Friday, May 23, 2014

BIRTH | May 23–Edward Lorenz, Chaos Theorist

Edward Norton Lorenz,
Chaos Theorist Who Gave
the Name to the Butterfly
This day in 1917 was born Edward Lorenz, in West Hartford, Connecticut. Originally trained in mathematics, he became a weather forecaster in the US Army, following which most of his career was spent in MIT's Meteorology Department.

He is responsible for chaos theory and for popularizing its explanation through what he called "the butterfly effect".  Lorenz originally used the image of a gull flapping in trying to explain how small actions in the atmosphere could trigger vast and unexpected changes.

The concept actually predates Lorenz's discovery and name. Sci-fi writers had been aware of this idea in time-travel or sci-fi stories, when a hero goes back in time. A seemingly insignificant choice ends up changing the course of history.

Lorenz showed how mathematics supports the idea that tiny changes can have huge effects. His discovery of this effect in the 1960s occurred when he tried to save time entering values in a computer weather-prediction program. He rounded off six decimal places to three. The resulting weather pattern was completely different. He changed the image from a gull to a butterfly in his 1972 presentation, "Does the flap of a butterfly's wings in Brazil set off a tornado in Texas?"

The paradigm shift caused by Lorenz's work rivals that of relativity theory. The complex or seemingly random behavior of many physical or biological systems does not require that equations which describe these systems are themselves highly complex or random. They indicate the presence of the fractal geometry of the strange attractor. Ian Stewart says in his book Does God Play Dice? (Stewart, 1997):
When I read [Lorenz’s] words I get a prickling at the back of my neck and my hair stands on end. He knew! Thirty-four years ago, he knew! And when I look more closely, I’m even more impressed. In a mere 12 pages, Lorenz anticipated several major ideas of non-linear dynamics before it became fashionable, before anyone else had realized that new and baffling phenomena such as chaos existed.
What Lorenz achieved can only be explained in terms of his predecessors. He studied highly truncated versions of the equations for the Rayleigh-Benard convection problem. Following a thread started by Saltzman, he theorized that much of the irregularity of these equations was contained in a three-dimensional core. He used linear stability analysis to focus on Rayleigh numbers that make the system linearly unstable. To investigate the non-linear behavior of this linearly unstable system, he coded his truncated equations on a digital computer. After an initial period, when the system evolved in a regular way away from an unstable fixed point, it then behaved completely irregularly.

Lorenz analyzed this irregularity. He plotted trajectories of the system in the three-dimensional state space on a geometric subset of state space. This geometry actually had the shape of a butterfly with two wings at the back (hence perhaps Lorenz's reference to a butterfly in 1972), but merged into a single layer at the front. Lorenz knew that the trajectories of a deterministic differential equation cannot merge. What looked like a single sheet at the front must really be two sheets together. But that meant that each sheet at the back was double, too. He wrote:
We conclude that there is an infinite complex of surfaces each extremely close to one or the other, of two merging surfaces.
Lorenz had discovered the fractal structure associated with the attractors of chaotic systems. He wanted to study a piece of the solution in greater detail. He re-ran the equations from a saved “dump”. But the numbers in the dump had been truncated and he found that the solution diverged totally from the original solution. The irregularity of the solutions also gave rise to an inherent unpredictability. Lorenz provides a layperson’s account of chaos theory in his book The Essence of Chaos, 1993.

He married Jane Loban in 1948 and they had two daughters and a son.

 (Thanks to Garrison Keillor for noting Lorenz's contribution. He is one of the sources for this summary.)