Top Document: Fractal Frequently Asked Questions and Answers Previous Document: Logistic equation Next Document: Iterated function systems and compression See reader questions & answers on this topic!  Help others by sharing your knowledge Q10: What is Feigenbaum's constant? A10: In a period doubling cascade, such as the logistic equation, consider the parameter values where perioddoubling 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 independent computations by Jay Hill and Keith Briggs, it has the value 4.669201609102990671853... 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 perioddoubling 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 2.502907875095. This constant is the scaling factor between x values at bifurcations. Feigenbaum says, "Asymptotically, the separation of adjacent elements of perioddoubled attractors is reduced by a constant value [alpha] from one doubling to the next". If d[n] is the algebraic distance between nearest elements of the attractor cycle of period 2^n, then d[n]/d[n+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. 435439. 3. K. Briggs, G. R. W. Quispel and C. Thompson, Feigenvalues for Mandelsets, _J. Phys._ A24 (1991), pp. 33633368. 4. M. Feigenbaum, The Universal Metric Properties of Nonlinear Transformations, _J. Stat. Phys_ 21 (1979), p. 69. 5. M. Feigenbaum, Universal Behaviour in Nonlinear Systems, _Los Alamos Sci_ 1 (1980), pp. 14. Reprinted in _Universality in Chaos_ , compiled by P. Cvitanovic. User Contributions:Comment about this article, ask questions, or add new information about this topic:Top Document: Fractal Frequently Asked Questions and Answers Previous Document: Logistic equation Next Document: Iterated function systems and compression Single Page [ Usenet FAQs  Web FAQs  Documents  RFC Index ] Send corrections/additions to the FAQ Maintainer: stepp@marshall.edu
Last Update March 27 2014 @ 02:11 PM
