Back | Next
Contents

CHAPTER 11
Chaos: The Unlicked Bear-Whelp

My original plan was to leave this chapter out of the book, as too technical. However, it was suggested to me that the science of chaos theory can be a fertile source of stories; more than that, it was pointed out that a story ("Feigenbaum Number," Kress, 1995) had been written drawing explicitly on an earlier article of mine on the subject. Faced with such direct evidence, I changed my mind. I did, however, remove most of the equations. I hope the result is still intelligible.

 

11.1 Chaos: new, or old? The Greek word "chaos" referred to the formless or disordered state before the beginning of the universe. The word has also been a part of the English language for a long time. Thus in Shakespeare's Henry VI, Part Three, the Duke of Gloucester (who in the next play of the series will become King Richard III, and romp about the stage in unabashed villainy) is complaining about his physical deformities. He is, he says, "like to a Chaos, or an unlick'd bear-whelp, that carries no impression like the dam." Chaos: something essentially random, an object or being without a defined shape.

Those lines were written about 1590. The idea of chaos is old, but chaos theory is a new term. Twenty years ago, no popular article had ever been written containing that expression. Ten years ago, the subject was all the rage. It was hard to find a science magazine without finding an article on chaos theory, complete with stunning color illustrations. Today, the fervor has faded, but the state of the subject is still unclear (perhaps appropriate, for something called chaos theory). Most of the articles seeking to explain what it is about are even less clear.

Part of the problem is newness. When someone writes about, say, quantum theory, the subject has to be presented as difficult, and subtle, and mysterious, because it is difficult, and subtle, and mysterious. To describe it any other way would be simply misleading. In the past sixty years, however, the mysteries have had time to become old friends of the professionals in the field. There are certainly enigmas, logical whirlpools into which you can fall and never get out, but at least the locations of those trouble spots are known. Writing about any well-established subject such as quantum theory is therefore in some sense easy.

In the case of chaos theory, by contrast, everything is new and fragmented; we face the other extreme. We are adrift on an ocean of uncertainties, guided by partial and inadequate maps, and it is too soon to know where the central mysteries of the subject reside.

Or, worse yet, to know if those mysteries are worth taking the time to explore. Is chaos a real "theory," something which will change the scientific world in a basic way, as that world was changed by Newtonian mechanics, quantum theory, and relativity? Or is it something essentially trivial, a subject which at the moment is benefiting from a catchy name and so enjoying a certain glamour, as in the past there have been fads for orgone theory, mesmerism, dianetics, and pyramidology?

I will defer consideration of that question, until we have had a look at the bases of chaos theory, where it came from, and where it seems to lead us. Then we can come back to examine its long-term prospects.

 

11.2 How to become famous. One excellent way to make a great scientific discovery is to take a fact that everyone knows must be the case—because "common sense demands it"—and ask what would happen if it were not true.

For example, it is obvious that the Earth is fixed. It has to be standing still, because it feels as though it is standing still. The Sun moves around it. Copernicus, by suggesting that the Earth revolves around the Sun, made the fundamental break with medieval thinking and set in train the whole of modern astronomy.

Similarly, it was clear to the ancients that unless you keep on pushing a moving object, it will slow down and stop. By taking the contrary view, that it takes a force (such as friction with the ground, or air resistance) to stop something, and otherwise it would just keep going, Galileo and Newton created modern mechanics.

Another case: To most people living before 1850, there was no question that animal and plant species are all so well-defined and different from each other that they must have been created, type by type, at some distinct time in the past. Charles Darwin and Alfred Russel Wallace, in suggesting in the 1850s a mechanism by which one form could change over time to another in response to natural environmental pressures, allowed a very different world view to develop. The theory of evolution and natural selection permitted species to be regarded as fluid entities, constantly changing, and all ultimately derived from the simplest of primeval life forms.

And, to take one more example, it was clear to everyone before 1900 that if you kept on accelerating an object, by applying force to it, it would move faster and faster until it was finally traveling faster than light. By taking the speed of light as an upper limit to possible speeds, and requiring that this speed to be the same for all observers, Einstein was led to formulate the theory of relativity.

