Feigenbaum's period-doubling constant

As a parameter is increased, a smooth one-humped map such as the logistic map $x \mapsto rx(1-x)$ goes through a cascade of period doublings on its way to chaos. The parameter intervals between successive doublings shrink by a universal factor, $\delta \approx 4.6692$. This constant has been computed to 10,000 decimals, yet it is not even known to be irrational, and no closed form is known.

Problem

For the logistic map $f_r(x) = rx(1-x)$, let $r_n$ be the parameter value at which the attracting cycle of period $2^n$ appears. Feigenbaum's constant is

$$\delta = \lim_{n \to \infty} \frac{r_{n-1} - r_{n-2}}{r_n - r_{n-1}} = 4.66920160910299067185320382046620161725\ldots$$

The same value arises for a broad class of one-parameter families of unimodal maps with a quadratic maximum. Find a closed-form expression for $\delta$ in terms of standard constants and special functions.

Background

Feigenbaum found the constants of period doubling and explained their universality in 1978–79 [1, 2]. The doubling operator $(Tg)(x) = g(g(g(1)\,x))/g(1)$ has an even fixed point $g$ with a quadratic maximum and $g(0) = 1$. The companion constant is $\alpha = 1/g(1)$, and $\delta$ is the largest eigenvalue of the doubling operator linearized at this fixed point [1, 2, 5]. Lanford gave a computer-assisted proof of Feigenbaum's conjectures in 1982 [3]. Together with $\alpha$, $\delta$ is one of the best-known universal constants of the transition to chaos.

What is known

• High-precision values: Briggs (1991) [4]; Broadhurst (1999), 1018 decimal places, as reported in [5]; Molteni (2016), 10,000 decimal places, with a method of lower time and space complexity than earlier ones [5]. HorizonMath's reference value has 110 digits.
• It has not been proved that $\delta$ is irrational or transcendental, though it is generally believed to be transcendental (OEIS A006890). No closed form is known. HorizonMath places the problem in its top difficulty tier, "possibly unsolvable".

What would count as progress

• A closed-form expression that agrees with $\delta$ to 20 significant digits, HorizonMath's acceptance threshold. Its evaluator checks this at 100-digit precision and rejects numerical shortcuts such as quadrature, root-finding, infinite series and fitted constants. Ten thousand known decimals make it easy to rule out chance agreement.
• Any proven arithmetic statement about $\delta$, such as irrationality, would be a major advance in its own right.

References

1. M. J. Feigenbaum, "Quantitative universality for a class of nonlinear transformations", J. Stat. Phys. 19 (1978), 25–52. doi:10.1007/BF01020332
2. M. J. Feigenbaum, "The universal metric properties of nonlinear transformations", J. Stat. Phys. 21 (1979), 669–706. doi:10.1007/BF01107909
3. O. E. Lanford III, "A computer-assisted proof of the Feigenbaum conjectures", Bull. Amer. Math. Soc. 6 (1982), 427–435. doi:10.1090/S0273-0979-1982-15008-X
4. K. Briggs, "A precise calculation of the Feigenbaum constants", Math. Comp. 57 (1991), 435–439. doi:10.1090/S0025-5718-1991-1079009-6
5. A. Molteni, "An efficient method for the computation of the Feigenbaum constants to high precision", 2016. arXiv:1602.02357

Source

Imported from HorizonMath feigenbaum_delta, curated by Erik Y. Wang, Sumeet R. Motwani et al. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion.