Communications in Mathematical Physics

, Volume 67, Issue 2, pp 93–108 | Cite as

Preturbulence: A regime observed in a fluid flow model of Lorenz

  • James L. Kaplan
  • James A. Yorke


This paper studies a forced, dissipative system of three ordinary differential equations. The behavior of this system, first studied by Lorenz, has been interpreted as providing a mathematical mechanism for understanding turbulence. It is demonstrated that prior to the onset of chaotic behavior there exists a preturbulent state where turbulent orbits exist but represent a set of measure zero of initial conditions. The methodology of the paper is to postulate the short term behavior of the system, as observed numerically, to establish rigorously the behavior of particular orbits for all future time. Chaotic behavior first occurs when a parameter exceeds some critical value which is the first value for which the system possesses a homoclinic orbit. The arguments are similar to Smale's “horseshoe”.


Differential Equation Neural Network Statistical Physic Complex System Fluid Flow 
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.
    Kaplan, J.L., Yorke, J.A.: The onset of chaos in a fluid flow model of Lorenz. Proc. NY Acad. Sci. (to appear)Google Scholar
  2. 2.
    Lorenz, E.N.: Deterministic nonperiodic flow. J. Atmos. Sci.20, 130–141 (1963)Google Scholar
  3. 3.
    Guckenheimer, J.: A strange attractor, in the Hopfbifurcation theorem and its applications. Marsden, J.E., McCracken, M. (ed.), pp. 368–381. Berlin, Heidelberg, New York: Springer 1976Google Scholar
  4. 4.
    Williams, R.F.: The structure of Lorenz attractors. Preprint (1977)Google Scholar
  5. 5.
    McLaughlin, J.B., Martin, P.C.: Transition to turbulence in a statistically stressed fluid system. Phys. Rev. A12, 186–203 (1975)Google Scholar
  6. 6.
    Curry, J.H.: Transition to turbulence in finite-dimensional approximations to the Boussinesq equations. Ph. D. Thesis, University of California, Berkeley (1976)Google Scholar
  7. 7.
    Smale, S.: A structurally stable differentiable homeomorphism with an infinite number of periodic points. Proc. Int. Symp. Nonlinear Vibrations, Vol. II (1961); Izdat. Akad. Nauk. Ukrain SSR, Kiev (1963)Google Scholar
  8. 8.
    Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc.73, 747–817 (1967)Google Scholar
  9. 9.
    Denjoy, A.: Sur les courbes défines par les équations différentielles à la surface du tore. J. Math. ser 9,11, 333–375 (1932)Google Scholar
  10. 10.
    Ruelle, D., Takens, F.: On the nature of turbulence. Commun. Math. Phys.20, 167–192 (1971);23, 343–344 (1971)Google Scholar
  11. 11.
    Gottschalk, W.H., Hedlund, G.A.: Topological dynamics, Vol. 36 (revised ed.). Providence, R.I.: Am. Math. Soc., Colloquium Pub. 1968Google Scholar
  12. 12.
    Nitecki, Z.: Differentiable dynamics. An introduction to the orbit structure of diffeomorphisms. Cambridge, Mass.: M.I.T. Press 1971Google Scholar
  13. 13.
    Robbins, K.A.: A new approach to subcritical instability and turbulent transitions in a simple dynamo. Math. Proc. Cambridge Phil. Soc.Google Scholar
  14. 14.
    Creveling, H.F., DePaz, J.F., Baladi, J.V., Schoenhals, R.J.: Stability characteristics of a single phase free convection loop. J. Fluid Mech.67, 65–84 (1975)Google Scholar
  15. 15.
    Marsden, J.E., McCracken, M.: The Hopf bifurcation and its applications. Berlin, Heidelberg, New York: Springer 1976Google Scholar
  16. 16.
    Ruelle, D.: The Lorenz attractor and the problem of turbulence. Proc. Conf. Quantum Dynamics Models and Mathematics, Bielefeld (1975)Google Scholar
  17. 17.
    Guckenheimer, J., Oster, G., Ipaktchi, A.: The dynamics of density dependent population models. J. Math. Biol.4, 101–147 (1977)Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • James L. Kaplan
    • 1
  • James A. Yorke
    • 2
  1. 1.Department of MathematicsBoston UniversityBostonUSA
  2. 2.Institute for Physical Science and Technology and Department of MathematicsUniversity of MarylandCollege ParkUSA

Personalised recommendations