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Two researchers have created a strikingly simple model of chaotic behavior, in which variations in initial conditions become so tangled and magnified by the system's dynamics that the outcome appears to be unpredictably random. The team, then based at the Massachusetts Institute of Technology (M.I.T.), did this by bouncing a tiny droplet on a soap film, using an inexpensive speaker to drive the miniature trampoline.
Physicist Tristan Gilet, then a visiting student at M.I.T. from the University of Liège in Belgium, and John Bush, an M.I.T. mathematician, were intrigued by recent "beautiful experiments looking at droplets bouncing on a fluid bath," Bush says. One of these experiments, which Gilet co-authored, showed droplets hovering and even rolling over an oil bath without falling in.
What the researchers performing those experiments could not do, Bush says, "was describe the bouncing dynamic particularly well, because it's more complex—they have to describe the flow within the droplet, in the underlying bath and in the intervening air layer." To simplify the system, Bush and Gilet decided to forgo the bath and look at the behavior of droplets on a film.
What they found was that the droplet's behavior was accurately described by a single mathematical equation, a clean agreement between observation and theory that Bush calls quite rare. "One simple equation basically describes the system exactly," Bush says. "Typically in physics one has a divergence between experiments and theory."
Controlling the film's vibration with a $100 speaker, the researchers were able to modulate the periodicity, or cyclical nature, of the droplet's bounce. And by increasing the amplitude of the film's vibration, Bush says, he and Gilet could make the period "longer and longer and longer, and eventually it gets so long that it becomes effectively infinite and transitions into a chaotic state." In other words, at that point the bouncing becomes essentially unpredictable, as any uncertainties in the initial state take over.
Chaos theory, Bush says, is "really simply a statement of lack of precision on the initial conditions of a system. So unless you know the exact initial conditions of a system, any uncertainty will be amplified and you'll lose predictive power." Such chaotic systems come into play in describing financial markets and weather patterns, as in the famed butterfly effect, in which the beating of a butterfly's wings can theoretically cause enough of an atmospheric disturbance to significantly alter later weather outcomes.
Matthew Hancock, a postdoctoral fellow in biomedical engineering at Brigham and Women's Hospital in Boston who did not co-author the paper but whose input Bush and Gilet acknowledge in the endnotes, says the study "describes an extremely elegant example of a chaotic system, which should soon appear in textbooks."
Hancock praises the experimenters for boiling down the study of chaos theory to a clear and demonstrable form. "Usually chaos is studied in equations that are some gross simplification of a physical system," he says. "Here, it emerges from an exact description of the dynamics."
