The 2,196th Meeting of the Society

October 14, 2005

The Theory of Chaos

James Yorke

University of Maryland

About the Lecture

Scientists were probably the last people to find out about chaos. Everyone knows our lives are all chaotic and unpredictable in the long run. The mother of a friend of mine once took a taxi, met the driver, and wound up marrying him. If she had taken a different taxi, my friend would never have existed. I often say that the most successful people are those who are good at plan B. Our predictions must be flexible. Franklin wrote the famous lines “For the want of a nail, the shoe was lost; for the want of a shoe the horse was lost; and for the want of a horse the rider was lost, being overtaken and slain by the enemy, all for the want of care about a horseshoe nail.” Others carried this story further so that losing the rider and his message lead to the loss of a battle, then a war, and finally a kingdom, all for the want of a horseshoe's nail. There is common science fiction theme of time travelers making small pivotal perturbations in the past that result in crucial changes in the present. In Ray Bradbury's 1952 short story, “A Sound of Thunder”, a time traveler goes back millions of years and accidentally steps on a butterfly, significantly changing the present day world.

Chaos is an area of science and mathematics that describes situations in which small changes can cascade into larger and larger long-term effects.

Of course scientists always knew that is chaotic, but few recognized until the last 30 years that scientific environments in which precise rules govern change can be quite unpredictable in the long run. It is not the complexity of our lives that cause chaos as much as the instability of our lives. Meteorologist Edward Lorenz, one of the founders of chaos theory, suggested in 1960 that the flap of a butterfly wing in Brazil might set off a tornado in Texas, implying that we can never know all the factors that determine our weather. At best we can only predict the details of the weather a few days ahead. Scientists have found that many situations are equally unstable. Computer models have greatly helped us understand how pervasive chaos is throughout science. Our group at the University of Maryland has aimed at telling scientists how to look for varieties of chaos, for specific phenomena common to many situations. But I continue to wonder, if nearly all scientists missed this pervasive phenomenon, what else might we all be missing now?

About the Speaker

JAMES A. YORKE earned his bachelors degree from Columbia University in 1963 and his doctoral degree in Mathematics from the University of Maryland in 1966. He stayed at the University of Maryland as a member of Institute for Physical Sciences and Technology (IPST). IPST was established in 1950 to perform interdisciplinary research in the sciences. Today he holds the title of Distinguished University Professor and also is a member of the Mathematics and Physics Departments.

He is perhaps best known to the general public for coining the mathematical term “chaos” with T.Y. Li in a 1975 paper “Period Three Implies Chaos”. “Chaos” is a mathematical concept in non-linear dynamics for systems that vary according to precise deterministic laws but appear to behave in random fashion. The objective of his chaos research is to describe those robust properties that are common in the dynamics of physical, biological, and chemical systems.

Professor Yorke has coauthored three books on chaos and a monograph on gonorrhea epidemiology, and supervised approximately 30 Ph.D. dissertations in the Depts. of Mathematics and Physics. Dr. Yorke's Curriculum Vitae lists of over 300 publications.

Minutes

President Robert Hershey called the 2,196th meeting to order at 8:15 pm October 10, 2005. The recording secretary read the minutes of the 2,195th meeting. They were approved.

Mr. Hershey introduced the speaker of the evening, Mr. James Yorke of the university of Maryland. Mr. York spoke on “The Theory of Chaos.”

“I'm here to tell you a little bit about chaos,” Mr. Yorke began, and a little chaos was demonstrated as a number of different people tried their individual approaches to get the sound working. Coincidentally, he noted that scientists were among the last to learn about chaos.

Chaos, he said, is all about small changes having large effects. There have been stories about minor changes in the far past having major changes in the present. There was an episode of “The Simpsons” in which Bart Simpson went back in time to the dinosaur period and stepped on a bug. Back in the present, he did not like the effects, so he went back and stepped on some more bugs. He went back and forth doing this. In one version, doughnuts had no holes. After squashing a lot of bugs, he got something that was fairly similar to the present, and fortunately, Bart left the current world in that condition.