It may make you famous, but it is a risky business, this offering of scientific theories that ask people to abandon their long-cherished beliefs about what "just must be so." As Thomas Huxley remarked, it is the customary fate of new truths to begin as heresies.

Huxley was speaking metaphorically, but a few hundred years ago he could have been speaking literally. Copernicus did not allow his work on the movement of the Earth around the Sun to be published in full until 1543, when he was on his deathbed, nearly 30 years after he had first developed the ideas. He probably did the right thing. Fifty-seven years later Giordano Bruno was gagged and burned at the stake for proposing ideas in conflict with theology, namely, that the universe is infinite and there are many populated worlds. Thirty-three years after that, Galileo was made to appear before the Inquisition and threatened with torture because of his "heretical" ideas. His work remained on the Catholic Church's Index of prohibited books for over two hundred years.

By the nineteenth century critics could no longer have a scientist burned at the stake, even though they may have wanted to. Darwin was merely denounced as a tool of Satan. However, anyone who thinks this issue is over and done with can go today and have a good argument about evolution and natural selection with the numerous idiots who proclaim themselves to be scientific creationists.

Albert Einstein fared better, mainly because most people had no idea what he was talking about. However, from 1905 to his death in 1955 he became the target of every crank and scientific nitwit outside (and often inside) the lunatic asylums.

Today we will be discussing an idea, contrary to common sense, that has been developing in the past twenty years. So far its proposers have escaped extreme censure, though in the early days their careers may have suffered because no one believed them—or understood what they were talking about.

 

11.3 Building models. The idea at the heart of chaos theory can be simply stated, but we will have to wind our way into it.

Five hundred years ago, mathematics was considered essential for bookkeeping, surveying, and trading, but it was not considered to have much to do with the physical processes of Nature. Why should it? What do abstract symbols on a piece of paper have to do with the movement of the planets, the flow of rivers, the blowing of soap bubbles, the flight of kites, or the design of buildings?

Little by little, that view changed. Scientists found that physical processes could be described by equations, and solving those equations allowed predictions to be made about the real world. More to the point, they were correct predictions. By the nineteenth century, the fact that manipulation of the purely abstract entities of mathematics could somehow tell us how the real world would behave was no longer a surprise. Sir James Jeans could happily state, in 1930, "all the pictures which science now draws of nature, and which alone seem capable of according with observational fact, are mathematical pictures," and " . . . the universe appears to have been designed by a pure mathematician."

The mystery had vanished, or been subsumed into divinity. But it should not have. It is a mystery still.

I would like to illustrate this point with the simplest problem of Newtonian mechanics. Suppose that we have an object moving along a line with a constant acceleration. It is easy to set up a situation in the real world in which an object so moves, at least approximately.

It is also easy to describe this situation mathematically, and to determine how the final position depends on the speed and initial position. When we do this, we find that a tiny change in initial speed or position causes a small change in final speed and position. We say that the solution is a continuous function of the input variables.

This is an especially simple example, but scientists are at ease with far more complex cases.

Do you want to know how a fluid will move? Write down a rather complex equation (to be specific, the three-dimensional time-dependent Navier-Stokes equation for compressible, viscous flow). Solve the equation. That's not a simple proposition, and you may have to resort to a computer. But when you have the results, you expect them to apply to real fluids. If they do not, it is because the equation you began with was not quite right—maybe we need to worry about electromagnetic forces, or plasma effects. Or maybe the integration method you used was numerically unstable, or the finite difference interval too crude. The idea that the mathematics cannot describe the physical world never even occurs to most scientists. They have in the back of their minds an idea first made explicit by Laplace: the whole universe is calculable, by defined mathematical laws. Laplace said that if you told him (or rather, if you told a demon, who was capable of taking in all the information) the position and speed of every particle in the Universe, at one moment, he would be able to define the Universe's entire future, and also its whole past.

