© 29 Sep 2011 Luther Tychonievich
Licensed under Creative Commons: CC BY-NC-ND 3.0
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Revealing the infinitesimal character of things.


Consider a smoothly polished hill with a perfectly round ball placed exactly at the summit of the hill. This is an unstable equilibrium: an equilibrium since it is balanced, but unstable since the balance will be lost after even the slightest perturbation. A ball in a basin is in stable equilibrium: after perturbations it will return to its original state. Humans usually like stable equilibria, though there are examples of human-designed unstable equilibria like the iconic long stare that is the first part of a wild-west gun dual.

Consider our ball and hill in a fictitious world with no perturbations. Let’s say the ball is initially off from the summit of the hill, but so close to the summit we cannot measure how off it is. Since it isn’t at the peak it will eventually roll down the hill, and once it does we can determine where it must have been initially. The behavior of the system tells us something bout the initial state that we couldn’t measure initially. As it continues to roll we can tell what direction it is going with ever increasing precision.

Chaos is the mathematical term for systems that exhibit this kind of detail-magnifying behavior not only for a single initial condition but continually throughout their life.

One of the more intuitive examples of chaos is balls bouncing on a floor that oscillating up and down. When a ball hits an upward-moving floor it will bounce higher than when it hits a downward moving floor. Two almost-identical balls will hit at slightly different times, meaning they’ll receive slightly different bounces, meaning next time they’ll hit a bit more off from one another, and so on. This is displayed in the following animation; the three balls start at heights differing by less than a tenth of a pixel.

Click the graphic multiple times to speed up the simulation.

400,300, Please wait for canvas to load.

One of the interesting elements of chaotic systems is that they let you measure things you can’t initially measure. The vibrating table could eventually distinguish between balls that are arbitrarily close to being identical. In addition, it doesn’t matter when the difference is inserted into the system.

Probably the most-talked-about chaotic system is the weather. The chaos of weather is the origin of the phrase “‍butterfly effect‍”: even a single flap of a butterfly’s wings will eventually change the behavior of the weather as a whole. Of course, there are millions of butterflies out there adding oodles of uncertainty to all our models; but even if we had two all-but-identical planets differing only in the actions of one butterfly, after a sufficient time there would be no correlation between the two planet’s weather. It’s chaotic.

Chaos is often misunderstood. It isn’t random (though it can magnify the impact of small-scale randomness). It also isn’t a magic switch; even the most clever butterfly couldn’t give us perfect weather every year. The large-scale behavior of a chaotic system is beyond any small-scale control. You can tweak which bumps are big and which are small, but you can’t magically change the character of the system. A wise butterfly might be able to create one mild winter, but it would be surrounded by apparently-uncontrolled weather on either side.

The existence of chaotic systems makes the world much more complicated than it would otherwise be. Without chaos, it is quite easy to say that some approximation is “‍good enough.‍” With it, though, that which is infinitesimally small today may completely change tomorrow.

When the balls at the peak of the hill I can never tell you it’s position accurately enough, nor is any stir of air close enough to still. In chaotic systems, the ball is always on the peak of the hill.

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