Chaos and the Solar System
by Paul Trow

Mathematical theories do not usually get much public attention.  A recent exception is the theory of chaos, which emerged on the scientific world in the 1970’s, and has since received extensive coverage by the media and in a number of books aimed at a general audience. The term “chaos” has even made its way into several popular science fiction movies. Part of the fascination with this theory has to do its claim that chaos – which is inherently more interesting than order - is a pervasive feature of the world around us. Chaos, we are told, is responsible for the unpredictability of the weather and the fluctuations of the stock market. We can observe it in ecological systems, the rhythms of the human body, and the turbulence of a mountain stream. Surprisingly, although chaos is all around us here on Earth, the mathematician Henri Poincare first discovered it, late in the nineteenth century, in the mathematics underlying the solar system.

Before explaining just what Poincare discovered, let us recall a little of the background the background that led up to it. In a sense, chaos theory is as a belated reaction to the principle of determinism, which dominated science for over three hundred years. Newton established this principle through his second law of motion, force equals mass times acceleration, from which it follows that if we know all of the forces acting on a system, we should – at least in theory - be able to predict how it will evolve over time. Determinism gave rise to the metaphor of the clockwork universe, which, when once wound up, will evolve forever in a predetermined manner.

Chaos theory, on the other hand, throws a bucket of cold water on the principle of determinism, by pointing out the inherent difficulty of predicting the long-term evolution of many systems. For example, although meteorologists may be able to forecast the weather a couple of days in advance, doing so a week in advance is problematic, and a month is out of the question. Although chaos theory does not directly contradict determinism, it does point out the inherent limitations of using scientific laws to predict the future. In this sense, chaos is more of an anti-theory than a theory – unlike previous scientific theories, it emphasizes the weakness of science, rather than its strength.

What chaos theory reveals is the underlying tension between determinism and randomness. This is an ancient dichotomy. Many societies have believed both that the future is foreordained and that chance plays an important role in life. We are familiar with many seemingly random events, such as rolling a pair of dice. Now, according to determinism, if we could somehow divine the exact forces acting on the dice as they roll, we could predict which numbers would come up. Since in practice we cannot do this, we are not surprised that the outcome appears to be random. But what about deterministic systems, in which all the forces can, in principle, be found?  Oddly enough, in many cases these can also have the appearance of randomness.

What exactly is chaos? We can give an example by a rigid pendulum, such as in a grandfather clock. If the pendulum swings freely, its motion will be perfectly regular and periodic, and if there were no friction, it would continue this way forever. The system is perfectly predictable – it is the opposite of chaos. But now suppose that we place an electric magnet at the base of the pendulum, and arrange for the magnet turn on momentarily at regular intervals – say once every second – at which time it exerts a magnetic force on the pendulum. This device is rather like a parent pushing a child on a swing, but unlike the parent who pushes in time with the swing, the magnet’s forces are out of phase with the pendulum. If the magnet exerts its force during the pendulum’s downward swing, it speeds the pendulum up – but if it does so during the upward swing, it slows the pendulum down. The question is simply this: what will happen to the pendulum? Will it swing regularly or irregularly? Can we predict its motion at all?

Since the device is such a simple deterministic system, and the magnet’s push occurs at regular intervals, we might guess that the pendulum’s motion would be periodic. In other words, after a while the motion would begin to repeat itself. Surprisingly, what actually happens is that the pendulum begins to swing irregularly, sometimes higher and sometimes lower, without any discernable pattern. We are no more able to predict how high the pendulum will go after a few swings than we are to predict the outcome of a roll of the dice. What this device shows is that a simple deterministic mechanism can generate what appears to be random motion – in other words, it is chaotic.

There is a historical irony in the fact that this simple device is chaotic. Galileo studied the motion of freely swinging pendulums, and discovered that their period is independent of the length of the pendulum. Indeed, this discovery, along with his famous analysis of falling bodies and projectiles, were some of the first quantitative descriptions of terrestrial motion. Galileo’s work was one of the two most significant influences on Newton’s thinking. The irony is that something so simple as a pendulum – the very symbol of the deterministic universe – can be altered so slightly as to produce chaos.  And the historical question this raises is why    it took almost three hundred for someone to recognize the possibility of chaotic motion.  I think it is safe to say that because Galileo and the other great genius of the scientific revolution were searching for order in the universe, they were blind to the existence of chaos all around them.

