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By Keith Briggs


Abstract. 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

Topics: 9
Year: 1989
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