Abstract
For suitably defined largeN, we express Feigenbaum's equation as a singular Schroder functional equation whose solution is obtained using a scaling ansatz. In the limit of infiniteN certain self-consistency conditions on the scaled Schroder solution lead to an essentially singular solution of Feigenbaum's equation with a length scale factor of α≅ 0.0333 and. a limiting feigenvalue of δ∞≃30.50, in agreement with Eckmann and Wittwer's value of α=0.0333831... and their conjectured estimate of δ∞≲30.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
M. J. Feigenbaum,J. Stat. Phys. 19:25 (1978);21:669 (1979).
J. B. McGuire and C. J. Thompson,J. Stat. Phys. 27:183 (1982).
P. Collet and J.-P. Eckmann,Iterated Maps on the Interval as Dynamical Systems (Birkhauser, Boston, 1980).
R. M. May,Naturè 261:459 (1976).
P. Collet, J.-P. Eckmann, and O. E. Lanford III,Commun. Math. Phys. 76:211 (1980).
M. Campanino, H. Epstein, and D. Ruelle,Commun. Math. Phys. 79:261 (1981).
O. E. Lanford III,Commun. Math. Phys. 96:521 (1984).
J.-P. Eckmann and P. Wittwer,Computer Methods and Bore Summability Applied to Feigenbaum's Equation (Lecture Notes in Physics, Vol. 227, Springer, Berlin, 1985).
H. Epstein,Commun. Math. Phys. 106:395 (1986).
R. Vilela Menders,Phys. Lett. 84A:1 (1981).
B. Hu and J. M. Mao,Phys. Rev. 25A:3259 (1982).
R. Delbourgo and B. G. Kenny,Phys. Rev. 33A:3292 (1986).
J. Groeneveld,Physica 138A:137 (1986).
J. P. van der Weele, H. W. Capel, and R. Kluiving,Physica 145A:425 (1987).
M. Kuczma,Functional Equations in a Single Variable (Warsaw, Poland, 1968), Chapter VI.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Thompson, C.J., McGuire, J.B. Asymptotic and essentially singular solutions of the Feigenbaum equation. J Stat Phys 51, 991–1007 (1988). https://doi.org/10.1007/BF01014896
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01014896