This toy is an example of what is called a forced oscillator – a fancy name for what happens when an external force is applied to any simple periodic motion. It is a phenomenon that occurs throughout nature – for example, in the human heart. Like the swing of the pendulum, the heart expands and contracts regularly. The brain provides an external stimulus to this periodic motion through the signals it sends to the heart to keep it pumping. Consequently, the heart does not beat perfectly regularly, but in a slightly chaotic manner – not so erratically that it fails to deliver blood throughout the body, but on the other hand, not so regularly that it becomes rigid and unable to respond to the unpredictable impulses. This mild irregularity may be beneficial, allowing the heart a looser, more flexible response, like a jazz musician playing slightly off the beat. The same kind of chaotic behavior can be observed in everything from the rise and fall of animal populations, to the irregular drops of water from a leaky faucet.

With the possible exception of the heartbeat, all of these examples can be studied without any highly sophisticated technology. So it is somewhat surprising that their chaotic properties were not recognized until very recently, and that chaos was not first discovered in a relatively simple system of this kind. To find the origins of chaos theory, we must look to the much older and more venerable question of the motion of the planets in the solar system.

The motion of the planets is a problem that has perplexed mathematicians and astronomers since antiquity. According to the Greek mathematician Pythagoras, the Earth and the planets revolved about a central fire, producing musical notes as they moved, from which we derive the expression “the harmony of the spheres.”  To account for the anomalies of planetary motion, later Greek astronomers were forced to add additional cycles to the original circular motion, producing the rather complicated theory of epicycles. Almost two thousand years later, Kepler realized that the observed motion could be explained much more simply if the orbit of each planet were an ellipse, the familiar oval-shaped curve studied by the ancient Greeks. For centuries the solar system was a place where people had come to expect order and harmony, and certainly not the discord of chaos.

The great breakthrough in describing the motion of the planets was, of course, due to Newton, who used his laws of gravitation and motion to deduce the fact that the planets have elliptical orbits.  In order to apply his laws, he had to invent an entirely new branch of mathematics, called calculus, which describes how physical quantities change over time – for example, velocity is the rate of change of position, and acceleration is the rate of change of velocity.  Equations relating quantities and their rates of change, called differential equations, have been used to express most of the laws of the physical sciences, and to model everything from stock prices to airplane wings (what we often hear referred to in the media as “computer models”). Undoubtedly the single most important example of a differential equation occurs in Newton's second law of motion, force equals mass times acceleration, which governs the motion of every object in the universe. For each planet in the solar system, the gravitational attraction of the sun and all the other planets gives rise to a collection of equations, which together determine its motion.  

These equations alone, however, are not enough to predict the exact motion of a planet: for that we must also solve them. A solution corresponds to a path, or trajectory, the planet will follow over time. This turned out to be a very difficult problem, which Newton was unable to answer in general. A great deal of mathematics and physics over the past three hundred years has been devoted to solving specific types of differential equations, motivated by different physical processes. Ideally, we would like to find an exact formula for a solution: for example, the path of a baseball in flight is a parabola (provided we ignore air resistance). But solutions of this kind are actually quite rare, and so usually we must resort to finding an approximate numerical solution. These are very effective in the short run, for practical problems such as putting a satellite into orbit. In the nineteenth century, mathematicians used numerical solutions to predict the existence of the planet Neptune before it was actually observed by telescope.

     Newton was able to solve a special case of planetary motion, in which there are just two bodies – say a sun and one planet - each exerting a gravitational force on the other. If the planet is caught by the gravitational attraction of the sun, its orbit will be an ellipse and its motion will be periodic, repeating the same path in equal time intervals.  The other possibility is that the planet escapes the sun’s gravitation, in which case its orbit will be an open-ended curve. The solution to the motion of a two-body system, by Newton and his eighteenth century mathematical successors, is one of the triumphs of Newtonian mechanics.

In our own solar system, which has many more than two bodies, things are much more complicated.  The planets follow orbits that are almost, but not exactly, ellipses, the discrepancy being due to the fact that each planet has its own gravitational field, which influences – or perturbs – the motion of all the others. Consequently, the planets’ orbits are not exactly periodic: they return to a slightly different position, and their time of revolution about the sun varies slightly, from year to year. Describing the motion of any planetary system (including purely imaginary ones that exist only on paper) is the subject of a branch of mathematics called celestial mechanics. Its problems are extremely difficult and have eluded some the greatest mathematicians in history.

