

A006890


Decimal expansion of Feigenbaum bifurcation velocity.
(Formerly M3264)


16



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

1,1


COMMENTS

"... These are related to properties of dynamical systems with 'perioddoubling' oscillations. The ratio of successive differences between perioddoubling 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(k1)}/{g(k+1)g(k)} of successive perioddoubling 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 (19442019).  Amiram Eldar, Jun 16 2021


REFERENCES

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. 6576
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.


LINKS

Brady Haran and Ben Sparks, 4.669, Numberphile video (2017).
C. A. Pickover, "Wonders of Numbers, Adventures in Mathematics, Mind and Meaning," Zentralblatt review.


EXAMPLE

4.669201609102990671853203820466201617258185577475768632745651343004134...


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STATUS

approved



