Alternate View Column AV-51
Keywords: chaos butterfly effect sensitivity perturbations steering control
Published in the March-1992 issue of Analog Science Fiction & Fact Magazine;
This column was written and submitted 8/25/91 and is copyrighted ©1991 by John G. Cramer.
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In this column, I want to tell you about some new ideas about the "management" of chaotic systems. It now appears possible that even when systems are inherently chaotic, it is possible steer them toward a desired state and even to stabilize them. This seems to me to be a very important conclusion, since much of the real world is chaotic. But before going into the new ideas, we should discuss chaos itself.
About a decade ago the concept of chaos burst upon scientific community as a new paradigm for viewing the certain of the workings of nature and the structures of mathematics. It embodied two key concepts: (1) that certain systems that are classified as "chaotic", while completely determined by initial conditions and the laws of physics, are nevertheless so unstable as to be inherently unpredictable; and (2) that the behavior of chaotic systems is not arbitrarily random, but instead shows regularities, repeating patterns, and self-similarities. The new science of chaos thus staked out its territory in the middle ground between order and randomness, a ground that in the real world is occupied by systems ranging from energy levels in nuclei, to turbulence in plasmas, to the spread of gypsy moths, to weather patterns of Earth and Jupiter, to the stock and commodities markets.
To many in the sciences the new understanding of chaotic systems came as a great relief, for it showed that some of the most daunting scientific problems were not just difficult, they were truly insoluble. Moreover, much data that had been shelved as unanalyzable could be recycled and used as evidence for chaotic behavior, and the systematic but unpredictable patterns and structures implicit in the data could be extracted and discussed.
The emergence of the ideas underlying the theory of chaos came from a number of directions: from meteorologists trying to predict and understand weather, from mathematicians investigating the solutions of non-linear equations and the convergence of recursive procedures, from engineers trying to understand the characteristics of fluids in turbulent flow, from economists and population biologists and astronomers, all trying to understand systems showing behavior that defied prediction.
From the weather forecasters came the Butterfly Effect. This is the notion that the causal relations that produce the weather are so inherently unstable that the disturbance from the flutter of a butterfly's wings in Oslo could result in a major hurricane in the Carolinas a few weeks later.
From the mathematicians came the concept of Strange Attractors. All the knowable characteristics of the state of a non-quantum-mechanical physical system can be represented by a single point in an abstract multi-dimensional space that mathematicians, physicists, and engineers call "phase space". For example, the "state" of a planet in the solar system might be characterized by 9 phase space dimensions representing its position, velocity, and rotation. When the system changes, the point representing the state of the system moves through this phase space, tracing out a trajectory.
In well-behaved predictable systems, this trajectory either moves to a point of stability and stops, or else it orbits in the same trajectory over and over. These stable orbits and points are called "attractors" in the jargon of classical dynamics. In the dynamics of chaotic systems it was first suspected and than proved that the trajectory of the system does not repeat, but it does trace out a sort of envelope within which the trajectory is confined. This is a "strange attractor", an infinitely long non-repeating trajectory that never crosses itself and is confined within a finite volume of phase space. Using the term coined by Benoit Mandelbrot, such a trajectory forms a fractal.
The ideas and analysis techniques of chaos have been applied in fields ranging from abstract physics and mathematics to economics and population biology. Mathematical techniques have been developed which demonstrate the chaotic nature of a given system and permit it to be classified. With this new understanding it is even possible to make certain predictions about a chaotic system, for example forecasting average behavior and the likelihood of extreme behavior.
One of the primary tenets of the science of chaos, however, is that it is not possible to predict the future (or past) state of a chaotic system from its present state over more than a very small time interval. The 19th century conceit that with the positions and velocities of all the particles in the universe we could completely calculate both the past and the future has been sunk deeper than the Titanic. In fact, this conventional wisdom concerning the unpredictability of chaos has perhaps gone too far. The current perception is that chaotic systems are not only unpredictable, they are also uncontrollable. That perception, as it turns out, is false.
