the Feigenbaum constant

When a whole zoo of different systems all approach chaos by the same period-doubling staircase, you might expect each to climb it at its own pace. Instead, Mitchell Feigenbaum discovered in the 1970s that the steps shrink by the same ratio for all of them, an exact number, appearing in the dripping faucet and the electronic circuit and the abstract equation alike. That number is the Feigenbaum constant.

Consider the successive parameter values r_n at which a system undergoes its nth period doubling (from period 2^(n-1) to 2^n). The gaps between consecutive doublings shrink geometrically, and the ratio of successive gaps approaches a universal limit: delta = lim (r_n - r_{n-1}) / (r_{n+1} - r_n) = 4.669201609... A second Feigenbaum constant, alpha = 2.502907875..., measures the geometric rescaling of the width of the attractor branches at each doubling. Crucially, these numbers do not depend on the specific system, only on the shape of the map near its maximum: for any smooth map with a quadratic (parabolic) maximum, delta and alpha are the same.

This universality is the deepest idea in the theory: it means the transition to chaos via period doubling is a critical phenomenon, analogous to a phase transition, with delta and alpha playing the role of critical exponents. Feigenbaum explained it with a renormalization-group argument, the same conceptual machinery used for continuous phase transitions, showing that the doubling operation has a fixed point in the space of maps. The value of delta was later confirmed to high precision in real fluid, chemical, and electronic experiments. The caveat: the specific numbers 4.6692 and 2.5029 belong to the quadratic-maximum universality class, and maps with a differently shaped maximum have their own, different constants.