From: feldmann4350@my-dejanews.com
Subject: Re: Feigenbaum's number
Date: Tue, 30 Mar 1999 06:35:03 GMT
Newsgroups: sci.math

In article <37010eea.54779102@news.newsguy.com>,
  quentin@inhb.co.nz (Quentin Grady) wrote:
>
> G'day G'day Folks,
>
>  In popular books on Chaos theory mention is made of Feigenbaum's
> number 4.669201609...
>
> Which mathematical series will generate this number?
>
>

No series here (well, if there is one, many people would be greatly
interested). In fact, the definition of F=4.669201609.. is quite involved:
take the sequence u(n+1)=k*u(n)*(1-u(n)); u(0)=0.5; note it converges for
0<k<k_1=2; then it oscillates and has two "limit values" for 2<k<k_2, then 4
limit values for k<k_3, the 8 for k<k_4, and so on ...(the "doubling period
way to chaos"). Now, F is the limit of (k_(i+1)-k_i)/(k_(i+2)-k_(i+1))... Any
good book on Chaos theory will tell you why F is important (main reason is it
is *universal*: in the initial calculation, Feigenbaum found the same F for
the quite different sequence u(n+1)=k*sin(u(n))...)

There are  known ways to calculate F to as many decimals as you like (the
direct definition is too difficult); but no simple analytical expression. Try
this adress to more information on F's calculation:
http://www.lacim.uqam.ca/pi/indexf.html (The Plouffe's invertor)


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From: "John Bailey" <jmb184@frontiernet.net>
Subject: Re: Feigenbaum?
Date: Sun, 9 May 1999 05:59:26 -0400
Newsgroups: sci.math

nee@beer.com wrote in message <3734C7D4.36A72C73@beer.com>...
>Feigenbaum numbers - what are they? (think it is related to
>iterations)...
>
from  the FAQ--for sci.fractals:

Q10: What is Feigenbaum's constant?

   A10: In a period doubling cascade, such as the logistic equation,
   consider the parameter values where period-doubling events occur (e.g.
   r[1]=3, r[2]=3.45, r[3]=3.54, r[4]=3.564...). Look at the ratio of
   distances between consecutive doubling parameter values; let delta[n]
   = (r[n+1]-r[n])/(r[n+2]-r[n+1]). Then the limit as n goes to infinity
   is Feigenbaum's (delta) constant.

   Based on computations by F. Christiansen, P. Cvitanovic and H.H. Rugh,
   it has the value 4.6692016091029906718532038... Note: several books
   have published incorrect values starting 4.66920166...; the last
   repeated 6 is a typographical error.

   The interpretation of the delta constant is as you approach chaos,
   each periodic region is smaller than the previous by a factor
   approaching 4.669...

   Feigenbaum's constant is important because it is the same for any
   function or system that follows the period-doubling route to chaos and
   has a one-hump quadratic maximum. For cubic, quartic, etc. there are
   different Feigenbaum constants.

   Feigenbaum's alpha constant is not as well known; it has the value
   2.50290787509589282228390287272909. This constant is the scaling
   factor between x values at bifurcations. Feigenbaum says,
   "Asymptotically, the separation of adjacent elements of period-doubled
   attractors is reduced by a constant value [alpha] from one doubling to
   the next". If d[a] is the algebraic distance between nearest elements
   of the attractor cycle of period 2^a, then d[a]/d[a+1] converges to
   -alpha.

   References:

    1. K. Briggs, How to calculate the Feigenbaum constants on your PC,
       Aust. Math. Soc. Gazette 16 (1989), p. 89.
    2. K. Briggs, A precise calculation of the Feigenbaum constants,
       Mathematics of Computation 57 (1991), pp. 435-439.
    3. K. Briggs, G. R. W. Quispel and C. Thompson, Feigenvalues for
       Mandelsets, J. Phys. A 24 (1991), pp. 3363-3368.
    4. F. Christiansen, P. Cvitanovic and H.H. Rugh, "The spectrum of the
       period-doubling operator in terms of cycles", J. Phys A 23, L713
       (1990).
    5. M. Feigenbaum, The Universal Metric Properties of Nonlinear
       Transformations, J. Stat. Phys 21 (1979), p. 69.
    6. M. Feigenbaum, Universal Behaviour in Nonlinear Systems, Los
       Alamos Sci 1 (1980), pp. 1-4. Reprinted in Universality in Chaos,
       compiled by P. Cvitanovic.

   Feigenbaum Constants
          http://www.mathsoft.com/asolve/constant/fgnbaum/fgnbaum.html