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Nonlinear Phenomena

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Posted by Meg Brooks on February 19, 2001 16:26:32 UTC

Megan:

Chaos, as discussed on The Infinite Mind, manifests itself even in seemingly simple equations describing familiar events in the macroscopic world. Unlike the randomness of "ignorance" scientists are often accustomed to, chaos is not caused by too many variables for a human mind to monitor. Under the right conditions, it seems, the simplest systems can generate randomness that is inherent and inexpugnable.

But unlike other deterministic systems, chaotic systems exhibit what physicists call "extreme sensitivity to initial conditions." Encountering a rock, for example, a stream of smoothly running water suddenly shatters into turbulence. Place a cork upstream and it will travel what appears to be a random path. Repeat the experiment, placing the cork as close as humanly possible to the same starting position, and it will follow a completely different trajectory. The slightest uncertainty in the initial condition is amplified so rapidly that prediction becomes impossible.

Or, as a mathematician would put it, the trajectories diverge exponentially. In a linear relationship, 10x=y, the output rises slowly as x increases from 0 to 6: 0, 10, 20, 30, 40, 40, 50, 60. But consider what happens when we feed the same input to an exponential equation: 10[exponent X]=y. The value of y starts at 1, then, as x increase, jumps to 10, 100, 1,000, 10,000, 1,000,000. Faced with a chaotic system, in which tiny perturbations are amplified exponentially, even the perspicacious physicist would have to know the initial conditions with infinite accuracy -- to the nth decimal point -- to foresee what it will do.

And even more striking is the fact that a system need not be as complex as a turbulent stream to exhibit chaos. In fact, mathematicians have shown that equations simple enough to solve on a pocket calculator could be chaotic: change the numbers you plug in by the tiniest amount and the output will be wildly different. Still, chaos is an example of what physicists call "nonlinear phenomena."

Indeed, chaos makes perfect predictability impossible. Even for the all-knowing mathematician, one is still defeated by uncertainty -- quantum and otherwise -- amplified again and again by chaotic collisions. But for those of us compelled to seek order in the world around us, chaos also holds out cause for hope. Even very simple systems can be chaotic. So if you look hard enough, behavior that seems random might turn out to be generated by a few simple equations. Unlike the pure randomness of quantum mechanics, chaos displays hidden harmonies.

B. L. Nelson
Benjamin_Nelson@WPBA.pbs.org
For transcripts to The Infinite Mind, call 1 888 350-MIND
Harvard School of Scientific Computation

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