Chaos - James Gleick [80]
No one realized it then, but Lorenz had looked at the same equation in 1964, as a metaphor for a deep question about climate. The question was so deep that almost no one had thought to ask it before: Does a climate exist? That is, does the earth’s weather have a long-term average? Most meteorologists, then as now, took the answer for granted. Surely any measurable behavior, no matter how it fluctuates, must have an average. Yet on reflection, it is far from obvious. As Lorenz pointed out, the average weather for the last 12,000 years has been notably different than the average for the previous 12,000, when most of North America was covered by ice. Was there one climate that changed to another for some physical reason? Or is there an even longer-term climate within which those periods were just fluctuations? Or is it possible that a system like the weather may never converge to an average?
Lorenz asked a second question. Suppose you could actually write down the complete set of equations that govern the weather. In other words, suppose you had God’s own code. Could you then use the equations to calculate average statistics for temperature or rainfall? If the equations were linear, the answer would be an easy yes. But they are nonlinear. Since God has not made the actual equations available, Lorenz instead examined the quadratic difference equation.
Like May, Lorenz first examined what happened as the equation was iterated, given some parameter. With low parameters he saw the equation reaching a stable fixed point. There, certainly, the system produced a “climate” in the most trivial sense possible—the “weather” never changed. With higher parameters he saw the possibility of oscillation between two points, and there, too, the system converged to a simple average. But beyond a certain point, Lorenz saw that chaos ensues. Since he was thinking about climate, he asked not only whether continual feedback would produce periodic behavior, but also what the average output would be. And he recognized that the answer was that the average, too, fluctuated unstably. When the parameter value was changed ever so slightly, the average might change dramatically. By analogy, the earth’s climate might never settle reliably into an equilibrium with average long-term behavior.
As a mathematics paper, Lorenz’s climate work would have been a failure—he proved nothing in the axiomatic sense. As a physics paper, too, it was seriously flawed, because he could not justify using such a simple equation to draw conclusions about the earth’s climate. Lorenz knew what he was saying, though. “The writer feels that this resemblance is no mere accident, but that the difference equation captures much of the mathematics, even if not the physics, of the transitions from one regime of flow to another, and, indeed, of the whole phenomenon of instability.” Even twenty years later, no one could understand what intuition justified such a bold claim, published in Tellus, a Swedish meteorology journal. (“Tellus! Nobody reads Tellus,” a physicist exclaimed bitterly.) Lorenz was coming to understand ever more deeply the peculiar possibilities of chaotic systems—more deeply than he could express in the language of meteorology.
As he continued to explore the changing masks of dynamical systems, Lorenz realized that systems slightly more complicated than the quadratic map could produce other kinds of unexpected patterns. Hiding within a particular system could be more than one stable solution. An observer might see one