[Editors] MIT ‘fluid trampoline’ offers glimpse of chaos

[Teresa Herbert]

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MIT professor discovers chaos on a ‘fluid trampoline’
--Latest milestone in MIT’s contributions to chaos theory
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For Immediate Release
THURSDAY, DEC. 18, 2008

Contact: Teresa Herbert, MIT News Office
E: therbert at mit.edu, T: 617-258-5403

Photo, Video and Graphic Available

CAMBRIDGE, Mass. -- A water drop placed on a soap film that vibrates  
up and down may bounce as if on a trampoline — but it’s much more than  
that, according to MIT mathematicians who say the ”fluid trampoline”  
is the simplest fluid system yet explored that exhibits chaotic  
behavior.

MIT math professor John Bush and visiting student Tristan Gilet built  
the system in the Applied Math Laboratory, then demonstrated that the  
drop bouncing may be accurately described with a single simple  
equation. They report their findings in a paper scheduled to appear in  
the Dec. 29 online edition of  Physical Review Letters.

Their study builds upon the pioneering work of the late Edward Lorenz,  
an MIT meteorologist who in 1963 discovered chaos in a simplified  
mathematical model of the atmosphere, now called the Lorenz equations.  
Known as the father of chaos theory, Lorenz passed away in April 2008  
after a distinguished career in MIT’s Department of Earth, Atmosphere  
and Planetary Sciences.

The trademark of chaotic systems is their sensitivity to initial  
conditions. Any uncertainty in the initial state of a chaotic system  
will soon be amplified, leading to a loss of predictive power over the  
system. The chaotic nature of the Earth’s atmosphere is responsible  
for the shortcomings of weather forecasts, which are notoriously  
untrustworthy beyond a few days.

Since Lorenz’s early work, chaos has been discovered in a wide variety  
of complex systems, from the beating heart to population dynamics,  
from planetary orbits to the stock market. An interesting  
philosophical question arises, says Bush: “What is the simplest  
physical system that exhibits chaotic behavior? What are the minimum  
ingredients for chaos?”

In the 1970s, MIT math professors Lou Howard and Willem Malkus  
developed the first mechanical chaotic oscillator in the Applied Math  
Laboratory, a water wheel whose motion is precisely described by the  
Lorenz equations. The original water wheel consists of a series of  
perforated Dixie cups fixed to a tilted wheel: When the cups are  
filled from above, the wheel may spin in an unpredictable, chaotic  
fashion.

Subsequently, chaos has been observed and studied in a number of  
simple systems, including a bouncing rubber ball, the double pendulum  
and the dripping faucet. While the latter system is the simplest fluid  
oscillator to study experimentally, Bush points out that the fluid  
trampoline is the simplest when one considers both ease of experiment  
and theory.

The form of bouncing on the fluid trampoline depends on the amplitude  
and frequency of the soap film vibration. At low amplitude, the drop  
bounces with the period of the forcing. Progressively increasing the  
amplitude leads to the bouncing period doubling then quadrupling.  
Ultimately, chaos emerges via a so-called period-doubling cascade. The  
authors demonstrate that the trajectory of the bouncing drop is  
accurately described by a single second-order differential equation  
that allows them to rationalize all of the observed bouncing behavior,  
including the period-doubling transitions to chaos.

Their study is the latest milestone in MIT’s long association with chaos
theory. Says Bush, “We have brought chaos back to its fluid mechanical  
roots at MIT.”

Gilet, a graduate student from the University of Liege in Belgium, was  
visiting MIT thanks to the financial support of the FNRS/FRIA and the  
Belgian government.

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Written by Anne Trafton, MIT News Office
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