In March, 1991, a paper entitled "Controlling a Chaotic System" appeared in the physics journal Physical Review Letters. A group of mechanical engineers at the University of Pennsylvania had chosen a particularly simple chaotic system, a closed water-filled circular loop of pipe that is held vertically, electrically cooled at the bottom, cooled by a water jacket at the top, and monitored for temperature difference between the "3-o'clock" and "9-o'clock" positions at the midpoints of the loop. The hot water from the bottom of the loop, being lighter, tries to rise and the cooled water at the top tries to sink. This can happen by either a clockwise or a counterclockwise circulation in the loop, which is symmetrically constructed so as not to favor either circulation over the other.
When the electrical heating is low, one circulation pattern or the other at random becomes established and continues. When the heating is increased, the system becomes chaotic and the flow pattern can be observed, from variations in midpoint temperature difference, to flow erratically and to change abruptly and irregularly from one circulation direction to the other. Thus they have produced a table-top model of the kind of chaos found in systems ranging from whirlpools and hurricanes to planetary rings and spiral galaxies.
The Pennsylvania group made a small change in their chaotic system. They changed the power providing the heating coil up or down by about 4% in response to any small rise or fall in the average midpoint temperature difference of the system. This small change, which has the effect of slightly by systematically favoring one circulation direction over the other, eliminated the chaotic behavior of the system. At power levels three times higher than that producing the onset of chaotic behavior in the unmodified loop, the loop with feedback added showed smooth laminar flow.
In May, 1991, another paper entitled "Taming Chaotic Systems with Weak Periodic Perturbations" appeared in Physical Review Letters. Another group of mechanical engineers, this one at Tel-Aviv University in Israel, used the mathematical model of a pendulum damped by friction and driven by AC and DC electrical forces. They demonstrated that the chaotic behavior of the pendulum could be steered to a desired state through the introduction into the system of weak but well-timed "kicks". By introducing a periodic perturbation with about 1% of the strength of the other forces in the system, they could drive the system to near-stability. The semi-stable end-point of the system depended primarily on the frequency small extra perturbation introduced. Notice that, unlike the water loop, there was no "feedback" in the control of the pendulum. The perturbing signal did not depend on some repeated measurement of the system, it simply gave it a regular kick with a selected frequency.
The moral that can be extracted from these two studies is that chaotic behavior, well known to be extraordinarily sensitive to small differences in initial conditions, is also very sensitive to tiny but systematic changes. Chaos can be removed, reduced, or steered by monitoring and feedback or by introducing periodic perturbations of a carefully chosen frequency.
What is the value of this new insight about chaos? It has many implications. It means, in particular, that we do not have to despair because some system in which we are interested in inherently chaotic. Even though the system in its pristine state many not be predictable, there is still the possibility that we can control or steer it with small perturbation. Consider the weather, for example. If a butterfly in Oslo can unthinkingly cause a hurricane in the Carolinas, then perhaps a few hundred blowers, carefully positioned and carefully controlled by the best computer models of the dynamics of weather, can prevent a hurricane in the Carolinas.
Or consider the stock market. Fortunes are made and lost daily through accurate or inaccurate forecasts of market trends. But it has been demonstrated by economists using the new techniques of chaos theory that the stock market is inherently chaotic and that most middle term and long term forecasting is doomed to failure. Moreover, it is well known that the stock market is easily perturbed. A purchase or sale, a leak of inside information, or even a false rumor can send the market into an abrupt rise or a tailspin.
What, then, is going to happen when this news has made its way from Physical Review Letters to Analog and then to Wall Street? What's going to happen when the computer traders realize that they can steer and control the fluctuations and trends of the market by introducing periodic perturbations (regular buy/sell orders or rumors) or use similar feedback-derived perturbations to drive the market into a predictable state from which money can be made?
As with every new gimmick on Wall Street, for a while someone is probably going to make a lot of money. But when the trick is learned and many operators are doing it, the effects will cancel out and the Market, "efficient" as always, will go its own way. This impact of physics and mathematics on economics and finance should be at least interesting to watch.
And remember, folks, you saw it first in Analog!
Chaos - Making a New Science, James Gleick, Viking, New York (1987);
"Controlling a Chaotic System", J. Singer, Y-Z Wang, and Haim H. Bau, Physical Review Letters 66, 1123 (1991);
"Taming Chaos with Weak Periodic Perturbations", Y. Braiman and I. Goldhirsch, Physical Review Letters 66, 2545 (1991).
This page was created by John G. Cramer on 7/12/96.