Chaos - James Gleick [98]
In Libchaber’s experiment, as it happened, the first wavelength to appear was about two seconds. The next bifurcation brought a subtle change. The roll continued to wobble and the bolometer temperature continued to rise and fall with a dominant rhythm. But on odd cycles the temperature started going a bit higher than before, and on even cycles a bit lower. In fact, the maximum temperature split in two, so that there were two different maximums and two minimums. The pen line, though hard to read, developed a wobble on top of a wobble—a metawobble. On the spectrum diagram, that showed up more clearly. The old frequency was still strongly present, since the temperature still rose every two seconds. Now, however, a new frequency appeared at exactly half the old frequency, because the system had developed a component that repeated every four seconds. As the bifurcations continued, it was possible to distinguish a strangely consistent pattern: new frequencies appeared at double the old, so that the diagram filled in the quarters and the eighths and the sixteenths, starting to resemble a picket fence with alternating short and tall pickets.
Even to a man looking for hidden forms in messy data, tens and then hundreds of runs were necessary before the habits of this tiny cell started to come clear. Peculiar things could always happen as Libchaber and his engineer slowly turned up the temperature and the system settled from one equilibrium into another. Sometimes transient frequencies would appear, slide slowly across the spectrum diagram, and disappear. Sometimes, the clean geometry notwithstanding, three rolls would develop instead of two—and how could they know, really, what was happening inside that tiny cell?
TWO WAYS OF SEEING A BIFURCATION. When an experiment like Libchaber’s convection cell produces a steady oscillation, its phase-space portrait is a loop, repeating itself at regular intervals (top left). An experimenter measuring the frequencies in the data will see a spectrum diagram with a strong spike for this single rhythm. After a period-doubling bifurcation, the system loops twice before repeating itself exactly (center), and now the experimenter sees a new rhythm at half the frequency—twice the period—of the original. New period-doublings fill in the spectrum diagram with more spikes.
IF LIBCHABER HAD KNOWN then of Feigenbaum’s discovery of universality, he would have known exactly where to look for his bifurcations and what to call them. By 1979 a growing group of mathematicians and mathematically inclined physicists were paying attention to Feigenbaum’s new theory. But the mass of scientists familiar with the problems of real physical systems believed that they had good reason to withhold judgment. Complexity was one thing in the one-dimensional systems, the maps of May and Feigenbaum. It was surely something else in the two– or three– or four-dimensional systems of mechanical devices that an engineer could build. Those required serious differential equations, not just simple difference equations. And another chasm seemed to divide those low-dimensional systems from systems of fluid flow, which physicists thought of as potentially infinite-dimensional systems. Even a cell like Libchaber’s, so carefully structured, had a virtual infinitude of fluid particles. Each particle represented at least the potential for independent motion. In some circumstances, any particle might be the locus of some new twist or vortex.
REAL-WORLD DATA CONFIRMING THEORY. Libchaber’s spectrum diagrams showed vividly the precise pattern of period-doubling predicted by theory. The spikes of new frequencies stand out clearly above the experimental noise. Feigenbaum’s scaling theory predicted not only when and where the new frequencies would arrive but also how strong they would be-their amplitudes.
“The notion that the actual relevant meat-and–potatoes motion in such a system boils down to maps—nobody understood that,” said Pierre Hohenberg of AT&T Bell Laboratories in New Jersey. Hohenberg became one of the very few physicists