Communications in Mathematical Physics

, Volume 77, Issue 1, pp 65–86 | Cite as

The transition to aperiodic behavior in turbulent systems

  • Mitchell J. Feigenbaum


Some systems achieve aperiodic temporal behavior through the production of successive half subharmonics. A recursive method is presented here that allows the explicit computation of this aperiodic behavior from the initial subharmonics. The results have a character universal over specific systems, so that all such transitions are characterized by noise of a universal internal similarity.


Neural Network Statistical Physic Complex System Nonlinear Dynamics Quantum Computing 
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  1. 1.
    Franceschini, V., Tebaldi, C.: Sequences of infinite bifurcation and turbulence in five-modes truncation of the Navier-Stokes equations. Istituto Matematico, Univ. di Modena preprintGoogle Scholar
  2. 2.
    Franceschini, V.: A Feigenbaum sequence of bifurcations in the Lorenz model, Istituto Matematico, Univ. di Modena preprint, to be published in J. Stat. Phys.Google Scholar
  3. 3.
    Computations by the author on Duffing's equation, following Ueda, Y.: J. Stat. Phys.20 (2), 181 (1979)Google Scholar
  4. 4.
    Holmes, P.: A nonlinear oscillator with a strange attractor. Department of Theoretical and Applied Mechanics, Cornell University (preprint)Google Scholar
  5. 5.
    Huberman, B., Crutchfield, J.: Chaotic states of anharmonic systems in periodic fields. Xerox Corp., Palo Alto Research Center (preprint)Google Scholar
  6. 6.
    Marzec, C.J., Spiegel, E.A.: A strange attractor. Astronomy Department, Columbia University (preprint)Google Scholar
  7. 7.
    Libchaber, A., Maurer, J.: Une expérience de Rayleigh-Bénard de géométrie réduite. École Normale Supérieure (preprint)Google Scholar
  8. 8.
    Feigenbaum, M.J.: Phys. Lett.74A, 375 (1979)Google Scholar
  9. 9.
    Collet, P., Eckmann, J.-P., Koch, H.: Period doubling bifurcations for families of maps onC n. Department of Physics, Harvard University (preprint)Google Scholar
  10. 10.
    Metropolis, N., Stein, M.L., Stein, P.R.: Combinatorial Theory15 (1), 25 (1973)Google Scholar
  11. 11.
    May, R., Oster, G.: Amer. Naturalist110 (974), 573 (1976). This paper independently of myself, discovers the first clue of a universal metric propertyGoogle Scholar
  12. 12.
    Feigenbaum, M.J.: Annual Report 1975–76, LA-6816-PR, Los AlamosGoogle Scholar
  13. 13.
    Feigenbaum, M.J.: J. Stat. Phys.19 (1), 25 (1978)Google Scholar
  14. 14.
    Feigenbaum, M.J.: J. Stat. Phys.21 (6) (1979)Google Scholar
  15. 15.
    Feigenbaum, M.J.: Lecture Notes in Physics93, 163 (1979)Google Scholar
  16. 16.
    Collet, P., Eckmann, J.-P., Lanford III, O.: Universal properties of maps on an interval (in preparation)Google Scholar
  17. 17.
    Collet, P., Eckmann, J.-P.: Bifurcations et groupe de renormalisation. IHES/P/78/250 (preprint)Google Scholar
  18. 18.
    Derrida, B., Gervois, A., Pomeau, Y.: J. Phys. A12, 269 (1979). This paper contains the first numerical observation of δ in a 2-dimensional mapGoogle Scholar

Copyright information

© Springer-Verlag 1980

Authors and Affiliations

  • Mitchell J. Feigenbaum
    • 1
  1. 1.Los Alamos Scientific Laboratory (T-DOT MS 452)University of CaliforniaLos AlamosUSA

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