A very brief history of universality in period doubling

In 1976, M.J. Feigenbaum got me interested in his Aug 1975 discovery of universality in one-dimensional iterative maps. The first published report on this work is dated Aug 1976 (Los Alamos Theoretical Division Annual Report 1975-1976, pp. 98-102, read it here), and it became rapidly widely known through many seminars given by Feigenbaum, both in US and Europe. His first paper, submitted to Advances in Mathematics in Nov 1976 was rejected. The second paper was submitted to SIAM Journal of Applied Mathematics in April 1977 and rejected in October 1977. Finally, J. Lebowitz published both papers without further referee pain (M. J. Feigenbaum, J. Stat. Phys. 19, 25 (1978) and 21, 6 (1979)). By 1978 others have published similar results, and by 1979 mathematicians also understood that the numerical methods we used to solve the universal equations were in fact convergent. They did the usual; they added names of two of my friends to the equation, for having rediscovered it in 1978 (remind me, who is the second person to discover general relativity?), and they changed letters around to make the equations unintelligible to physicists.

Following Feigenbaum's functional formulation of the problem, in spring 1976 I derived the equation for the period doubling fixed point function (not a big step - it is the limit of his functional recursion sequence), which has since played a key role in the theory of transitions to turbulence. Since then we have generalized the universal equations to period n-tuplings; constructed universal scaling functions for all winding numbers in circle maps, and established universality of the Hausdorff dimension of the critical staircase.

Thanks to my diaries we can be ridiculously precise about the dates. For hyperlinks to some of the references click here.

What came next? We can quote Freeman Dyson: As usual when you discover something new, the response comes in three waves. First, this is nonsense. Second, this is trivial. Third, this is important, and we did it before you did.

Here one has to focus on what Feigenbaum's contribution was, otherwise this is yet another walk through the woods, like all histories of nonlinear dynamics: Anybody who ever ran into a nonlinear problem had pondered a parabola sooner or later (x^0, x^1 and then - boom! x^2 is a big trouble; WKB; extremal points in path integrals; etc., etc). Lots of people have thought about and understood period doubling sequences. In all that work, the focus was on bifurcations and topological sequences of bifurcations.

Feigenbaum's contribution was to create a new field of mathematics, renormalization theory of dynamical systems, that occupies many good mathematicians (Dennis Sullivan, a couple of Fields medals, ...), and for a while occupied many physicists that had ran out of Widom-Fisher-Kadanoff-Wilson renormalization flow problems. Period doubling and the infinity of related renormalization problems are a different kind of beast from the simple fixed point renormalization flows.

For physicists, chemists, biophysicists, ... the theory would have remained a curiosity, were it not for the beautiful experiment by Libchaber and Maurer (1981), and many others that followed. Crucial insights came from Collet and Eckmann (1980) and Collet, Eckmann and Koch (1980) who explained how the dynamics of dissipative system (such as a viscous fluid) can become become 1-dimensional.

Predrag Cvitanović, Atlanta, July 6, 2019