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