This state of affairs left open a fundamental question: is the solar system stable? In other words, will the planets stay in roughly their current orbits, or will the cumulative effects of small perturbations change their orbits substantially over time, possibly causing a planet to crash into the sun, or leave the solar system forever? Numerical solutions do not give the answer, because the errors involved tend to multiply over time, making them useless for long-term predictions. In general, a system is stable if a small change to its state will have only a small effect on its motion.  As a simple example, a marble resting in the bottom of a bowl is stable because if you give it a small push, it will roll around near the bottom.  On the other hand, a marble balanced at the top of an upside down bowl is unstable: a small push will cause it to roll off the bowl, far from its initial state.  In planetary motion, a two-body system is stable: if one of the planets receives a small push, say from a collision with a meteor, the new orbit will be another ellipse, very close to the original orbit. A system with more than two bodies, on the other hand, may not be stable: a small push can cause a planet to move far from its original orbit after time.  Although our own solar system has appeared to be stable during the few thousand years in which people have been observing the heavens (the blink of an eye on a cosmic time scale), what will happen to it in the long run remains an unsolved problem.

Let me hasten to assure you that you do not need to worry about the stability of the solar system, as even if it were unstable, it would probably take millions of years for the Earth’s orbit to change appreciably - despite the following satire, which Swift directed at eighteenth century mathematicians, thinly disguised as the scholars on the floating island of Laputa in Gulliver’s Travels

"The People are under continual Disquietudes, never enjoying a Minute’s Peace of Mind; and their Disturbances proceed from Causes which very little affect the rest of Mortals.  Their Apprehensions arise from several Changes they dread in the Celestial Bodies. For Instance; that the Earth by the continual Approaches of the Sun towards it, must in Course of Time be absorbed or swallowed up."

 Despite Swift’s scorn for mathematics, the stability of solutions to differential equations has important consequences here on Earth. For example, it can determine whether a bridge, swaying and twisting in a strong wind, will collapse (an event that actually occurred in 1940, to the Tacoma Narrows Bridge in Washington State.

Toward the end of the nineteenth century, the unresolved question of the stability of the solar system set the stage for the discovery of chaos. In 1887, King Oscar II of Sweden, who had a strong interest in mathematics, arranged an international mathematical competition involving four major problems, one of which was to solve the equations of motion for any planetary system. A committee of five of the most distinguished mathematicians of the time was to select the winner, who would take home a prize of 2,500 crowns. Henri Poincaré, a young mathematician who had already been working on problems in celestial mechanics for a number of years (and whose cousin Raymond would later become President of France), decided to take up the challenge.

As other mathematicians had done before, Poincaré considered a special case in which there are just three planetary bodies (the so-called three-body problem). Poincaré, however, tried a novel approach to the problem: rather than trying to explicitly solve the equations of motion, as mathematicians had always done previously, he looked at the qualitative behavior of planetary orbits - for example, whether they were periodic or followed more irregular paths. This approach had a liberating effect, enabling him to see possibilities that others had overlooked. What he discovered was quite unexpected: the motion of a planet in a three-body system can be very wild and unpredictable indeed. Its orbit can follow an apparently random curve, winding back around itself over and over again, like a long and tangled string. As he described such curves:

When one tries to depict the figure formed by these two curves and their infinity of intersections, each of which corresponds to a doubly asymptotic solution, these intersections form a kind of net, web, or infinitely tight mesh; neither of the two curves can ever cross itself, but must fold back on itself in a very complex way … Nothing can give us a better idea of the complexity of the three-body problem.

To get a rough idea of how such complicated orbits might occur, imagine a small asteroid, moving back and forth between two larger bodies - call them planets A and B.  Given the right conditions, it is possible for the asteroid to alternate between the two planets, spending some of its time revolving around A, and some revolving around B, like a bee flitting back and forth between two flowers.  If we track which planet the asteroid goes around at each revolution, we will get a sequence of A’s and B’s which can look statistically like a sequence of random coin tosses.