The twentieth century, and the introduction by Heisenberg of the Uncertainty Principle, weakened that statement, because it showed that it was impossible to know precisely the position and speed of a body. Nonetheless, the principle that mathematics can exactly model reality is usually still unquestioned.

It should be, because it is absolutely extraordinary that the pencil and paper scrawls that we make in our studies correspond to activities in the real world outside.

Now, hidden away in the assumption that the world can be described by mathematics there is another one; one so subtle that most people never gave it a thought. This is the assumption that chaos theory makes explicit, and then challenges. We state it as follows:

Simple equations must have simple solutions.

There is no reason why this should be so, except that it seems that common sense demands it. And, of course, we have not defined "simple."

Let us return to our accelerating object, where we have a simple-seeming equation, and an explicit solution. One requirement of a simple solution is that it should not "jump around" when we make a very small change in the system it describes. For example, if we consider two cases of an accelerated object, and the only difference between them is a tiny change in the original position of the object, we would expect a small change in the final position. And this is the case. That is exactly what was meant by the earlier statement, that the solution was a continuous function of the inputs.

But now consider another simple physical system, a rigid pendulum (this was one of the first cases where the ideas of chaos theory emerged). If we give the pendulum a small push, it swings back and forward. Push it a little harder, and a little harder, and what happens? Well, for a while it makes bigger and bigger swings. But at some point, a very small change to the push causes a totally different type of motion. Instead of swinging back and forward, the pendulum keeps on going, right over the top and down the other side. If we write the expression for the angle as a function of time, in one case the angle is a periodic function (back and forth) and in the other case it is constantly increasing (round and round). And the change from one to the other occurs when we make an infinitesimal change in the initial speed of the pendulum bob. This type of behavior is known as a bifurcation in the behavior of the solution, and it is a worrying thing. A simple equation begins to exhibit a complicated solution. The solution of the problem is no longer a continuous function of the input variables.

At this point, the reasonable reaction might well be, so what? All that we have done is show that certain simple equations don't have really simple solutions. That does not seem like an earth-shaking discovery. For one thing, the boundary between the two types of solution for the pendulum, oscillating and rotating, is quite clear-cut. It is not as though the definition of the location of the boundary itself were a problem.

Can situations arise where this is a problem? Where the boundary is difficult to define in an intuitive way? The answer is, yes. In the next section we will consider simple systems that give rise to highly complicated boundaries between regions of fundamentally different behavior.

 

11.4 Iterated functions. Some people have a built-in mistrust of anything that involves the calculus. When you use it in any sort of argument, they say, logic and clarity have already departed. The solutions for examples I have given so far implied that we write down and solve a differential equation, so calculus was needed to define the behavior of the solutions. However, we don't need calculus to demonstrate fundamentally chaotic behavior; and many of the first explorations of what we now think of as chaotic functions were done without calculus. They employed what is called iterated function theory. Despite an imposing name, the fundamentals of iterated function theory are so simple that they can be done with an absolute minimum knowledge of mathematics. They do, however, benefit from the assistance of computers, since they call for large amounts of tedious computation.

Consider the following very simple operation. Take two numbers, x and r. Form the value y=rx(1-x).

Now plug the value of y back in as a new value for x. Repeat this process, over and over.

For example, suppose that we take r=2, and start with x=0.1. Then we find y=0.18.

Plug that value in as a new value for x, still using r=2, and we find a new value, y=0.2952.

Keep going, to find a sequence of y's, 0.18, 0.2952, 0.4161, 0.4859, 0.4996, 0.5000, 0.5000 . . .

In the language of mathematics, the sequence of y's has converged to the value 0.5. Moreover, for any starting value of x, between 0 and 1, we will always converge to the same value, 0.5, for r=2.

Here is the sequence when we begin with x=0.6:

0.4800, 0.4992, 0.5000, 0.5000 . . .

Because the final value of y does not depend on the starting value, it is termed an attractor for this system, since it "draws in" any sequence to itself.

The value of the attractor depends on r. If we start with some other value of r, say r=2.5, we still produce a convergent sequence. For example, if for r=2.5 we begin with x=0.1, we find successive values: 0.225, 0.4359, 0.6147, 0.5921, 0.6038, 0.5981, . . . 0.6. Starting with a different x still gives the same final value, 0.6.

