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A006890 Decimal expansion of Feigenbaum bifurcation velocity.

%I A006890 M3264 #80 Mar 16 2025 10:44:10
%S A006890 4,6,6,9,2,0,1,6,0,9,1,0,2,9,9,0,6,7,1,8,5,3,2,0,3,8,2,0,4,6,6,2,0,1,
%T A006890 6,1,7,2,5,8,1,8,5,5,7,7,4,7,5,7,6,8,6,3,2,7,4,5,6,5,1,3,4,3,0,0,4,1,
%U A006890 3,4,3,3,0,2,1,1,3,1,4,7,3,7,1,3,8,6,8,9,7,4,4,0,2,3,9,4,8,0,1,3,8,1,7,1,6
%N A006890 Decimal expansion of Feigenbaum bifurcation velocity.
%C A006890 "... These are related to properties of dynamical systems with 'period-doubling' oscillations. The ratio of successive differences between period-doubling bifurcation parameters approaches the number 4.669... Period doubling has been discovered in many physical systems before they enter the chaotic regime. Feigenbaum numbers have not been proved to be transcendental but are generally believed to be. ..." [Pickover]
%C A006890 The Feigenbaum delta constant is the convergence ratio {g(k)-g(k-1)}/{g(k+1)-g(k)} of successive period-doubling thresholds g(k) in the continuous map x(n+1)=f(x(n),g) of an interval onto itself. - _Lekraj Beedassy_, Jan 07 2005
%C A006890 The above statement is only valid for functions f satisfying some properties, e.g., having a single locally quadratic maximum. See, e.g., the MathWorld link for more details. - _M. F. Hasler_, May 01 2018
%C A006890 Named after the American mathematical physicist Mitchell Jay Feigenbaum (1944-2019). - _Amiram Eldar_, Jun 16 2021
%D A006890 Michael F. Barnsley, Fractals Everywhere, New Edition, Prof. of Math., Australian National University, Dover Publications, Inc., Mineola, NY, 2012, page 314.
%D A006890 John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 208.
%D A006890 Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 65-76
%D A006890 Clifford A. Pickover, (1993) 'The fifteen most famous transcendental numbers.' "Journal of Recreational Mathematics," 25(1):12.
%D A006890 Clifford A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Chapter 44, 'The 15 Most Famous Transcendental Numbers,' Oxford University Press, Oxford, England, 2000, pages 103 - 106.
%D A006890 Clifford A. Pickover, The Math Book, Sterling, NY, 2009; see p. 462.
%D A006890 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A006890 Ian Stewart, Nature's Numbers, Chapter 8, Do Dice Play God?, Weidenfeld & Nicolson, 1995.
%H A006890 Harry J. Smith, <a href="/A006890/b006890.txt">Table of n, a(n) for n = 1..1019</a>
%H A006890 Keith Briggs, <a href="http://dx.doi.org/10.1090/S0025-5718-1991-1079009-6">A precise calculation of the Feigenbaum constants</a>, Math. Comp., Vol. 57, No. 195 (1991), pp. 435-439.
%H A006890 B. Derrida, A. Gervois and Y. Pomeau, <a href="http://dx.doi.org/10.1088/0305-4470/12/3/004">Universal metric properties of bifurcations</a>, J. Phys. A, Vol. 12 (1979), pp. 269-296.
%H A006890 Brady Haran and Phillip Moriarty, <a href="http://www.youtube.com/watch?v=S7E-EIjA2EM">A magic number</a> (video) (2009).
%H A006890 Brady Haran and Ben Sparks, <a href="https://www.youtube.com/watch?v=ETrYE4MdoLQ">4.669</a>, Numberphile video (2017).
%H A006890 Sibyl Kempson, <a href="https://doi.org/10.1162/PAJJ_a_00115">Restless Eye: Text for the Advanced Beginner Group</a>, PAJ: A Journal of Performance and Art, Volume 34, Number 3, September 2012 (PAJ 102).
%H A006890 A. Krowne, <a href="https://planetmath.org/feigenbaumconstant">Feigenbaum constant</a>, PlanetMath.org.
%H A006890 Robert P. Munafo, <a href="http://www.mrob.com/pub/muency/feigenbaumconstant.html">Feigenbaum Constant</a>.
%H A006890 C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," <a href="http://www.zentralblatt-math.org/zmath/en/search/?q=an:0983.00008&amp;format=complete">Zentralblatt review</a>.
%H A006890 Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap33.html">Feigenbaum constants</a>.
%H A006890 Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/feigenbaum.txt">Feigenbaum constants to 1018 decimal places</a>.
%H A006890 Gianluca Simonetto, <a href="https://thesis.unipd.it/handle/20.500.12608/45498">Chaos and universality in non-linear dynamics: the logistic map</a>, Univ. Padova (Italy, 2023, in Italian).
%H A006890 Judi Thurlby, <a href="https://pure.port.ac.uk/ws/portalfiles/portal/43264172/Thesis_Final_Judi_Thurlby_September_2021.pdf">Rigorous calculations of renormalisation fixed points and attractors</a>, PhD thesis, U. Portsmouth, (2021). 400 digits in section 3.8.
%H A006890 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FeigenbaumConstant.html">Feigenbaum Constant</a>.
%H A006890 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FeigenbaumConstantApproximations.html">Feigenbaum Constant Approximations</a>.
%H A006890 Wikipedia, <a href="http://en.wikipedia.org/wiki/Feigenbaum_constant">Feigenbaum constant</a>.
%e A006890 4.669201609102990671853203820466201617258185577475768632745651343004134...
%Y A006890 Cf. A159766 and A069544 (continued fraction), A069261 (Egyptian fraction), A108952 (1/delta), A102817 (Gamma(delta^2)).
%Y A006890 Cf. A006891 (Feigenbaum reduction parameter), A218453.
%K A006890 cons,nonn,nice
%O A006890 1,1
%A A006890 _N. J. A. Sloane_, _Colin Mallows_, _Jeffrey Shallit_
%E A006890 Additional comments from _Robert G. Wilson v_, Dec 29 2000