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Turbulence: Determinism and Chaos

Published online by Cambridge University Press:  15 February 2018

Y. Pomeau
Y. Pomeau*
Affiliation:
Gif sur Yvette, France

Extract

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After the article of Ruelle and Takens, there has been recently much interest in the problem of the “onset of turbulence”. That is, instead of trying to understand the structure of a well established turbulence flow, one studies the way in which a flow “jumps” from a quiet stable laminar state to a turbulent state when its Reynolds (or Rayleigh) number increases.

Type
IX. Turbulence
Information
International Astronomical Union Colloquium , Volume 38: Problems of Stellar Convection , 1977 , pp. 337 - 348
Copyright
Copyright © 1976

References

1. Ruelle, D., Takens, F., Comm. Math. Phys. 2038, 167 (1971)Google Scholar
2. Halmos, P.R., Evgodia theory, Chelsea Pub. Comp. New York (1956)Google Scholar
3. May, R., Nature 26138, 459 (1976)Google Scholar
4. Li, T.Y. and Yorke, J.A., Am. Math. Month. 8238, 985 (1975)Google Scholar
5. Rikitake, T., “Electromagnetism and the earth interior” Elsevier (1968)Google Scholar
6. Laj, C., Pomeau, Y., in preparationGoogle Scholar
7. Monin, Yaglom, “Statistical Fluid Mechanics”, Vol. 1-238, MIT Press (1971)Google Scholar
8. Kubo, I., notes from the Nagoya Univ., Hopf, E., “Ergodentheorie”, Spinger Verlag, Berlin (1937), Smorodinsky, M. , “Ergodic Theory,Entropy”, Springer Verlag Lectures Notes in math. 2 14, Berlin (1971)Google Scholar
9. Smale, S., Bull. AMS 7338, 747 (1967)Google Scholar
10. Martin, P.C. and Mc Laughlin, J.B., Phys. Rev. Lett. 3338 1189 (1974), Phys. Rev. A12, 186 (1975)Google Scholar
11. Lorenz, E.N., J. Atmo. Sci. 2038, 130 (1963)Google Scholar
12. Thorn, R. “Modèle mathématique de la morphogènése”, 10–18, Paris (1974)Google Scholar
13. Andronov, A. and Pontryagin, L., Dokl. Akad. Nauk. SSSR 1438, 247 (1937)Google Scholar
14. Ruelle, D., in “Turbulence and Navier-Stokes equation” Lecture Notes in Math, 565, Springer Verlag, Berlin (1975), Lanford, O., III lectures at IHES (1975), Williams, R.B. “On the Structure of the Lorenz Attractors”, preprintGoogle Scholar
15. Arnold, V., Avez, A., “Ergodic Problems of Classical Mechanics”, Benjamin, New York (1969)Google Scholar
16. Betchov, R. and Criminale, W.O. Jr. “Stability of Parallel Flows”, Acad. Press, New York (1966)Google Scholar
17. et Lifchitz, Landau, “Mécanique des Fluides” chap. III, §27, ed. Mir. (Moscou) (1971)Google Scholar
18. Caldwell, D.R.. J. of Fluid Mech. 6438, 347 (1974)CrossRefGoogle Scholar
19. Bogoliubov, N.N., Mitropolskii, J.A., Samoĭlenko, A.M., “Methods of accelerated convergence in non linear mechanics” Spinger Verlag, Berlin (1976)Google Scholar
20. Arnold, V.I., Small divisors I, Izv. Akad. Nauk. SSSR Ser. Mat. 2538 (1), 21 (1961), Small divisors II, Usp. Mat. Nauk 18 (5), 13 (1963) ; 18 (6), 91 (1963), M.R. Hermann, Thesis, Orsay (1976)Google Scholar
21.Reference 19, p. 154Google Scholar
22.Reference 9, p. 788, Shub, M., Thesis, Univ. of Calif. Berkeley (1967)Google Scholar
23. Gollub, J.P., Hulbert, S.L., Dolny, G.M. and Swinney, H.L., to appear in “Photon correlation, spectroscopy and velocimetry“ Ed. Pyke, E.R. and Cummins, H.Z., Plenum Press (1976)Google Scholar
24. Ibanez, J.L., Pomeau, Y., to be publishedGoogle Scholar
25.Ref. 3, and Stefan, P., preprint IHES (Bures-sur-Yvette) Dec. 1976, A.N. Šarkovskiy, Urk. Math. t. 16, 1 (1964), B. Derrida and Y. Pomeau, to be publishedGoogle Scholar