Feigenbaum's period-doubling constant

Find a closed form for Feigenbaum's period-doubling constant δ ≈ 4.6692016091, known to 10,000 digits but not proven irrational.

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	1	+# Feigenbaum's period-doubling constant
	2	+
	3	+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.
	4	+
	5	+## Problem
	6	+
	7	+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
	8	+
	9	+$$
	10	+\delta = \lim_{n \to \infty} \frac{r_{n-1} - r_{n-2}}{r_n - r_{n-1}} = 4.66920160910299067185320382046620161725\ldots
	11	+$$
	12	+
	13	+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.
	14	+
	15	+## Background
	16	+
	17	+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.
	18	+
	19	+## What is known
	20	+
	21	+- 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.
	22	+- It has not been proved that $\delta$ is irrational or transcendental, though it is generally believed to be transcendental ([OEIS A006890](https://oeis.org/A006890)). No closed form is known. HorizonMath places the problem in its top difficulty tier, "possibly unsolvable".
	23	+
	24	+## What would count as progress
	25	+
	26	+- 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.
	27	+- Any proven arithmetic statement about $\delta$, such as irrationality, would be a major advance in its own right.
	28	+
	29	+## References
	30	+
	31	+1. M. J. Feigenbaum, "Quantitative universality for a class of nonlinear transformations", J. Stat. Phys. 19 (1978), 25–52. [doi:10.1007/BF01020332](https://doi.org/10.1007/BF01020332)
	32	+2. M. J. Feigenbaum, "The universal metric properties of nonlinear transformations", J. Stat. Phys. 21 (1979), 669–706. [doi:10.1007/BF01107909](https://doi.org/10.1007/BF01107909)
	33	+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](https://doi.org/10.1090/S0273-0979-1982-15008-X)
	34	+4. K. Briggs, "A precise calculation of the Feigenbaum constants", Math. Comp. 57 (1991), 435–439. [doi:10.1090/S0025-5718-1991-1079009-6](https://doi.org/10.1090/S0025-5718-1991-1079009-6)
	35	+5. A. Molteni, "An efficient method for the computation of the Feigenbaum constants to high precision", 2016. [arXiv:1602.02357](https://arxiv.org/abs/1602.02357)
	36	+
	37	+## Source
	38	+
	39	+Imported from [HorizonMath feigenbaum_delta](https://github.com/ewang26/HorizonMath/blob/4ef0b61a8a6d4c5ded661fab7c2e896af84abe6e/data/problems_full.json#L334-L344), 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.