For anyone who is familiar with a programming language such as C or even BASIC (Have you noticed how computers are used less and less to compute?), I recommend playing this game for yourself. The whole program is only a dozen lines long. Suggestion: Run the program in double precision, so you don't get trouble with round-off errors. Warning: Larking around with this sort of thing will consume hours and hours of your time.

The situation does not change significantly with r=3. We find the sequence of values: 0.2700, 0.5913, 0.7250, 0.5981, 0.7211 . . . 0.6667. This time it takes thousands of iterations to get to a final converged value, but it makes it there in the end. Even after only a dozen or two iterations we can begin to see it "settling-in" to its final value.

There have been no surprises so far. What happens if we increase r a bit more, to 3.1? We might expect that we will converge, but even more slowly, to a single final value.

We would be wrong. Something very odd happens. The sequence of numbers that we generate has a regular structure, but now the values alternate between two different numbers, 0.7645, and 0.5580. Both these are attractors for the sequence. It is as though the sequence cannot make up its mind. When r is increased past the value 3, the sequence "splits" to two permitted values, which we will call "states," and these occur alternately.

Let us increase the value of r again, to 3.4. We find the same behavior, a sequence that alternates between two values.

But by r=3.5, things have changed again. The sequence has four states, four values that repeat one after the other. For r=3.5, we find the final sequence values: 0.3828, 0.5009, 0.8269, and 0.8750. Again, it does not matter what value of x we started with, we will always converge on those same four attractors.

Let us pause for a moment and put on our mathematical hats. If a mathematician is asked the question, Does the iteration y=rx(1-x) converge to a final value?, he will proceed as follows:

Suppose that there is a final converged value, V, towards which the iteration converges. Then when we reach that value, no matter how many iterations it takes, at the final step x will be equal to V, and so will y. Thus we must have V=rV(1-V).

Solving for V, we find V=0, which is a legitimate but uninteresting solution, or V=(r-1)/r. This single value will apply, no matter how big r may be. For example, if r=2.5, then V=1.5/2.5=0.6, which is what we found. Similarly, for r=3.5, we calculate V=2.5/3.5=0.7142857.

But this is not what we found when we did the actual iteration. We did not converge to that value at all, but instead we obtained a set of four values that cycled among themselves. So let us ask the question, what would happen if we began with x=0.7142857, as our starting guess? We certainly have the right to use any initial value that we choose. Surely, the value would simply stay there?

No, it would not.

What we would find is that on each iteration, the value of y changes. It remains close to 0.7142857 on the first few calculations, then it—quite quickly—diverges from that value and homes in on the four values that we just mentioned: 0.3828, 0.5009, etc. In mathematical terms, the value 0.7142857 is a solution of the iterative process for r=3.5. But it is an unstable solution. If we start there, we will rapidly move away to other multiple values.

Let us return to the iterative process. By now we are not sure what will happen when we increase r. But we can begin to make some guesses. Bigger values of r seem to lead to more and more different values, among which the sequence will oscillate, and it seems as though the number of these values will always be a power of two. Furthermore, the "splitting points" seem to be coming faster and faster.

Take r=3.52, or 3.53, or 3.54. We still have four values that alternate. But by r=3.55, things have changed again. We now find eight different values that repeat, one after the other. By r=3.565, we have 16 different values that occur in a fixed order, over and over, as we compute the next elements of the sequence.

It is pretty clear that we are approaching some sort of crisis, since the increments that we can make in r, without changing the nature of the sequence, are getting smaller and smaller. In fact, the critical value of r is known to many significant figures. It is r=3.569945668. . . . As we approach that value there are 2n states in the sequence, and n is growing fast.

What happens if we take r bigger than this, say r=3.7? We still produce a sequence—there is no difficulty at all with the computations—but it is a sequence without any sign of regularity. There are no attractors, and all values seem equally likely. It is fair to say that it is chaos, and the region beyond the critical value of r is often called the chaos regime.

