In his awesome Discworld Series, Terry Pratchett often included twisted, hysterical versions of modern science and scientific myths. One of my favorites among these is the quantum weather butterfly. In this post we are going to talk about the famous butterfly effect and the odd notion of deterministic chaos as well as one of the biggest human misunderstandings of an obvious mathematical truth.
How did butterflies end up in this?
Let’s begin with the experiment that led to coining the term “butterfly effect”. It’s easy to see that, if you can predict the weather, you are in far better shape than if you cannot. Improved weather prediction, from better mathematical models of weather to orbiting satellites, are a reason that the number of people who died in the recent collection of hurricanes is small when compared to the historical statistics; far more people had enough warning to make it to shelter. Bottom line, there are a lot of reasons for wanting to predict the weather, from agriculture to civil defense.
Some time ago, a shiny new model of super computer with the most advanced weather model of the times was being tested. The researchers start with current conditions, as measured by hundreds of weather stations, and then run complex linked differential equation models of heat, moisture, wind velocity, air pressure, etc. into the future to predict the weather. The following experiment changed our view of the world. The researchers ran the model forward 12 weeks, reset everything, changed one wind velocity measurement by a tiny amount (the flap of a butterfly’s wing) and ran forward again. One of these runs ended with calm in Asia while the other predicted a huge typhoon.
This is an example of chaos.
What is chaos? A system is said to exhibit chaos if it depends in a very sensitive way on its starting points or, in math speak, if it exhibits sensitive dependence on initial conditions. An earlier Occupy Math, called Chaos on Purpose looked at logistic chaos.
Suppose we have an island where at most 1,000 goats can live. The measured reproductive rate of the goats on a giant open continent averages 3.8 goats per goat in the original population, per year. The growth model on the island limits goat reproduction on the island because food runs low. The goats will tend to overeat and later many goats starve, after which the plants recover. The population of goats jumps around a lot (most of the deaths take place during the lean winter season).
Running a logistic model experiment for these goats for 20 years tells us that an initial population of 500 goats leads to 809 goats, an initial population of 510 goats leads to 781 goats, an initial population of 520 goats leads to a population of 608 goats, and an initial population of 530 goats leads to a population 373 goats, all after 20 years have passed. Small changes in the initial population lead to bigger differences 20 years later. It also looks, from these four examples, like small increases in the initial population lead to much larger decreases in the 20-years-later populations, but this is more-or-less an illusion. All four experimental populations dropped below 200 and exceeded 900 in the course of the 20 years.
Why isn’t the variation in goats just randomness?
If these were real wild goats subject to predation by wolves and humans, with the ability to fall off a cliff occasionally, and with the potential to drown or die of exposure, then the numbers would involve a lot of randomness — but these were idealized mathematical goats! A single equation was used to compute next years population and it used only the number of goats this year for its input. There was no randomness included in the model at all. None. This means that really complicated jump-all-over-the-place behavior can arise in a dead-simple model with only one input. This shows that chaos (sensitive dependence on initial conditions) can arise in absurdly simple systems. Of course, it also arises in really complicated systems, like the weather.
If one butterfly can trigger a typhoon, can’t we control the weather?
No, not really. Chaotic systems amplify small differences into big differences, but they do not amplify all the small differences and the effect of a small difference remains unpredictable. You cannot stop the typhoon by swatting the responsible butterfly because you cannot figure out which of the ten million butterflies is responsible. It might even be several of them. On the other hand, a butterfly extermination program might kill the butterfly whose flap would have prevented the typhoon. This is the lesson: chaotic systems are unpredictable even in the absence of any randomness. They amplify differences that start below our ability to measure into unpredictable, large differences in the not-too-distant future. It’s important to note that the weather for the rest of the week is fairly predictable most of the time — the problem with the butterfly effect experiment was that twelve weeks is forever in “weather” time.
So it is hopeless?
Yes! Totally hopeless — if you think “hope” means that you can control everything. The weather is not totally impossible to deal with. People survive typhoons all the time, it just means you need well-built houses and shelters, emergency supplies, and up-to-date civil defense plans backed by regular drills. Chaos is the motivating example of this post, but the real target is the illusion of control. People naturally overestimate their degree of control over a situation.
In one of Dick Francis’ novels a character observes, “If you die in America, it’s your fault.” Americans obsess over exercise, nutrition, weight-loss, GMOs, chemicals, “bad” areas, and a dozen other things in a hundred counter-productive and self-deceiving ways. The idea that you might die of bad luck is not given much, if any, weight. This is a powerful example of the illusion of control. People think they can hold off death himself by correct action — and that they can attract him by failing to act correctly. Occupy Math is certain that you can improve your odds of living longer — but there is an unpredictable component that cannot be avoided.
What does this have to do with chaos?
Chaos is a big source of unpredictability, randomness is another. Finally, unpredictability arises from complexity. The world is a huge, complex place and its current state advances under the influence of chaos and randomness. The mathematical discipline of statistics and its good friend, the field of risk analysis, seek to find the practical limits of control. Oddly, the Boy Scouts nailed the most important point in their motto: Be Prepared.
Ethiopia has had a couple of really big famines in recent history. Before these events, famine was a good deal rarer. What made the difference? A government that firmly believed that centrally planning everything was the key to human happiness took over the country. This government also thought that the traditional beliefs of indigenous peoples were superstitions (well, some were) and that they were therefore worthless (D’oh! Not even close to correct). One of these traditional beliefs was that it was a really good idea to maintain a food reserve, e.g. grain in lined, dry, covered pits. Since central planning always screws up (it relies heavily on the illusion of control), there were crop failures not caused by anything other than mismanagement. The government opened the reserves to keep people fed and maintain its own popularity. A few years later a big drought hit and there were images of famine, death, and starving babies on Western news programs.
There are many examples that show the illusion of control is, potentially, deadly.
You cannot control the world or the future. You can influence it, you can prepare for it, and you should do both if you can. Training in mathematics initially makes you think you can predict everything. Early in the development of modern mathematics, during the Enlightenment, mathematical models started making some really good predictions. People got all excited and decided the universe was predictable — they coined the term clockwork universe for this belief. Shortly after that, both chaos and the true randomness that underlies quantum mechanics were discovered, ending any hope that the universe is a big clock. In the modern day, studying mathematics helps you to understand the limits of prediction and modeling; this in turn lets you get the most out of them.
This Occupy Math has presented a few examples and tried to explain that, while you don’t have control and will never have control, you do have choices and the ability to put aside reserves of time, money, food and other things. One of the best things to store against an uncertain future is the good will of other people. The reason for this is explored in an earlier Occupy Math on cooperation and its benefits. Do you know of examples where the illusion of control has led people astray? Please comment or tweet!
I hope to see you here again,
University of Guelph,
Department of Mathematics and Statistics