Mr. Yorke turned to a more serious example, the first mission to a comet. The European Space agency planned an expedition to Halley's Comet, and the United States space people wanted to be first. There was nothing in the budget for a spacecraft. As it happened, there was a spacecraft parked in a halo orbit between the earth and sun, although it had only 200 pounds of fuel aboard.

A very creative person, Robert Farquhar, figured out that by accelerating that spacecraft by only ten miles an hour at a certain point in its orbit, it would leave its orbit and take a circuitous route through space. Turned and accelerated by gravity within 100 km of the moon, it would pass through the tail of a comet 60 million miles away. Thus, they were able to move that spacecraft to a desired position at a certain time 60 million miles away from where it started with less than 200 pounds of fuel. They needed to make slight corrections on route, since a very small error, .001 of the acceleration, would have put the spacecraft a very great distance from the target. They made 37 rocket burns to successively approach the direction and acceleration needed.

Mr. Yorke then gave a much simpler demonstration, a pendulum hanging from a pendulum. When they swing, energy is transferred from one to the other in seemingly unpredictable ways and the rotation of the two appears erratic. We observers were asked to clap when we guessed the smaller pendulum would no longer rotate over its fulcrum on the larger pendulum. Laughter, groans, and clapping produced more chaos. Although the pendulum movement was in some ways chaotic, they did run down. Chaos usually refers to things that go on and on.

As another example of a small change with a big effect, he told the story of a paper given by W. Jason Morgan in 1968 that provided the foundation for all subsequent work on plate tectonics. One H. Menard, then the Presidential Science Advisor, chaired the session. Menard thought the paper was foolishness and tried to prevent and substantially slowed its publication. Others capitalized on Morgan's ideas to such an extent that Morgan's role in it was almost lost. Only because, years later, a single reprint from the session was found in France was Morgan recognized as the source.

He discussed also the planned economy idea, a major feature of the Communist government of the Soviet Union. They knew, theoretically, the number of shoes, in which styles, in which sizes, that would be produced at different times, and they knew it years in advance. Theoretically they knew almost everything about their economy. The problem with this idea was that it did not work very well. The seemingly more chaotic, free world approach of just providing a corral in which competition can occur seems to produce more goods in more appropriate quantities. It allows for on-course adjustments to plans.

He discussed an early jet airplane demonstration flight. It exploded, killing one of the chancellors of the Massachusetts Institute of Technology. This happened because the windows had square corners, which concentrated the forces in the metal so it would tear, allowing the windows to explode. They got into this problem because they had a plan to produce this airplane in a certain period of time. In order to meet the plan, they reached a point where they stopped testing.

Cars used to be sold on their new features. Each new model was deliberately packed with new features. The problem with this was that each new feature seemed to have a 50-50 chance of breaking down.

Because of chaos, he said, you have to allow for things to break down. Allow a plan B to develop.

A spirited and long discussion followed the address. People talked about tilt-a-whirls, differences between the natural world and the laboratory, chaos in quantum mechanics, spiders in the South American rain forest, and other things.

Mr. Yorke clarified that no situations are boundaryless. Chaos refers to the fact that we do not know the boundaries. The small things that determine the big things do represent boundaries; we just do not know how they will affect them. But you don't open a door and wind up on Mars.

Asked if we can plan for chaos, he said, “Don't drink and drive.” He also said it is very hard to make something work the first time. He recommends allowing things to break down. Then develop plans B and C and D and E.

Mr. Hershey presented a plaque to Mr. Yorke commemorating the occasion. He invited visitors to apply for membership. He invited everyone to enjoy the reception after the meeting. He made the parking announcement. Finally, he adjourned the 2,196th meeting at 9:41 pm to the social hour.

The weather: Mild and pleasant
Temperature: 20°C
Attendance: 54
Respectfully submitted,

Ronald O. Hietala,
Recording secretary