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Chaos
J. C. Sprott
University of Wisconsin -- Madison
Many important problems in physics, such as the motion of the planets around the Sun or the oscillations of a mass suspended by a spring, have well-understood solutions. These solutions come from fundamental equations such as Newton's laws and can be expressed as simple mathematical functions such as sine and cosine. The laws of classical physics are deterministic in the sense that they allow us to predict what will happen next based on the present state of the system. This kind of detailed quantitative prediction has proved much more difficult in other fields, such as the life sciences. Even within physics there are many processes such as the turbulent motion of a fluid or the motion of molecules in gas that defy simple description and seem governed more by chance and probability than by determinism. These systems typically involve very complicated equations or a large number of simple equations.
What has come as a surprise to most scientists in recent years is the fact that simple systems, governed by simple laws, can exhibit behavior of such complexity as to appear random. Such systems are said to exhibit "chaos," which we may define as the apparently random behavior of a deterministic system. These systems cannot be "solved" in the usual sense of finding a mathematical expression that gives the state of the system at some arbitrary future time. Many examples of chaos have been discovered in recent years, and some processes previously thought to be random have now been explained in terms of relatively simple models.
The Three-Body Problem
An example of chaos that bothered Newton 300 years ago and was appreciated by Henri Poincaré (a French mathematician) 100 years ago is the famous three-body problem in which, for example, a planet orbits a pair of stars. The motion of a single planet around a star is a simple ellipse, as was understood by Kepler even before the time of Newton. But when a third body is added, the motion, as shown in Figure 1 is of great complexity and never repeats. This complexity persists even if both stars are held fixed and the planet moves in a plane about the stars.
Despite valiant attempts by the best mathematicians, this problem has never been solved except by a computer that laboriously calculates each successive position in terms of the previous position using tiny time steps. Digital computers are an indispensable tool for studying problems of this type, and the complicated results vividly illustrate why such problems will probably never be solved by the usual algebraic methods.
A characteristic of chaos is an extreme sensitivity to initial conditions. If we were to repeat the three-body calculation with the initial position or velocity of the planet only slightly different (say one part in a million), after a few orbits around the stars, the motion in the two cases would bear no resemblance to each other. One orbit could be near one star and the other near the other. Since there are always limits to our measuring ability, the implication is that long-term prediction of chaotic phenomena is impossible, even when the underlying physical laws are simple and deterministic. The meteorologist's inability to make accurate long-range weather predictions is a consequence of the chaotic motion of the atmosphere. Such sensitivity to initial conditions has been dubbed "the butterfly effect" since it is theoretically possible for a butterfly flapping its wings in Brazil to set off tornadoes in Texas.
The Driven Pendulum
The same system can often exhibit both regular and chaotic behavior, with the transition determined by a small change in some parameter. Consider the simple pendulum, whose motion is so predictable that we can build clocks based on it. Since a real pendulum always has some friction, it is necessary to give it a periodic push to keep it going. If the push is gentle, the angle that the pendulum makes with the vertical is a sinusoidal function of time. One way to display the motion is to plot the angular velocity of the pendulum as a function of its angle. The result is an ellipse, as shown in Figure 2. The motion of the pendulum is reminiscent of Kepler's elliptical orbits.
Now if we push the pendulum slightly harder, but still periodically, so that it occasionally goes over the top, the phase-space plot looks very different, as shown in Figure 3, and if we watch long enough, the trajectory will fill in a large region of the plane. Although the motion looks random and unpredictable, it is in fact deterministic and subject to rules that are not apparent in Figure 3.
Suppose that instead of looking at the entire trajectory, we flash a strobe light occasionally and place a dot on the graph where the trajectory is at that instant. For example, if the strobe light flashes every time the pendulum is pushed, a plot such as Figure 4 emerges after many dots have been collected. Such a plot is called a "Poincaré section," and it reveals dramatically and beautifully both the complexity and the structure that characterize the motion. If the motion had been periodic, the Poincaré section would have been just a few isolated dots. If it had been random, a region of the plane would have been solidly filled with points.
