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R7 ≈ 3.569891

R8 ≈ 3.569934

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R∞ ≈ 3.569946

Here, R1 corresponds to period 21 (= 2), R2 corresponds to period 22 (= 4), and in general, Rn corresponds to period 2n. The symbol ∞ (“infinity”) is used to denote the onset of chaos—a trajectory with an infinite period.

Feigenbaum noticed that as the period increases, the R values get closer and closer together. This means that for each bifurcation, R has to be increased less than it had before to get to the next bifurcation. You can see this in the bifurcation diagram of Figure 2.11: as R increases, the bifurcations get closer and closer together. Using these numbers, Feigenbaum measured the rate at which the bifurcations get closer and closer; that is, the rate at which the R values converge. He discovered that the rate is (approximately) the constant value 4.6692016. What this means is that as R increases, each new period doubling occurs about 4.6692016 times faster than the previous one.

This fact was interesting but not earth-shaking. Things started to get a lot more interesting when Feigenbaum looked at some other maps—the logistic map is just one of many that have been studied. As I mentioned above, a few years before Feigenbaum made these calculations, his colleagues at Los Alamos, Metropolis, Stein, and Stein, had shown that any unimodal map will follow a similar period-doubling cascade. Feigenbaum’s next step was to calculate the rate of convergence for some other unimodal maps. He started with the so-called sine map, an equation similar to the logistic map but which uses the trigonometric sine function.

Feigenbaum repeated the steps I sketched above: he calculated the values of R at the period-doubling bifurcations in the sine map, and then calculated the rate at which these values converged. He found that the rate of convergence was 4.6692016.

Feigenbaum was amazed. The rate was the same. He tried it for other unimodal maps. It was still the same. No one, including Feigenbaum, had expected this at all. But once the discovery had been made, Feigenbaum went on to develop a mathematical theory that explained why the common value of 4.6692016, now called Feigenbaum’s constant, is universal—which here means the same for all unimodal maps. The theory used a sophisticated mathematical technique called renormalization that had been developed originally in the area of quantum field theory and later imported to another field of physics: the study of phase transitions and other “critical phenomena.” Feigenbaum adapted it for dynamical systems theory, and it has become a cornerstone in the understanding of chaos.

It turned out that this is not just a mathematical curiosity. In the years since Feigenbaum’s discovery, his theory has been verified in several laboratory experiments on physical dynamical systems, including fluid flow, electronic circuits, lasers, and chemical reactions. Period-doubling cascades have been observed in these systems, and values of Feigenbaum’s constant have been calculated in steps similar to those we saw above. It is often quite difficult to get accurate measurements of, say, what corresponds to R values in such experiments, but even so, the values of Feigenbaum’s constant found by the experimenters agree well within the margin of error to Feigenbaum’s value of approximately 4.6692016. This is impressive, since Feigenbaum’s theory, which yields this number, involves only abstract math, no physics. As Feigenbaum’s colleague Leo Kadanoff said, this is “the best thing that can happen to a scientist, realizing that something that’s happened in his or her mind exactly corresponds to something that happens in nature.”

Mitchell Feigenbaum (AIP Emilio Segre Visual Archives, Physics Today Collection)

Large-scale systems such as the weather are, as yet, too hard to experiment with directly, so no one has directly observed period doubling or chaos in their behavior. However, certain computer models of weather have displayed the period-doubling route to chaos, as have computer models of electrical power systems, the heart,