Poincaré’s discovery was surprising because it contradicted age-old assumptions about the motion of the planets. Beginning with the early Greeks, who thought that the planets moved in circles, astronomers had long believed that planetary motion was built up from simple motion. The theories of Kepler and Newton reinforced this belief. Before Poincaré, no one had imagined that such complicated, unpredictable motion could occur in the solar system. It must have come as a shock to him to realize that the motion of a planet could appear to be as random as that in a pinball machine. What he had seen hidden within his equations was the first glimpse of chaos. He might have recalled Keats’ lines:

    Then felt I like some watcher of the skies
    When a new planet swims into his ken;

In the end, Poincaré did not completely solve the problem of planetary motion, but his qualitative approach gave important new insights into the behavior of planetary orbits, and into solutions to differential equations in general. His work also led to an entirely new branch of mathematics called topology, which deals with properties of form and shape rather than number, and which has had a profound influence on twentieth century mathematics and physics. Like many problems in mathematics, the new ideas and methods generated by the study of planetary motion were as important as the original problem itself. In 1890, Poncaré submitted a memoir, entitled On the Problem of Three Bodies and the Equations of Dynamics, to the committee appointed to judge the competition. Recognizing the fundamental importance of Poincaré’s work, the committee awarded him the prize.

Describing the motion of the planets may seem as esoteric now as it did to Swift in the eighteenth century. What possible relevance can an imaginary solution to hypothetical mathematical problem have to us? One answer is that the same kinds of equations that govern the solar system also apply to events here on Earth. As a result, the complexity that Poincaré discovered in planetary orbits can also appear in terrestrial events, and that is what makes so many of them highly unpredictable.

To understand how chaos limits our ability to predict the future, we must look a little more closely at solutions to differential equations. A differential equation can have many solutions, corresponding to different possible motions in the system. To find a specific solution, we must know the initial conditions, such as the position and velocity of an object at a given instant.  Different initial conditions will yield different solutions: if the direction of the Earth’s orbit were to suddenly change, say by a collision with an asteroid, then the resulting orbit would also be changed. It turns out that in many systems, a small change in initial conditions will lead to large changes in the solution after a period of time. If the original change is too small to be detected, then the evolution of the system will be effectively unpredictable. This property, called sensitive dependence on initial conditions, is one of the essential features of chaos.

As Poincaré had realized, sensitive dependence on initial conditions is what makes weather prediction so difficult more than a few days in advance.  Models for the weather are based upon complicated systems of differential equations, involving many variables - temperature, atmospheric pressure, humidity and so forth – which can never be measured entirely accurately. Consequently, the initial conditions that are recorded for these equations – i.e. the current measurements of the variables - are not exactly the true initial conditions. So, even using world’s fastest computers, the computer-generated solution will soon begin to diverge from the true solution corresponding to the real weather. The net effect is that the forecast a week in advance is likely to be wrong, and furthermore the inaccuracy, being built into the mathematics, is unavoidable.   

Although Poincaré had identified the basic ideas of chaos by the end of the nineteenth century, the time was not right for the emergence of chaos theory in the scientific world. Perhaps because of the growing specialization in the sciences, scientists outside the field of celestial mechanics (and related mathematical areas) did not become aware of his work until much later. It would take another eighty years for its significance to be recognized.

In the early 1960’s, a meteorologist named Edward Lorenz began experimenting with a mathematical model of convection – i.e. hot air rising – a process very important to the weather. One version of his model consisted of three deceptively simple equations.  Unlike Poincaré, Lorenz had access to one of the earliest computers, with which he could plot numerical solutions to these equations. What the computer plot revealed was a strange curve in three-dimensional space, winding around two centers in a seemingly arbitrary fashion. A two-dimensional projection of it can be seen in the figure below (which is a more recent version of Lorenz’s original picture).

The Lorenz Attractor

Although Lorenz was unaware of Poincaré’s earlier work, the complexity of this curve was similar to that of the chaotic solutions Poincaré had found in planetary motion seventy years earlier.

Lorenz let the computer run for several hours, and then, to check his results, he ran the same simulation again. To save time, he decided to enter the value of the solution from the half-way point of the first run, and let it evolve from there.  To his great surprise, the new plot was not the same as the first: in fact, as time went on, it diverged more and more, until it bore no resemblance to the original solution.  Suddenly he realized that the numbers he had inserted in the second run were not exactly those produced at the half-way point of first run: he had rounded them off to the nearest thousandth.  In essence, he had generated a slightly different set of initial conditions at the half-way point, and in doing so, had accidentally rediscovered sensitive dependence on initial conditions. He concluded, as had Poincaré, that the weather might be inherently unpredictable.

