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A006890
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Decimal expansion of Feigenbaum bifurcation velocity.
(Formerly M3264)
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16
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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, 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, 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
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OFFSET
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1,1
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COMMENTS
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"... 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]
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
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
Named after the American mathematical physicist Mitchell Jay Feigenbaum (1944-2019). - Amiram Eldar, Jun 16 2021
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REFERENCES
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Michael F. Barnsley, Fractals Everywhere, New Edition, Prof. of Math., Australian National University, Dover Publications, Inc., Mineola, NY, 2012, page 314.
Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, pp. 65-76
Clifford A. Pickover, (1993) 'The fifteen most famous transcendental numbers.' "Journal of Recreational Mathematics," 25(1):12.
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.
Clifford A. Pickover, The Math Book, Sterling, NY, 2009; see p. 462.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Ian Stewart, Nature's Numbers, Chapter 8, Do Dice Play God?, Weidenfeld & Nicolson, 1995.
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LINKS
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Brady Haran and Ben Sparks, 4.669, Numberphile video (2017).
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.
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EXAMPLE
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4.669201609102990671853203820466201617258185577475768632745651343004134...
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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