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%I M3264 #73 Apr 23 2023 22:42:55
%S 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 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 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 Decimal expansion of Feigenbaum bifurcation velocity.
%C "... 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 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 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 Named after the American mathematical physicist Mitchell Jay Feigenbaum (1944-2019). - _Amiram Eldar_, Jun 16 2021
%D Michael F. Barnsley, Fractals Everywhere, New Edition, Prof. of Math., Australian National University, Dover Publications, Inc., Mineola, NY, 2012, page 314.
%D Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 65-76
%D Clifford A. Pickover, (1993) 'The fifteen most famous transcendental numbers.' "Journal of Recreational Mathematics," 25(1):12.
%D 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 Clifford A. Pickover, The Math Book, Sterling, NY, 2009; see p. 462.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D Ian Stewart, Nature's Numbers, Chapter 8, Do Dice Play God?, Weidenfeld & Nicolson, 1995.
%H Harry J. Smith, <a href="/A006890/b006890.txt">Table of n, a(n) for n = 1..1019</a>
%H 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 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 Brady Haran and Phillip Moriarty, <a href="http://www.youtube.com/watch?v=S7E-EIjA2EM">A magic number</a> (video) (2009).
%H Brady Haran and Ben Sparks, <a href="https://www.youtube.com/watch?v=ETrYE4MdoLQ">4.669</a>, Numberphile video (2017).
%H 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 A. Krowne, <a href="https://planetmath.org/feigenbaumconstant">Feigenbaum constant</a>, PlanetMath.org.
%H Robert P. Munafo, <a href="http://www.mrob.com/pub/muency/feigenbaumconstant.html">Feigenbaum Constant</a>.
%H 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&format=complete">Zentralblatt review</a>.
%H Simon Plouffe, <a href="http://www.worldwideschool.org/library/books/sci/math/MiscellaneousMathematicalConstants/chap33.html">Feigenbaum constants</a>.
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/feigenbaum.txt">Feigenbaum constants to 1018 decimal places</a>.
%H 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 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 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FeigenbaumConstant.html">Feigenbaum Constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FeigenbaumConstantApproximations.html">Feigenbaum Constant Approximations</a>.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Feigenbaum_constant">Feigenbaum constant</a>.
%e 4.669201609102990671853203820466201617258185577475768632745651343004134...
%Y Cf. A159766 and A069544 (continued fraction), A069261 (Egyptian fraction), A108952 (1/delta), A102817 (Gamma(delta^2)).
%Y Cf. A006891 (Feigenbaum reduction parameter), A218453.
%K cons,nonn,nice
%O 1,1
%A _N. J. A. Sloane_, _Colin Mallows_, _Jeffrey Shallit_
%E Additional comments from _Robert G. Wilson v_, Dec 29 2000
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