Can You Specify the Chaos?

In mathematical terms, chaos describes situations that start out almost identically but while following the exact same processes, wind up in wildly different places.

One of the frequently used descriptions is "the butterfly effect," which shows hows something as small as the air disturbed when a butterfly flaps its wings can cascade into a storm on the other side of the planet. It means that the same air movements that fuel or even intensify storms are intimately interconnected and can be affected by even the smallest changes.

As this story at Science a GoGo points out, the measure of chaos is something called the Lyapunov exponent. If the Lyupanov exponent of an equation or system is positive, then that means that the very similar starting points will diverge widely over time. This doesn't even have to be that complicated. Think about two cars leaving from a parking lot at the same speed, with one traveling north and the other north by northwest. They begin right next to each other, but even though their paths differ by just 22.5 degrees, after just a few minutes they are already some distance apart.

But, as that same story notes, the Lyupanov exponent method tests for chaos in solutions to a model or an equation, rather than in the equation itself. In other words, current definitions don't offer a way to determine if a particular model is always going to produce chaotic solutions, unless it involves four or more preschoolers and a supermarket checkout candy display. That one will always be chaos.

The sneaky chaos models were ones that often involved what are called "forced systems," which are systems that tend to chaos but which continue to have external forces acting on them that might tend to reduce the level of the chaos. Water flowing through a pipe is an example. Suppose you had some water and you knew the position of every molecule of H2O in it. If you set that water flowing, it isn't very long before the math of predicting where the molecules will go gets so complicated as they bounce off each other and whatnot that you have a completely chaotic system. But water flowing through a pipe forces some limits on the system. While you may not know where any one molecule is going to be, you've got a pretty confined space for it to operate and so there are a lot of places it won't be.

So mathematicians at the University of Maryland who go by the name "The Chaos Group" decided on another definition, which would include those kinds of systems. If their suggestion takes hold, then it could offer scientists a good way of checking to see if a system responds to some level of control. One of the features of a completely chaotic system is that there is no way to anticipate every result of an input, as every change makes more changes, which make more changes, and so on.

The new definition and the possibility of controlling chaotic systems to some degree could lead to some interesting developments. But it probably won't mean controlling the weather, which is unfortunate for us but pretty good news for that stinkin' butterfly, since smashing him wouldn't help get rid of the storms.