner but I don't remember any details. See The New York Times.

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