Chaos - James Gleick [20]
But unpredictability was not the reason physicists and mathematicians began taking pendulums seriously again in the sixties and seventies. Unpredictability was only the attention-grabber. Those studying chaotic dynamics discovered that the disorderly behavior of simple systems acted as a creative process. It generated complexity: richly organized patterns, sometimes stable and sometimes unstable, sometimes finite and sometimes infinite, but always with the fascination of living things. That was why scientists played with toys.
One toy, sold under the name “Space Balls” or “Space Trapeze,” is a pair of balls at opposite ends of a rod, sitting like the crossbar of a T atop a pendulum with a third, heavier ball at its foot. The lower ball swings back and forth while the upper rod rotates freely. All three balls have little magnets inside, and once set in motion the device keeps going because it has a battery-powered electromagnet embedded in the base. The device senses the approach of the lowest ball and gives it a small magnetic kick each time it passes. Sometimes the apparatus settles into a steady, rhythmic swinging. But other times, its motion seems to remain chaotic, always changing and endlessly surprising.
Another common pendulum toy is no more than a so-called spherical pendulum—a pendulum free to swing not just back and forth but in any direction. A few small magnets are placed around its base. The magnets attract the metal bob, and when the pendulum stops, it will have been captured by one of them. The idea is to set the pendulum swinging and guess which magnet will win. Even with just three magnets placed in a triangle, the pendulum’s motion cannot be predicted. It will swing back and forth between A and B for a while, then switch to B and C, and then, just as it seems to be settling on C, jump back to A. Suppose a scientist systematically explores the behavior of this toy by making a map, as follows: Pick a starting point; hold the bob there and let go; color the point red, blue, or green, depending on which magnet ends up with the bob. What will the map look like? It will have regions of solid red, blue, or green, as one might expect—regions where the bob will swing reliably to a particular magnet. But it can also have regions where the colors are woven together with infinite complexity. Adjacent to a red point, no matter how close one chooses to look, no matter how much one magnifies the map, there will be green points and blue points. For all practical purposes, then, the bob’s destiny will be impossible to guess.
Traditionally, a dynamicist would believe that to write down a system’s equations is to understand the system. How better to capture the essential features? For a playground swing or a toy, the equations tie together the pendulum’s angle, its velocity, its friction, and the force driving it. But because of the little bits of nonlinearity in these equations, a dynamicist would find himself helpless to answer the easiest practical questions about the future of the system. A computer can address the problem by simulating it, rapidly calculating each cycle. But simulation brings its own problem: the tiny imprecision built into each calculation rapidly takes over, because this is a system with sensitive dependence on initial conditions. Before long, the signal disappears and all that remains is noise.
Or is it? Lorenz had found unpredictability, but he had also found pattern. Others, too, discovered suggestions of structure amid seemingly random behavior. The example of the pendulum was simple enough to disregard, but those who chose not to disregard it found a provocative message. In some sense, they realized, physics understood perfectly the fundamental mechanisms of pendulum motion but could not