Feigenbaum constants

Briggs, K. (1991) A precise calculation of the Feigenbaum constants. Mathematics of Computation 57, 435. 

For b > b3,b > h4, etc., successive losses of stability of the process (3.83) take place the sequence xB has 23, 24 attracting points. The described phenomenon is called period doubling. Finally, for the value b = = 0.892486... the process has the nature of the 2 -cycle. Feigenbaum has shown that the successive values of the parameter b, for which a qualitative change in nature of the process occurs, satisfy the dependence (<5 is called the Feigenbaum constant) ... 

S. lb the astonishment of scientists, the value of S turned out to be universal , i.e. characteristie for many very d erent mathematieal problems and, therefore, reached a status similar to that of the numbers -it and e. The numbers -it and e satisfy the exact relation —1, but so far no similar relation was found for the Feigenbaum constant. There is an appnmmate relation (used by physicists in phase transition theory) which is satisfied it + tan e = 4.669201932 <= 6. 

This is estimated to give 1-2 for the parameter values at the accumulation point of the bifurcations. Thus, the value of the Feigenbaum constant 5, in particular, should differ from the standard value 4.669.... ... 

M. J. Feigenbaum discovered a remarkable regularity in period doublings whereby the ratio of the parameter intervals between successive doublings approaches a universal constant 4.669. See M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations. J. Stat. Phys. 19,25-52 (1978). 

Moreover, the values of parameter for which period-doubling bifurcations occur before the onset of aperiodic oscillations define a sequence (Decroly, 1987a,b) characterized by a value close to that obtained by Feigenbaum (1978) for one of the universal constants associated with the cascade of period-doubling bifurcations leading to chaos. 

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