This may look like a very special case, because all the calculations were done based on one particular function, y=rx(1-x). However, it turns out that the choice of function is much less important than one would expect. If we substituted any up-and-down curve between zero and one we would get a similar result. As r increases, the curve "splits" again and again. There is a value of r for which the behavior becomes chaotic.

For example, suppose that we use the form y=r.sin(x)/4 (the factor of 4 is to make sure that the maximum value of y is the same as in the first case, namely, 1/4). By the time we reach r=3.4 we have four different values repeating in the sequence. For r=3.45 we have eight attractors. Strangest of all, the way in which we approach the critical value for this function has much in common with the way we approached it for the first function that we used. They both depend on a single convergence number that tells the rate at which new states will be introduced as r is increased. That convergence number is 4.669201609 . . . , and is known as the Feigenbaum number, after Mitchell Feigenbaum, who first explored in detail this property of iterated sequences. This property of common convergence behavior, independent of the particular function used for the iteration, is called universality. It seems a little presumptuous as a name, but maybe it won't, in twenty years time.

This discussion of iterated functions may strike you as rather tedious, very complicated, very specialized, and a way of obtaining very little for a great deal of work. However, the right way to view what we have just done is this: we have found a critical value, less than which there is a predictable, although increasingly complicated behavior, and above which there is a completely different and chaotic behavior. Moreover, as we approach the critical value, the number of possible states of the system increases very rapidly, and tends to infinity.

To anyone who has done work in the field of fluid dynamics, that is a very suggestive result. For fluid flow there is a critical value below which the fluid motion is totally smooth and predictable (laminar flow) and above which it is totally unpredictable and chaotic (turbulent flow). Purists will object to my characterizing turbulence as "chaotic," since although it appears chaotic and disorganized as a whole, there is a great deal of structure on the small scale since millions of molecules must move together in an organized way. However, the number of states in turbulent flow is infinite, and there has been much discussion of the way in which the single state of laminar flow changes to the many states of turbulent flow. Landau proposed that the new states must come into being one at a time. It was also assumed that turbulent behavior arose as a consequence of the very complicated equations of fluid dynamics.

Remember the "common sense rule": Simple equations must have simple solutions. And therefore, complicated behavior should only arise from complicated equations. For the first time, we see that this may be wrong. A very simple system is exhibiting very complicated behavior, reminiscent of what happens with fluid flow. Depending on some critical variable, it may appear totally predictable and well-behaved, or totally unpredictable and chaotic. Moreover, experiments show that in turbulence the new, disorganized states come into being not one by one, but through a doubling process as the critical parameter is approached. Maybe turbulence is a consequence of something in the fluid flow equations that is unrelated to their complexity—a hidden structure that is present even in such simple equations as we have been studying.

This iterated function game is interesting, even suggestive, but to a physicist it was for a long time little more than that. Physics does not deal with computer games, went the argument. It deals with mathematical models that describe a physical system, in a majority of cases through a series of differential equations. These equations are solved, to build an idea of how Nature will behave in any given circumstance.

The trouble is, although such an approach works wonderfully well in many cases, there are classes of problems that it doesn't seem to touch. Turbulence is one. "Simple" systems, like the dripping of water from a faucet, can be modeled in principle, but in practice the difficulties in formulation and solution are so tremendous that no one has ever offered a working analysis of a dripping tap.

The problems where the classical approach breaks down often have one thing in common: they involve a random, or apparently random, element. Water in a stream breaks around a stone this way, then that way. A snowflake forms from supersaturated vapor, and every one is different. A tap drips, then does not drip, in an apparently random way. All these problems are described by quite different systems of equations. What scientists wanted to see was physical problems, described by good old differential equations, that also displayed bifurcations, and universality, and chaotic behavior.

They had isolated examples already. For example, the chemical systems that rejoice in the names of the Belousov-Zhabotinsky reaction and the Brusselator exhibit a two-state cyclic behavior. So does the life cycle of the slime mold, Dictyostelium discoideum. However, such systems are very tricky to study for the occurrence of such things as bifurcations, and involve all the messiness of real-world experiments. Iterated function theory was something that could be explored in the precise and austere world of computer logic, unhindered by the intrusion of the external world.

