On the nature of turbulence

Abstract

A mechanism for the generation of turbulence and related phenomena in dissipative systems is proposed.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Abraham, R., Marsden, J.: Foundations of mechanics. New York: Benjamin 1967.

    Google Scholar 

  2. 2.

    Bass, J.: Fonctions stationnaires. Fonctions de corrélation. Application à la représentation spatio-temporelle de la turbulence. Ann. Inst. Henri Poincaré. Section B5, 135–193 (1969).

    Google Scholar 

  3. 3.

    Brunovsky, P.: One-parameter families of diffeomorphisms. Symposium on Differential Equations and Dynamical Systems. Warwick 1968–69.

  4. 4.

    Hirsch, M., Pugh, C. C., Shub, M.: Invariant manifolds. Bull. A.M.S.76, 1015–1019 (1970).

    Google Scholar 

  5. 5.

    —— —— -- Invariant manifolds. To appear.

  6. 6.

    Hopf, E.: Abzweigung einer periodischen Lösung von einer stationären Lösung eines Differentialsystems. Ber. Math.-Phys. Kl. Sächs. Akad. Wiss. Leipzig94, 1–22 (1942).

    Google Scholar 

  7. 7.

    Kelley, A.: The stable, center-stable, center, center-unstable, and unstable manifolds. Published as Appendix C of R. Abraham and J. Robbin: Transversal mappings and flows. New York: Benjamin 1967.

    Google Scholar 

  8. 8.

    Landau, L. D., Lifshitz, E. M.: Fluid mechanics. Oxford: Pergamon 1959.

    Google Scholar 

  9. 9.

    Leray, J.: Sur le mouvement d'un liquide visqueux emplissant l'espace. Acta Math.63, 193–248 (1934).

    Google Scholar 

  10. 10.

    Moser, J.: Perturbation theory of quasiperiodic solutions of differential equations. Published in J. B. Keller and S. Antman: Bifurcation theory and nonlinear eigenvalue problems. New York: Benjamin 1969.

    Google Scholar 

  11. 11.

    Smale, S.: Differentiable dynamical systems. Bull. Am. Math. Soc.73, 747–817 (1967).

    Google Scholar 

  12. 12.

    Thom, R.: Stabilité structurelle et morphogénèse. New York: Benjamin 1967.

    Google Scholar 

  13. 13.

    Williams, R. F.: One-dimensional non-wandering sets. Topology6, 473–487 (1967).

    Google Scholar 

  14. 14.

    Berger, M.: A bifurcation theory for nonlinear elliptic partial differential equations and related systems. In: Bifurcation theory and nonlinear eigenvalue problems. New York: Benjamin 1969.

    Google Scholar 

  15. 15.

    Fife, P. C., Joseph, D. D.: Existence of convective solutions of the generalized Bénard problem which are analytic in their norm. Arch. Mech. Anal.33, 116–138 (1969).

    Google Scholar 

  16. 16.

    Krasnosel'skii, M.: Topological methods in the theory of nonlinear integral equations. New York: Pergamon 1964.

    Google Scholar 

  17. 17.

    Rabinowitz, P. H.: Existence and nonuniqueness of rectangular solutions of the Bénard problem. Arch. Rat. Mech. Anal.29, 32–57 (1968).

    Google Scholar 

  18. 18.

    Velte, W.: Stabilität und Verzweigung stationärer Lösungen der Navier-Stokesschen Gleichungen beim Taylorproblem. Arch. Rat. Mech. Anal.22, 1–14 (1966).

    Google Scholar 

  19. 19.

    Yudovich, V.: The bifurcation of a rotating flow of fluid. Dokl. Akad. Nauk SSSR169, 306–309 (1966).

    Google Scholar 

Download references

Author information

Affiliations

Authors

Additional information

The research was supported by the Netherlands Organisation for the Advancement of Pure Research (Z.W.O.).

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Ruelle, D., Takens, F. On the nature of turbulence. Commun.Math. Phys. 20, 167–192 (1971). https://doi.org/10.1007/BF01646553

Download citation

Keywords

  • Neural Network
  • Statistical Physic
  • Complex System
  • Nonlinear Dynamics
  • Quantum Computing