A Precise Calculation of the Feigenbaum Constants
نویسنده
چکیده
The Feigenbaum constants arise in the theory of iteration of real functions. We calculate here to high precision the constants a and S associated with period-doubling bifurcations for maps with a single maximum of order z , for 2 < z < 12. Multiple-precision floating-point techniques are used to find a solution of Feigenbaum's functional equation, and hence the constants. 1. History Consider the iteration of the function (1) fßZ(x) = l-p\x\z, z>0; that is, the sequence (2) *(+i =/„,*(*/)> i'=l,2,...; x0 = 0. In 1979 Feigenbaum [8] observed that there exist bifurcations in the set of limit points of (2) (that is, in the set of all points which are the limit of some infinite subsequence) as the parameter p is increased for fixed z. Roughly speaking, if the sequence (2) is asymptotically periodic with period p for a particular parameter value p (that is, there exists a stable p-cycle), then as p is increased, the period will be observed to double, so that a stable 2/>cycle appears. We denote the critical /¿-value at which the 2J cycle first appears by Pj. Feigenbaum also conjectured that there exist certain "universal" scaling constants associated with these bifurcations. Specifically, (3) «5 = lim ZlZJhzi 7-00 pJ+x ftj exists, and ô2 is about 4.669. Similarly, if rf. is the value of the nearest cycle element to 0 in the 2J cycle, then (4) az = lim y;-oo dJ+x exists, and a2 is about -2.503 . Received November 22, 1989; revised September 10, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 11Y60, 26A18, 39A10, 65Q05. ©1991 American Mathematical Society 0025-5718/91 $1.00 + $.25 per page
منابع مشابه
Hausdorff Dimension and Conformal Measures of Feigenbaum Julia Sets
1.1. Statement of the results. One of the first questions usually asked about a fractal subset of R is whether it has the maximal possible Hausdorff dimension, n. It certainly happens if the set has positive Lebesgue measure. On the other hand, it is easy to construct fractal sets of zero measure but of dimension n. Moreover, this phenomenon is often observable for fractal sets produced by conf...
متن کاملIntroduction. — The concepts Scaling and Universality have played an essential role
Introduction. — The concepts Scaling and Universality have played an essential role in the description of statistical systems [1]. Recently a multisite interaction system on Husimi tree approximation was investigated [2]. First, it was shown, that this approach yields good approximation for the phase diagrams, which closely match the exact results obtained on a Kagome lattice [3]. Second, a mul...
متن کاملOn the statistical analysis of Feigenbaum constants
We present statistical analysis of blocks in the binary expansions of Feigenbaum constants a and d for the logistic map. The analysis is carried out on both 1016 and 3400 bit expansions. A w test is applied for lumping data and a serial test is applied on gliding data. Contrary to a previous research by Karamanos and Kotsireas, our test results did not indicate any evidence to reject randomness...
متن کاملGeometry of the Feigenbaum Map
We show that the Feigenbaum-Cvitanović equation can be interpreted as a linearizing equation, and the domain of analyticity of the Feigenbaum fixed point of renormalization as a basin of attraction. There is a natural decomposition of this basin which enables to recover a result of local connectivity by Jiang and Hu for the Feigenbaum Julia set.
متن کامل