While googling for something else, I stumbled on the fact the Butterfly Effect was so-named by a meteorologist who, in 1961, stumbled on something which surprised him…but I find surprisingly stupid (to be a surprise to one in the sciences).
The story goes that this meteorologist was running some modeling equation, and his vintage 1961 compute spit out sheets and sheets of numeric solutions carried out to 6 decimal places. But, it ran slow, so he ran the program again using only 3 decimal places. Also, to save time, he started the calc (of many iterations) in the middle using one row of (now truncated) values from the original set of outputs. He was surpised how different the results were, and he posed “Could the Flap of a Butterfly’s Wing Trigger a Tornado in Texas?”
Well, this meteorologist missed the obvious! Surely, the concept of mathematical error and sig figs existed in 1961! Why should this have surprised him? It’s like following an angle of 29.9888 dgrs instead of a true 30.0000 dgr angle for a mile…and you’ll see just how far off the mark you are!
So, shouldn’t the meteorologist have recognized the “error” in his surprise reaction? - Jinx
They thought there were important variables and insignificant ones, and if they could figure which were the important ones and measure them accurately, they could predict the weather to a reasonable degree. This technique works just fine in most fields, but not in chaotic ones like the weather.
This has nothing whatsoever to do with not knowing significant figures. It’s a problem common to all iterative equations, ones in which you plug in the result to use as the source of the next round of calculations. Using different starting conditions, which can be expressed as a varying number of significant figures, will create unpredictable results. I.e. more precision in the starting conditions does not help you to a better understanding of the final results. These are random (or quasi-random). This was the genesis of chaos theory.
If that wasn’t the page you saw, I recommend reading the whole thing. It’s a good starting point into a vastly complex subject.
The “butterfly” does not “cause” the tornado as such. But the researcher is correct in showing that when modelling something like the weather even missing something as insignificant a a butterfly flapping its wings in the model will, with enough iterations, introduce enough error that they miss predicting a tornado in Texas (assuming they could otherwise expect perfect weather prediction).
Like SmackFu mentioned this lies at the heart of modelling chaotic systems.
The entire thing about chaotic systems is that the error increases super-linearly. So say you made a measurement of 29.9, then you might get it to run for 5 days with an acceptable amount of deviation. Now say you measured it to 29.999, you might get 8 days. With 29.99999999 you might get 11 days and with 29.999999999999999 you might get 14 days. No matter how accurate you try and go, you still get a pathetically short range prediction.
The story is real. The meteorologist E. N. Lorenz reports his findings in two 1963 papers: Deterministic non-periodic flow in Journal of Atmospheric Science, 20, 130-141; and The predictability of hydrodynamic flow in Trans. NY Acad. Sci., Series II, 25, 409-432. The title of his 1972 talk, “Predictability: does the flap of a butterfly’s wings in Brazil set off a tornado in Texas?”, lends an air of credibility to the “whimsical description”, even if that description had literary precedent, e.g., Ray Bradbury’s short story A Sound of Thunder.
With some chaotic systems, it isn’t just the iteration that makes them unpredictable, but that the differences in starting conditions result in an entirely different set of interactions occurring in entirely different systems. A good analogy of this (or I thought so, anyway) is rainfall on watersheds in the Andes; there are locations where a raindrop will find its way into the Pacific ocean or the Atlantic, depending on whether it falls here or a few feet to the west.
Having read Glieck’s book (many times, uif you include the audio version), I recall the description in there. He stresses that the problem, as noted above, isn’t the mere propagation of error because of rounding, but the enormous, disproportionate change duer to minute differences in starting conditions – the term used is sensitive dependence on initial conditions. The point wasn’t that small differences in the starting points caused errors – he expected that, as Gleickl’s book makes clear. It’s that, after a few cycles the system evolved in ways completely different from the originasl development. This is a result outside the experience of classical beghavior. The error doesn’t merely propagate, but completely changes the result in a fairly short time. It’s the combination of nonlinear systems (which cause such errors to escalate) and self-referential algorithms. As pointed out in the book, it pretty much dooms the dream of weather control, which had hoped to be able to topple weather systems at strategically chosen “branch points” and force the weather system into one or another relatively stable states. One thing that chaos theory makes clear is that such stable states don’t exist, and tiny changes – like the breeze caused by the flap of a butterfly’s wings – could cause enormous, practically unpredictable changes in outcome.
The Bradbury story didn’t have to do with this effect, which hadn’t been discovered yet. It was that even killing a butterfly in the distant past could have ramifications in the present. It’s sdtrictly fortuitous that a butterfly was involved in both metaphors. Bradbury could’ve used a cockroach in his story, for the effect it had. But it’s not typical of Bradbury’s imagery. Bradbury didn’t call this “the butterfly effect”, but the movie of that name is clearly invoking Bradbury’s effect, rather than the Chaos effect.
This is true, but it is important to note that just because a system is non-linear, chaotic, highly perturbative, et cetera doesn’t mean that it is unstable, i.e. prone to cycling in an out-of-control, self-destructive manner. People often conflate the notion of chaotic systems–those which are, by their nature, unpredictable–with inherent instability, perhaps in part because of the (entirely coincidental) title of the afformentioned Ray Bradbury story and the groundbreaking work Lorentz (Edward, not the unrelated physicist Hendrik) did on his eponymous attractors. A system can be increasingly nondeterministic in it’s “final” state but yet remain within very exact boundaries. A single butterfly, or even a million, flapping as hard as they can aren’t going to cause a hurricane; a system so sensitive (and with such an implied high energy state) as to react to such a small input in such a dramatic way is likely to become unstable regardless. However, it is virtually impossible to predict the paths of the air molecules disturbed by the beating of a single butterfly’s wings, even over a very short time span, owing to the turbulence of air at the wing tips. Chaotic, but not unstable.
[post=6529072]Here[/post]'s an old thread where I got kind of cranky on the topic of “the butterfly effect” and misapprehensions thereof.
I’d expect them to be unrelated, since they have different last names. The mathematician is Lorenz, not Lorentz.
Oh, and “farse” is in fact a valid word according to the OED. Though what the butterfly effect has to do with “the hortatory or explanatory passages in the vernacular interpolated between the Latin sentences in chanting the lesson or epistle” is beyond me.
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