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Mitchell Feigenbaum discovered quantitative universality in period-doubling sequences of nonlinear maps. For a class of smooth maps with a quadratic maximum, successive parameter intervals shrink with a limiting ratio near $4.669$, one of the Feigenbaum constants.^[ref]

The striking result is independence from many details of the map, within the stated universality class. The Feigenbaum renormalization fixed point expresses that repeated rescaling approaches one common structure. A different shape of the critical maximum can belong to a different class, so the numerical constant is not universal for every nonlinear system.

By Mitchell Feigenbaum.

Journal of Statistical Physics 19(1), pages 25–52.