We must get to that external and real world eventually, but before we do so, let's take a look at another element of iterated function theory. This one has become very famous in its own right (rather more so, in my opinion, than it deserves to be for its physical significance, but perhaps justifiably most famous for its artistic significance).

The subject is fractals, and the contribution to art is called the Mandelbrot Set.

11.5 Sick curves and fractals. Compare the system we have just been studying with the case of the pendulum. There we had a critical curve, rather than a critical value. On the other hand, the behavior on both sides of the critical curve was not chaotic. Also, the curve itself was well-behaved, meaning that it was "smooth" and predictable in its shape.

Is there a simple system that on the one hand exhibits a critical curve, and on the other hand shows chaotic behavior?

There is. It is one studied in detail by Benoit Mandelbrot, and it gives rise to a series of amazing objects (one hesitates to call them curves, or areas).

We just looked at a case of an iterated function where only one variable was involved. We used x to compute y, then replaced x with y, and calculated a new y, and so on. It is no more difficult to do this, at least in principle, if there are two starting values, used to compute two new values. For example, we could have:

y=(w2-x2)+a
z=2wx+b

and when we had computed a pair (y,z) we could use them to replace the pair (w,x). (Readers familiar with complex variable theory will see that I am simply writing the relation z=z2+c, where z and c are complex numbers, in a less elegant form.)

What happens if we take a pair of constants, (a,b), plug in zero starting values for w and x, and let our computers run out lots of pairs, (y,z)? This is a kind of two-dimensional equivalent to what we did with the function y=rx(1-x), and we might think that we will find similar behavior, with a critical curve replacing the critical value.

What happens is much more surprising. We can plot our (y,z) values in two dimensions, just as we could plot speeds and positions for the case of the pendulum to make a phase space diagram. And, just as was the case with the pendulum, we will find that the whole plane divides into separate regions, with boundaries between them. The boundaries are the boundary curves of the "Mandelbrot set," as it is called. If, when we start with an (a,b) pair and iterate for (y,z) values, one or both of y and z run off towards infinity, then the point (a,b) is not a member of the Mandelbrot set. If the (y,z) pairs settle down to some value, or if they cycle around a series of values without ever diverging off towards infinity, then the point (a,b) is a member of the Mandelbrot set. The tricky case is for points on the boundary, since convergence is slowest there for the (y,z) sequence. However, those boundaries can be mapped. And they are as far as can be imagined from the simple, well-behaved curve that divided the two types of behavior of the pendulum. Instead of being smooth, they are intensely spiky; instead of just one curve, there is an infinite number.

The results of plotting the Mandelbrot set can be found in many articles, because they have a strange beauty unlike anything else in mathematics. Rather than drawing them here, I will refer you to James Gleick's book, Chaos: Making a New Science (Gleick, 1987), which shows some beautiful color examples of parts of the set. All this, remember, comes from the simple function we defined, iterated over and over to produce pairs of (y,z) values corresponding to a particular choice of a and b. The colors seen in so many art shows, by the way, while not exactly a cheat, are not fundamental to the Mandelbrot set itself. They are assigned depending on how many iterations it takes to bring the (y,z) values to convergence, or to a stable repeating pattern.

The Mandelbrot set also exhibits a feature known as scaling, which is very important in many areas of physics. It says, in its simplest terms, that you cannot tell the absolute scale of the phenomenon you are examining from the structure of the phenomenon itself.

That needs some explanation. Suppose that you want to know the size of a given object—say, a snowflake. One absolute measure, although a rather difficult one to put into practice, would be to count the number of atoms in that snowflake. Atoms are fundamental units, and they do not change in their size.

