Math in the News
Chaotic Motion of a Fluid Trampoline
January 23, 2009
A water drop bouncing up and down on a vibrating soap film may be the simplest fluid example of chaotic motion yet explored. Mathematician John Bush of the Massachusetts Institute of Technology and Tristan Gilet of the University of Liege, Belgium, claim that a single simple equation accurately describes the complex motions of their "fluid trampoline."
Their discovery builds upon the work of Edward Lorenz, who in 1963 described chaos in a simplified mathematical model of the atmosphere, now called the Lorenz equations. Researchers later identified chaotic motion in a variety of simple systems, from bouncing rubber balls and double pendulums to dripping faucets. Bush wondered, "What is the simplest physical system that exhibits chaotic behavior? What are the minimum ingredients for chaos?"
The trademark of any chaotic system is its sensitivity to initial conditions. Uncertainty in such a system's initial state will eventually be amplified, which leads to a loss of predictive power over the system.
In the duo's experiment, a drop of water bounces on a soap film, which becomes a fluid trampoline as the drop vibrates up and down. The form the bouncing takes depends on the amplitude and frequency of the soap film's vibration. At low amplitude, the drop bounces with the period of the forcing. Ever increasing amplitude results in the bouncing period doubling and then quadrupling. Finally, chaos emerges via a "period-doubling cascade."
The researchers showed that one second-order differential equation reveals all of the observed bouncing behavior, including the period-doubling transitions to chaos.
The findings will appear in Physical Review Letters.
Source: Massachusetts Institute of Technology, Dec. 22, 2008.