Feigenbaum function

From Wikipedia, the free encyclopedia
Jump to navigation Jump to search

In the study of dynamical systems the term Feigenbaum function has been used to describe two different functions introduced by the physicist Mitchell Feigenbaum:

Functional equation[edit]

The functional equation arises in the study of one-dimensional maps that, as a function of a parameter, go through a period-doubling cascade. The functional equation is the mathematical expression of the universality of period doubling. The equation is used to specify a function g and a parameter λ by the relation

with the initial conditions

  • g(0) = 1,
  • g′(0) = 0, and
  • g′′(0) < 0

For a particular form of solution with a quadratic dependence of the solution near x=0, the inverse 1/λ=2.5029... is one of the Feigenbaum constants.

Scaling function[edit]

The Feigenbaum scaling function provides a complete description of the attractor of the logistic map at the end of the period-doubling cascade. The attractor is a Cantor set, and just as the middle-third Cantor set, it can be covered by a finite set of segments, all bigger than a minimal size dn. For a fixed dn the set of segments forms a cover Δn of the attractor. The ratio of segments from two consecutive covers, Δn and Δn+1 can be arranged to approximate a function σ, the Feigenbaum scaling function.

See also[edit]

References[edit]

  • Weisstein, Eric W. "Feigenbaum Function". MathWorld.
  • Feigenbaum, M. (1978). "Quanitative universality for a class of nonlinear transformations". Journal of Statistical Physics. 19 (1): 25–52. Bibcode:1978JSP....19...25F. doi:10.1007/BF01020332. MR 0501179.
  • Feigenbaum, M. (1979). "The universal metric properties of non-linear transformations". Journal of Statistical Physics. 21 (6): 669–706. Bibcode:1979JSP....21..669F. doi:10.1007/BF01107909. MR 0555919.
  • Feigenbaum, Mitchell J. (1980). "The transition to aperiodic behavior in turbulent systems". Communications in Mathematical Physics. 77 (1): 65–86. Bibcode:1980CMaPh..77...65F. doi:10.1007/BF01205039.
  • Epstein, H.; Lascoux, J. (1981). "Analyticity properties of the Feigenbaum Function". Commun. Math. Phys. 81 (3): 437–453. Bibcode:1981CMaPh..81..437E. doi:10.1007/BF01209078.
  • Feigenbaum, Mitchell J. (1983). "Universal Behavior in Nonlinear Systems". Physica. 7D: 16–39. Bibcode:1983PhyD....7...16F. doi:10.1016/0167-2789(83)90112-4. Bound as Order in Chaos, Proceedings of the International Conference on Order and Chaos held at the Center for Nonlinear Studies, Los Alamos, New Mexico 87545,USA 24–28 May 1982, Eds. David Campbell, Harvey Rose; North-Holland Amsterdam ISBN 0-444-86727-9.
  • Lanford III, Oscar E. (1982). "A computer-assisted proof of the Feigenbaum conjectures". Bull. Am. Math. Soc. 6 (3): 427–434. doi:10.1090/S0273-0979-1982-15008-X. MR 0648529.
  • Campanino, M.; Epstein, H.; Ruelle, D. (1982). "On Feigenbaums functional equation ". Topology. 21 (2): 125–129. doi:10.1016/0040-9383(82)90001-5. MR 0641996.
  • Lanford III, Oscar E. (1984). "A shorter proof of the existence of the Feigenbaum fixed point". Commun. Math. Phys. 96 (4): 521–538. Bibcode:1984CMaPh..96..521L. doi:10.1007/BF01212533.
  • Epstein, H. (1986). "New proofs of the existence of the Feigenbaum functions". Commun. Math. Phys. 106 (3): 395–426. Bibcode:1986CMaPh.106..395E. doi:10.1007/BF01207254.
  • Eckmann, Jean-Pierre; Wittwer, Peter (1987). "A complete proof of the Feigenbaum Conjectures". J. Stat. Phys. 46 (3/4): 455. Bibcode:1987JSP....46..455E. doi:10.1007/BF01013368. MR 0883539.
  • Stephenson, John; Wang, Yong (1991). "Relationships between the solutions of Feigenbaum's equation". Appl. Math. Lett. 4 (3): 37–39. doi:10.1016/0893-9659(91)90031-P. MR 1101871.
  • Stephenson, John; Wang, Yong (1991). "Relationships between eigenfunctions associated with solutions of Feigenbaum's equation". Appl. Math. Lett. 4 (3): 53–56, . doi:10.1016/0893-9659(91)90035-T. MR 1101875.
  • Briggs, Keith (1991). "A precise calculation of the Feigenbaum constants". Math. Comp. 57 (195): 435–439. Bibcode:1991MaCom..57..435B. doi:10.1090/S0025-5718-1991-1079009-6. MR 1079009.
  • Tsygvintsev, Alexei V.; Mestel, Ben D.; Obaldestin, Andrew H. (2002). "Continued fractions and solutions of the Feigenbaum-Cvitanović equation". Comptes Rendus de l'Académie des Sciences, Série I. 334 (8): 683–688. doi:10.1016/S1631-073X(02)02330-0.
  • Mathar, Richard J. (2010). "Chebyshev series representation of Feigenbaum's period-doubling function". arXiv:1008.4608 [math.DS].
  • Varin, V. P. (2011). "Spectral properties of the period-doubling operator". KIAM Preprint. 9.