Toying with a touch of chaos.

Toying with a touch of chaos

Gyrating like a stiff but daring gymnast, the Space Ball moves in mysterious ways. This simple toy's erratic oscillations recently attracted the attention of engineer Alan Wolf, who started to explore its movements for signs of chaos--motions that can be described by simple equations and yet are quite unpredictable (SN: 7/30/83, p. 76).

Most chaotic systems that people investigate are expensive to set up and rather complicated to study, says Wolf, who works at the Cooper Union School of Engineering in New York City. "This is a [low-priced] toy that generates high-quality chaos data,' he says. "I can easily collect experimental data from it.' Wolf presented his findings this week in Washington, D.C., at a meeting of the American Physical Society.

"A very hot issue is the attempt, in looking at experimental systems, to reconcile them with theory,' says Wolf. "The theory is farther along than the experimental work.' Simple models like the Space Ball and several other similar systems provide a useful way of testing competing theories and of learning how to define the amount of chaos present (SN: 5/26/84, p. 328).

The Space Ball, which is made in Taiwan, runs on a 9-volt battery hidden in the toy's base. This activates an electromagnet that, in turn, "kicks' another electromagnet in the lower ball. "Basically, it's very efficient electric motor,' says Wolf.

For his experiments, Wolf removed the battery and added a power supply that can feed in anything from 0 to 40 volts. Wolf alters the Space Ball's motion by changing the voltage or the toy's starting position. "Sometimes, I get nothing,' he says. "Sometimes, I can get it to come to rest. At other times, I can get it doing simple periodic motion [like a pendulum] or more complicated periodic motions. And I can get chaos.

"It'll do things like practically stop for a period of time, and you think it's ready to quit,' Wolf says. "Then it starts spinning rapidly, then it slows down, then it spins rapidly but at a different rate. You can watch it for a week and there may be no two periods of five minutes during which it does anything roughly similar.'

Wolf has come up with two equations, based on Newton's laws of motion, that describe the toy's behavior. Solving the equations on a computer gives numbers that match the Space Ball's observed motions. "In a sense,' says Wolf, "that's a proof that it's chaotic as opposed to being poorly built.'

His aim, however, is to quantify the amount of chaos present, not just in the toy but also in any system that may be suspected of exhibiting chaotic behavior. He has developed a computer program, running on a microcomputer, that calculates a quantity called the Lyapunov exponent.

This number provides an estimate of how long the behavior of a system is predictable. For a nonchaotic system, that exponent would be infinite because its future behavior is completely predictable. In chaotic systems, a tiny difference in starting conditions leads to widely divergent and, as a result, unpredictable behavior. The Lyapunov exponent puts a number on how fast this divergence occurs.

"Engineers and scientists have discovered a whole new regime of dynamics,' says Francis C. Moon of Cornell University in Ithaca, N.Y., "and we're trying to categorize these different regimes. We want to know when these things occur and what the characteristics of this chaos are. Simple models help us test the criteria.'

Another simple but useful model, also presented at this week's meeting, is the work of graduate student Nicholas B. Tufillaro of Bryn Mawr (Pa.) College. His mechanical device consists of a small, vibrating table (constructed from a loudspeaker) and a ball that is constrained to bounce vertically on the table's surface.

As in the case of the Space Ball, a simple set of equations describes the physical system. At the same time, says Tufillaro, "the bouncing-ball system exhibits the whole zoo of nonlinear phenomena shown by far more complex and less comprehensible systems.'

In the bouncing-ball apparatus, changing the table's frequency or amplitude alters the ball's motion. At certain frequencies, the ball's motion becomes extremely erratic. Thus, this model allows researchers to study how a physical system shifts into chaos. Moreover, because the ball makes a click every time it hits the table, listeners can actually hear the sound of chaos.

The bouncing-ball system also has educational value, says Tufillaro. Some people still attribute what is often labeled as chaos to factors like background "noise' instead of believing that it results from the nature of the motion itself. Showing these skeptics a simple system that actually works as predicted mathematically can be very convincing.

Photo: The Space Ball is an electrically driven toy that can exhibit the erratic oscillations characteristic of chaos.