Chaos Squared

By Robert Naeye|Tuesday, March 01, 1994
Ordinary chaos makes some things unpredictable. A new kind of chaos is even more insidious: it makes some experiments unrepeatable.

The special status of scientific knowledge comes from its testability, and the essence of scientific testability is reproducibility: if a researcher performs an experiment, the same person or someone else should be able to duplicate the results by performing the same experiment in exactly the same way. But two physicists working in the emerging science of chaos have cast doubt on this age-old assumption. Certain experiments simply cannot be reproduced, say John Sommerer of Johns Hopkins and Edward Ott of the University of Maryland, because their starting conditions cannot be re-created perfectly--and even slight imperfections can change the course of the experiment.

Sommerer and Ott found this effect by analyzing the behavior of a hypothetical particle moving in a bowl-shaped force field. The experiment could be done in various ways, but you can picture it as a marble careering around a real bowl. Rather than do the experiment physically, Sommerer and Ott analyzed its mathematical description, which turns out to be very simple. They simulated many repetitions of the experiment on a computer. Their simulated particle faced two possible types of outcome: it could either rattle around the bottom of the bowl-shaped field, or it could zoom out the top. Which path the particle took depended only on its initial position and velocity.

Indeed, Sommerer and Ott found that the outcome depended on the exact initial placement of the particle. If the particle starts at position X with velocity V, it will remain trapped in the bowl. But even the most meticulous experimenter trying to repeat the trial by placing the particle in the same spot and giving it the same velocity might find the particle flying out of the bowl. You can never get exact precision in reproducing the initial conditions, and any error, no matter how small, can lead to different outcomes, explains Sommerer. Your experiment is vulnerable to all the surroundings, even somebody walking by next door or the position of the moon. All these influences can add up to infinitesimal perturbations that you can’t measure, so you think you’re doing the same thing every time. But it’s enough to perturb the system to a completely different outcome.

Physical systems that are sensitive to initial conditions are old hat now; they’re called chaotic. But what Sommerer and Ott have found is a much more extreme form of chaos. Although it is impossible to predict the evolution of an ordinary chaotic system in detail--the weather is the standard example--it is possible to say that it will evolve toward an attractor: a collection of states that fall within certain bounds. In the case of the weather, the attractor is Earth’s climate; meteorologists can’t predict every detail of the weather, but they can be confident it will not suddenly turn Venusian, with temperatures soaring to 850 degrees.

In Sommerer and Ott’s experiment, however, there are two attractors, one consisting of various trajectories inside the bowl, the other of trajectories that lead outside it. And what Sommerer and Ott have discovered is that it is impossible to forecast not only the precise trajectory the particle will take, as in ordinary chaos, but even which attractor it will choose--whether it will remain in the bowl or fly away. Theirs is the first physical system known to behave in this way.

If all systems exhibited such behavior, human beings never would have developed science. To experience the effect, though, a system has to be chaotic to begin with, and it must also exhibit one of several kinds of symmetry, such as the spatial symmetry of a bowl-shaped force field. This is certainly not the most common way for systems to behave, says Sommerer. But it doesn’t look like it’s by any means completely pathological. It may have been observed in the past and just written off as an error on the part of the experimenter.

This means there are some situations in which applying the usual scientific method is going to lead to an awful lot of frustration, he goes on. You may ask what this knowledge is good for. It’s like asking what smallpox is good for. It’s not good for anything; it’s bad. But it’s very important to know about it because if you run into it, you’re in big trouble.
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