The Lorenz Attractor and the Problem of Turbulence

  • David Ruelle
Conference paper
Part of the Acta Physica Austriaca book series (FEWBODY, volume 16/1976)


Turbulence in the flow of liquids is a fascinating phenomenon. This may partly explain the conceptual confusion which exists in the scientific literature as to the nature of this phenomenon. My own opinion, and that of some other people, is that turbulence at low Reynolds numbers corresponds to a mathematical phenomenon observed in the study of solutions of differential equations
$${{dx} \mathord{\left/ {\vphantom {{dx} {dt = X\left( x \right)}}} \right. \kern-\nulldelimiterspace} {dt = X\left( x \right)}}$$
The equation just written has to be understood as a time evolution equation in several dimensions. The mathematical phenomenon referred to is that in many cases, solutions of (1) have an asymptotic behavior when t → ∝ which appears erratic, chaotic, “turbulent”, and the solutions depend in a sensitive manner on initial condition.


Steady State Solution Unstable Manifold Time Correlation Function Sensitive Dependence Lebesgue Measure Zero 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R. Bowen and D. Ruelle. The ergodic theory of Axiom A flows. Inventiones math. To appear.Google Scholar
  2. 2.
    R. P. Feynman, R. B. Leighton, and M. Sands. The Feynman lectures on physics. 2. Addison-Wesley, Reading, Mass., 1964.Google Scholar
  3. 3.
    J. P. Gollub and H. Swinney. Onset of turbulence in a rotating fluid. Preprint.Google Scholar
  4. 4.
    E. Hopf. A mathematical example displaying features of turbulence. Commun. pure appl. Math. 1, 303–322 (1948).MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Ju.I. Kifer. On small random perturbations of some smooth dynamical systems. Izv. Akad. Nauk SSSR. Ser. mat. 38 No5, 1091–1115 (1974). English translation: Math. USSR Izvestija 8, 1083-1107 (1974).MathSciNetGoogle Scholar
  6. 6.
    L. D. Landau and E. M. Lifshitz. Fluid Mechanics. Pergamon, Oxford, 1959.Google Scholar
  7. 7.
    T.-Y. LI and J. Yorke. Period three implies chaos. Preprint.Google Scholar
  8. 8.
    E. N. Lorenz. Deterministic nonperiodic flow. J. atmos. Sci. 20, 130–141 (1963).ADSCrossRefGoogle Scholar
  9. 9.
    J. B. Mclaughlin and P. C. Martin. Transition to turbulence of a statically stressed fluid system. Phys. Rev. Lett. 33, 1189–1192 (1974), Phys. Rev. A 12, 186-203 (1975).ADSCrossRefGoogle Scholar
  10. 10.
    D. Ruelle. A measure associated with Axiom A attractors. Amer. J. Math. To appear.Google Scholar
  11. 11.
    D. Ruelle And F. Takens. On the nature of turbulence. Commun. Math. Phys., 20, 167–192 (1971), 23, 343-344 (1971).MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    Ia. G. Sinai. Gibbs measures in ergodic theory. Russ. Math. Surveys 166, 21–69 (1972).MathSciNetADSCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • David Ruelle
    • 1
  1. 1.Institute for Advanced StudyPrincetonUSA

Personalised recommendations