Fractals
The object in Figure 4 is called a "strange attractor." It is an attractor because no matter how the pendulum is started, the motion is quickly attracted to the darkly colored region. It is strange because the attractor is neither a line nor a surface but rather an object with fractional (noninteger) dimension. It resembles a turbulent fluid or a blob of salt-water taffy that has been repeatedly stretched and folded. The stretching causes nearby points to separate, accounting for the sensitivity to initial conditions. The folding produces structure on even the smallest scale. Such objects have been called "fractals," and they are often self-similar, which means that they look the same at any magnification.
Fractals seem to be very common in nature. Natural objects like clouds, rivers, and trees are neither lines nor surfaces nor solids, but rather somewhere in between. The length of a river or a coastline depends on the length of the ruler used to measure it. A small, puffy cloud a few meters across looks the same as a cloud that covers a continent. Fractal patterns generated by computer represent an exciting new art form.
A way to quantify chaos and to determine the complexity of the underlying equations is to calculate the fractal dimension. Truly random processes require an infinite number of variables and equations to specify them, and they produce objects with an infinite number of dimensions when plotted in phase space. Many interesting chaotic phenomena have low dimension.
The Logistic Equation
A particularly simple example of chaos arises from the logistic equation
which has been used to model population growth. The variable X is restricted to the range of 0 to 1 and might represent the population of some species of bug in successive seasons relative to the maximum possible number. For different values of the growth rate R in the range of 0 to 4, successive iteration of the logistic equation yields solutions that either die out, oscillate among various distinct values of X, or fluctuate chaotically. With R = 4, the logistic equation is fully chaotic and provides a dramatic illustration of a simple deterministic equation whose long-term behavior is unpredictable.
The transition from regular to chaotic behavior exhibited by the logistic equation has been observed in many physical systems. For example, a leaky faucet drips at regular intervals if the drip rate is small, but the drips become chaotic as the rate is increased. Similar behavior has been observed in electrical circuits, lasers, and convecting fluids.
Current Research
An important (and yet unsolved) problem is to predict the conditions under which a system will exhibit chaos. For systems governed by ordinary differential equations (such as Newton's second law), it appears that at least three variables and at least one nonlinear term are required. In the three-body problem, there are four variables (two components each of velocity and position), and the nonlinearity comes from the fact that the gravitational force is proportional to 1/r2. In the driven pendulum there are three variables (angular velocity, angle, and time), and the nonlinearity is provided by the gravitational torque, which is proportional to the sine of the angle. However, there are systems that satisfy these requirements yet do not exhibit chaos.
Since it is now known that simple equations can have complicated and unpredictable solutions, it is worthwhile to reexamine processes that exhibit complicated behavior in the hope that they can be understood in terms of simple models. Quantities that fluctuate in an apparently random manner arise in fields as diverse as meteorology, seismology, ecology, epidemiology, medicine, and economics, to name just a few. If such fluctuations turn out to be chaotic rather than random, then it might be possible to learn something about the underlying causes and perhaps to make improved short-term predictions.
The study of chaos is still in its infancy. The hope is that it will someday provide tools that are as useful for understanding complicated phenomena as are the tools described in this text for understanding simple phenomena.
Suggested Readings
Gleick, J., Chaos: Making a New Science, New York, Viking, 1987.
Stewart, I.,Does God Play Dice?: The Mathematics of Chaos, New York, Blackwell, 1989.
Crutchfield, J. P., J. D. Farmer, N. H. Packard, and R. S. Shaw, "Chaos," Scientific American, Dec. 1986, p. 46.
Hofstadter, D. R.,"Strange Attractors: Mathematical Patterns Delicately Poised Between Order and Chaos," Scientific American, Nov. 1981, p. 22.
Peitgen, H. O., and P. Richter, The Beauty of Fractals: Images of Complex Dynamical Systems, New York, Springer-Verlag, 1986.
Barnsley, M. F., R. L. Devaney, B. B. Mandelbrot, H. O. Peitgen, D. Saupe, and R. F. Voss, The Science of Fractal Images, New York, Springer-Verlag, 1988.
Questions
Problems
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Figure Captions
Figure 1 The chaotic motion of a planet orbiting a pair of identical stars.
Figure 2 Angular velocity versus angle for a pendulum driven at small amplitude.
Figure 3 Angular velocity versus angle for a pendulum driven at large amplitude.
Figure 4 Poincaré section for a driven pendulum showing the fractal structure of a strange attractor.