Perhaps because Lorenz’s ideas were so new and strange, and the mathematics so unfamiliar, his work was largely ignored by other meteorologists.  It would take another ten years for his ideas to be noticed, and when they were, it was by researchers in an entirely different field, that of turbulence.

We all know turbulence when we see it, in the wild eddies and whirlpools of a mountain stream, but its mathematical description is very complicated and still far from complete.  For the engineer, the effect of turbulence is usually harmful. It disrupts the flow of air over an airplane wing and reduces the efficiency of moving vehicles and large oil pipelines; so understanding turbulence is a problem of real practical importance.  One can view the onset of turbulence in the wake of a canoe or rowboat. When the boat begins to move slowly, water slides around it leaving a smooth wake.  But as the boat moves faster, circular eddies begin to form, becoming more pronounced as the speed increases. The smooth flow disintegrates and the wake fills with complex motion; order turns into disorder.

In the early 1970’s, two mathematicians, David Ruelle and Floris Takens, proposed that the transition from a smooth flow to a turbulent one occurs suddenly, at the onset of chaos.  This contradicted earlier theories in fluid dynamics, according to which turbulence appears in stages. Ruelle and Takens conjectured that turbulence involves something they called a strange attractor, which is essentially the very complicated shape that a chaotic solution (such as Lorenz’s) settles down to over time. A strange attractor is a bizarre geometric object, which can only be roughly represented by computer pictures, and whose presence is indicative of chaos. Although the theory of turbulence is still being developed, there is a lot of experimental evidence to support the theory of strange attractors.

After the work of Ruelle and Takens became publicized, and word of it spread through the scientific grapevine, scientists began to recognize chaos in many other areas. Today, the ideas of chaos theory are being applied in a wide variety of fields, not only physics, chemistry and biology, but also ecology, physiology, epidemiology and others. The history of chaos theory, which has involved many people working independently in different areas, illustrates the fact that science often progresses by fits and starts, without planning or direction. Had scientists outside the field of celestial mechanics realized the significance of Poincaré’s work when it was first published, chaos theory might have made its debut much earlier.

Why chaos theory is just emerging now, three hundred years after Newton, is a difficult question. Certainly, one reason is that the invention of computers enabled scientists, beginning with Lorenz, to study chaotic solutions to differential equations in much greater depth than was previously possible, and also to visualize chaos, by drawing the complex pictures that arise from chaotic systems. Many scientists have commented on the importance of forming mental images to their work. In the study of chaos, the human mind seems to have needed some assistance.

A more speculative explanation is that for a very long time, scientists focused their attention on systems for which they could write down and solve differential equations, which they did with great success. As these solutions tended to have relatively simple behavior (compared to chaos), this created the false impression that such behavior was typical in nature. In a sense, scientists wore blinders that prevented them from noticing chaotic systems. One could even argue that scientists pursued problems that have nice, tidy solutions because of their desire to find simplicity in nature. The relationship between science and the external world is a tangled one, as the astronomer Eddington has observed:

"We have found a strange foot-print on the shores of the unknown.  We have devised profound theories, one after another, to account for its origin.  At last, we have succeeded in reconstructing the creature that made the foot-print.  And Lo! It is our own."

The real significance of chaos theory is that it gives us a very different view of the world, in which randomness and unpredictability are much more prevalent than was formerly realized. Chaos theory opens a new window onto the richness and complexity of nature. We can make a rough analogy between chaos theory and the theory of evolution (although I certainly would not compare the social or scientific impact of evolution with that of chaos theory). Like evolution, chaos theory provides a new way of thinking about the natural world. After Darwin, people no longer regarded plant and animal species as unchanging; after the emergence of chaos theory, they no longer expect that scientific theories will always be able to predict the long-term behavior of complex systems with a high degree of accuracy. For now, the implications of this point of view are mainly confined to science, but chaos theory may eventually come to have a broader influence on the world of ideas, just as Newton's theories influenced the intellectual culture of the seventeenth and eighteenth centuries, and Darwin's influenced that of the nineteenth and twentieth centuries. Whether chaos theory will have as far-reaching an influence as the ideas of Newton and Darwin remains to be seen.

Copyright 2004  by Paul Trow