But suppose that instead of the number of atoms, you tried to use a different measure, say, the total area of the snowflake. That sounds much easier than looking at the individual atoms. But you would run into a problem, because as you look at the surface of the snowflake more and more closely, it becomes more and more detailed. A little piece of a snowflake has a surface that looks very much like a little piece of a little piece of a snowflake; a little piece of a little piece resembles a little piece of a little piece of a little piece, and so on. It stays that way until you are actually seeing the atoms. Then you at last have the basis for an absolute scale.

Mathematical entities, unlike snowflakes, are not made up of atoms. There are many mathematical objects that "scale forever," meaning that each level of more detailed structure resembles the one before it. The observer has no way of assigning any absolute scale to the structure. The sequence-doubling phenomenon that we looked at earlier is rather like that. There is a constant ratio between the distances at which the doublings take place, and that information alone is not enough to tell you how close you are to the critical value in absolute terms.

Similarly, by examining a single piece of the Mandelbrot set it is impossible to tell at what level of detail the set is being examined. The set can be examined more and more closely, forever, and simply continues to exhibit more and more detail. There is never a place where we arrive at the individual "atoms" that make up the set. In this respect, the set differs from anything encountered in nature, where the fundamental particles provide a final absolute scaling. Even so, there are in nature things that exhibit scaling over many orders of magnitude. One of the most famous examples is a coastline. If you ask "How long is the coastline of the United States?" a first thought is that you can go to a map and measure it. Then it's obvious that the map has smoothed the real coastline. You need to go to larger scale maps, and larger scale maps. A coastline "scales," like the surface of a snowflake, all the way down to the individual rocks and grains of sand. You find larger and larger numbers for the length of the coast. Another natural phenomenon that exhibits scaling is—significantly—turbulent flow. Ripples ride on whirls that ride on vortices that sit on swirls that are made up of eddies, on and on.

There are classes of mathematical curves that, like coastlines, do not have a length that one can measure in the usual way. A famous one is called the "Koch curve" and although it has weird properties it is easy to describe how to make it.

Take an equilateral triangle. At the middle of each side, facing outward, place equilateral triangles one third the size. Now on each side of the resulting figure, place more outward-facing equilateral triangles one third the size of the previous ones. Repeat this process indefinitely, adding smaller and smaller triangles to extend the outer boundary of the figure. The end result is a strange figure indeed, rather like a snowflake in overall appearance. The area it encloses is finite, but the length of its boundary turns out to be 3x4/3x4/3x4/3 . . . , which diverges to infinity. Curves like this are known as pathological curves. The word "pathological" means diseased, or sick. It is a good name for them.

There is a special term reserved for the boundary dimension of such finite/infinite objects, and it is called the Hausdorff-Besicovitch measure. That's a bit of a mouthful. The boundaries of the Mandelbrot set have a fractional Hausdorff-Besicovitch measure, rather than the usual dimension (1) of the boundary of a plane curve, and most people now prefer to use the term coined by Mandelbrot, and speak of fractal dimension rather than Hausdorff-Besicovitch dimension. Objects that exhibit such properties, and other such features as scaling, were named fractals by Mandelbrot.

Any discussion of chaos has to include the Mandelbrot set, scaling, and fractals, because it offers by far the most visually attractive part of the theory. I am not convinced that it is as important as Feigenbaum's universality. However, it is certainly beautiful to look at, highly suggestive of shapes found in Nature and—most important of all—it tends to show up in the study of systems that physicists are happy with and impressed by, since they represent the result of solving systems of differential equations.

11.6 Strange attractors. This is all very interesting, but in our discussion so far there is a big missing piece. We have talked of iterated functions, and seen that even very simple cases can exhibit "chaotic" behavior. And we have also remarked that physical systems often exhibit chaotic behavior. However, such systems are usually described in science by differential equations, not by iterated functions. We need to show that the iterated functions and the differential equations are close relatives, at some fundamental level, before we can be persuaded that the results we have obtained so far in iterated functions can be used to describe events in the real world.

