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Chaos: lethal butterflies (bugs) in computers

April 6, 2013

After talking about chaos in a previous post, I’m finally getting around to reading the 1988 bestseller Chaos by James Gleick. The first chapter is named “The Butterfly Effect”. Before I complain about Gleick’s chapter title (which inspired the title of my post), let me point out some great items that Gleick includes which I didn’t know about.

Gleick can tell a good story, and the Lorenz story seems to be a real life Butterfly Effect (which Gleick ironically doesn’t seem to point out, but anyway…). Many people may know how Lorenz stumbled onto chaos theory by typing the number 0.506 instead of 0.506127. The history prior to that is just as interesting, I think. For example, he started his career as a mathematician, but World War II apparently sucked him into meteorology where he began to study weather forecasting. What are the chances, huh?

The other part of the first chapter that was fun to read was the historical aspect of the computer hardware. As I’ll discuss shortly, Lorenz created a controversy by claiming that mathematics (and particularly computers) are incapable of predicting the weather for longer than was currently possible. Gleick makes an interesting observation that John von Neumann was famous for bringing computers to bear on weather forecasting. So it is interesting to me that Lorenz so quickly discovered a theoretical limitation as well.

I also enjoyed thinking about the actual hardware that Lorenz used. Gleick refers to Lorenz as having used a Royal McBee. That was the company name, and the actual model used by Lorenz apparently was the LGP-30. Among its interesting features were an oscilloscope-based numerical display and a laborious booting procedure involving paper tape! Now, I’m just old enough to have actually used a teletype interface. So it was fun to learn that Lorenz was creating graphs of his output by printing a single character on each row of the streaming teletype output. However, what really makes Lorenz heroic, in my opinion, is that he managed to work through his findings so carefully during a time when, as Gleick points out, numerical error was the first explanation that came to everyone’s mind when confronted with Lorenz’s results.

Finally, I want to vent my frustration about two things. The first is a minor issue: Gleick refers to a mechanical analogy of chaos that he calls the “Lorenzian Waterwheel”. The waterwheel was conceived and developed by Willem Malkus, not Lorenz (search YouTube for “chaos waterwheel” to see plenty). Ironically, Gleick actually talks about Malkus but does not give him credit for the waterwheel!

My second, and biggest, frustration relates to Gleick’s title for the first chapter: “The Butterfly Effect”. My main complaint is that he may be largely responsible for a horrible misuse of the term in popular culture. Maybe you’ve seen the 2004 movie “The Butterfly Effect”, or you remember the chaos expert named Ian Malcom in the 1993 movie “Jurassic Park”. The term was made famous by Lorenz’s own 1972 presentation paper titled “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?”  (I found a PDF here) Here is Lorenz’s point: long range weather prediction is impossible because we can’t measure the entire globe with enough resolution. I believe Gleick understands the point, but I don’t think he clearly explained why Lorenz used the analogy. The butterfly effect is not an analogy of cause and effect. Instead it’s an analogy of the opposite: how cause and effect may be impossible to determine. Below I have a quote from the original presentation, but first I’ll explain the misunderstanding. The butterfly is only one of countless variables. It has no significance by itself because the entire system of variables is necessary for the outcome to occur. The seemingly monumental implication of cause and effect is just an illusion that results from focusing on a tiny piece of the puzzle (the butterfly) and ignoring the rest (the weather throughout all of Earth). Lorenz is equally famous for finding an example of unpredictability that has just a few variables (literally only 3), but the relationship between those variables is not at all disparate or shocking.

Also frustrating is how the butterfly analogy has become entangled with the concept of sensitivity to initial conditions. Gleick seems to have set a precedent for this by literally declaring that they are the same. Certainly they are related: if the butterfly didn’t flap its wings, the outcome might be different, but that wasn’t the whole of Lorenz’s point. The real intent of the analogy was that a system that large, with that many variables and that broad range of scale, is beyond our practical ability to predict. Maybe Ashton Kutcher actually understood this in his movie “The Butterfly Effect”! After reading Gleick, I discovered that Peter Dizikes wrote a commentary in 2008 for the Boston Globe titled “The meaning of the butterfly: Why pop culture loves the ‘butterfly effect,’ and gets it totally wrong”. Apparently I’m not the only one who finds the term “butterfly effect” to be misused. It’s not unlike how the term “chaos” is also misused in popular culture.

So that you can judge for yourself, here is an excerpt from Lorenz’s 1972 presentation: “Here generally, I am proposing that over the years minuscule disturbances neither increase nor decrease the frequency of occurrence of various weather events such as tornados; the most that they may do is to modify the sequence in which these events occur. The question which really interests us is whether they can do even this–whether, for example, two particular weather situations differing by as little as the immediate influence of a single butterfly will generally after sufficient time evolve into two situations differing by as much as the presence of a tornado. In more technical language, is the behavior of the atmosphere unstable with respect to perturbations of small amplitude? …Since we do not know exactly how many butterflies there are, nor where they are all located, let alone which ones are flapping their wings at any instant, we cannot, if the answer to our question is affirmative, accurately predict the occurrence of tornados at a sufficiently distant future time. Here significantly, our general failure to detect systems even as large as thunderstorms when they slip between weather stations may impair our ability to predict the general weather pattern even in the near future.”

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