He's not a fortune teller, but if you could turn your journey to work each morning into a mathematical equation, this research engineer might be able to tell you which days' commute would go predictably and which would be chaotic. For example, the phone call that delays your departure by 15 seconds could have ramifications an order beyond what you might think if you end up missing your bus, then lose a big contract and half your income just because you were late.
Dr. Mark Johnson, a principal research engineer in the Department of Mechanical Engineering, has come up with a method for predicting whether a physical system will be chaotic or stable by applying a mathematical test to the equation that describes the system.
"Knowing whether or not a system is chaotic can help you understand its characteristics and save a lot of time," he said.
Scientists and engineers use mathematics to describe all sorts of normal physical phenomena, like blood flowing through veins, a tractor-trailer hauling lumber on a freeway or the movement of planets in a solar system.
Some of those processes are completely predictable -- they're the same over and over again, and a slight initial change produces only a small change in the end. Others might appear at first glance to be predictable, but minor alterations can bring about enormous differences in outcome. And still others are completely random.
Chaos, in the scientific sense, deals with the second group -- those natural processes that are very sensitive to small changes.
In the scenario above, that phone call might appear to be an ordinary variable in the equation describing your morning, but in this instance, the 15-second delay it causes has a dramatic effect on the system.
While various methods have been proposed for separating chaos from ordinary noise (or errors) in experimental data, Dr. Johnson tries to find out if a mathematical equation is chaotic by applying a mathematical technique that tests for randomness. He explained that a test like this could help researchers in industrial settings, such as chemical engineers who might need a chaotic process to achieve a certain end.
"Any type of industrial process where you have to mix highly viscous solutions will normally be enhanced by chaos, so you might want to design a process to be chaotic," he said.
One analogy of how this works is to imagine kneading a spot of red dye into bread dough. The stretching and folding required to knead dough make it a chaotic process. That dot of red dye will be evenly dispersed throughout the dough fairly quickly, producing a pale-pink loaf of bread. A non-chaotic mixing process might be less efficient.
To date, Dr. Johnson has used his method to test eight understood systems with clear results.
"As yet, there's no theoretical proof. It's worked on the processes I've tested it on, but all you need to do is find one case to disprove it," said Dr. Johnson, whose work in fluid mechanics led him to apply a test for randomness in fluid motions to other types of systems. He describes the procedure in a paper scheduled to appear in the June 1 issue of Physica D. His co-author is James Habyarimana, a senior in civil and environmental engineering who worked on the project as a summer UROP.
"I'd been to a number of talks where people described their chaotic systems," said Dr. Johnson. "Whenever I asked how they knew the process was chaotic, the best response they could give was that the equation had a positive Lyapunov exponent, which is a mathematical concept commonly used as an operational definition for chaos. Yet, by starting with a very well-studied chaotic system --the logistic equation for population growth -- I was able to construct an example of a chaotic equation that didn't fit that mold. So I tried to devise a better operational test."
Chaos, a characteristic of some natural processes that has been recognized widely for only a few decades, is an unpredictability inherent in a system that should otherwise be stable according to classical models of physics. Before MIT Professor Emeritus Edward N. Lorenz first wrote about the phenomenon in 1963, chaos was generally written off as noise in data and ignored.
Professor Lorenz, whose initial research on chaos was in meteorology, used the example of a butterfly to describe the effect that small changes can make on a chaotic system. The flap of a butterfly's wings in one hemisphere, through a cumulative series of effects, could conceivably produce a hurricane on the other side of the world.
Chaotic systems tend to behave predictably only at very short or very long time intervals; in between, they appear to be random. This characteristic can be referred to as disorder underlying an apparently stable pattern or, paradoxically, as pattern underlying apparent disorder. It explains why weather can be predicted only a few days at a time, but can also be expected to stay within certain parameters in the long run.
While Dr. Johnson looks at the equations that describe natural systems to determine if chaos is present, some theorists start with data, then test for chaos and draw conclusions. For instance, Dr. Chi-Sang Poon, a research scientist in the Harvard-MIT Division of Health Sciences and Technology, found a method to separate chaos from noise in data sets and then showed that the method could potentially diagnose heart disease.
A version of this article appeared in MIT Tech Talk on April 15, 1998.