Let us return to one simple system, the pendulum, and examine it in a little more detail. First let's recognize the difference between an idealized pendulum and one in the real world. In the real world, every pendulum is gradually slowed by friction, until it sits at the bottom of the swing, unmoving. This is a single point, termed an attractor for pendulum motion, and it is a stable attractor. All pendulums, unless given a periodic kick by a clockwork or electric motor, will settle down to the zero angle/zero speed point. No matter with what value of angle or speed a pendulum is started swinging, it will finish up at the stable attractor. In mathematical terminology, all points of phase space, neighbors or not, will approach each other as time goes on.

A friction-free pendulum, or one that is given a small constant boost each swing, will behave like the idealized one, swinging and swinging, steadily and forever. Points in phase space neither tend to be drawn towards each other, nor repelled from each other.

But suppose that we had a physical system in which points that began close together tended to diverge from each other. That is the very opposite of the real-world pendulum, and we must first ask if such a system could exist.

It can, as we shall shortly see. It is a case of something that we have already encountered, a strong dependence on initial conditions, since later states of the system differ from each other a great deal, though they began infinitesimally separated. In such a case, the attractor is not a stable attractor, or even a periodic attractor. Instead it is called a strange attractor.

This is an inspired piece of naming, comparable with John Archibald Wheeler's introduction of the term "black hole." Even people who have never heard of chaos theory pick up on it. It is also an appropriate name. The paths traced out in phase space in the region of a strange attractor are infinitely complex, bounded in extent, never repeating; chaotic, yet chaotic in some deeply controlled way. If there can be such a thing as controlled chaos, it is seen around strange attractors.

We now address the basic question: Can strange attractors exist mathematically? The simple pendulum cannot possess a strange attractor; so far we have offered no proof that any system can exhibit one. However, it can be proved that strange attractors do exist in mathematically specified systems, although a certain minimal complexity is needed in order for a system to possess a strange attractor. We have this situation: Simple equations can exhibit complicated solutions, but for the particular type of complexity represented by the existence of strange attractors, the system of equations can't be too simple. To be specific, a system of three or more nonlinear differential equations can possess a strange attractor; less than three equations, or more than three linear equations, cannot. (The mathematical statement of this fact is simpler but far more abstruse: A system can exhibit a strange attractor if at least one Lyapunov exponent is positive.)

If we invert the logic, it is tempting to make another statement: Any physical system that shows an ultra-sensitive dependence on initial conditions has a strange attractor buried somewhere in its structure.

This is a plausible but not a proven result. I am tempted to call it the most important unsolved problem of chaos theory. If it turns out to be true, it will have a profound unifying influence on numerous branches of science. Systems whose controlling equations bear no resemblance to each other will share a structural resemblance, and there will be the possibility of developing universal techniques that apply to the solution of complicated problems in a host of different areas. One thing in common with every problem that we have been discussing is nonlinearity. Nonlinear systems are notoriously difficult to solve, and seem to defy intuition. Few general techniques exist today for tackling nonlinear problems, and some new insight is desperately needed.

If chaos theory can provide that insight, it will have moved from being a baffling grab-bag of half results, interesting conjectures, and faintly seen relationships, to become a real "new science." We are not there yet. But if we can go that far, then our old common sense gut instinct, that told us simple equations must have simple solutions, will have proved no more reliable than our ancestors' common sense instinctive knowledge that told them the Earth was flat. And the long-term implications of that new thought pattern may be just as revolutionary to science.

Today, we are in that ideal time for writers, where what can be speculated in chaos theory far exceeds what is known. I still consider the ultimate importance of chaos theory as not proven, but it has certainly caused a change of outlook. Today you hear weather forecasters referring to the "butterfly effect," in which a butterfly flapping its wings in the East Indies causes a hurricane in the Caribbean—a powerful illustration of sensitive dependence on initial conditions.

Science fiction writers long ago explored the idea of the sensitive dependence on initial conditions in time travel stories. In "A Sound of Thunder" (Bradbury, 1952), a tiny change in the past produces a very different future.

Is time really like that? Or would small changes made in the past tend to damp out over time, to produce the same present? Stating this another way, if we were to rerun the history of the planet, would the same life-forms emerge, or a totally different set? This is a hot topic at the moment in evolutionary biology.

 

Back | Next
Contents
Framed