Remote Access Bulletin of the American Mathematical Society

Bulletin of the American Mathematical Society

ISSN 1088-9485(online) ISSN 0273-0979(print)

 
 

 

Differentiable dynamical systems and the problem of turbulence


Author: David Ruelle
Journal: Bull. Amer. Math. Soc. 5 (1981), 29-42
MSC (1980): Primary 76A, 58F
DOI: https://doi.org/10.1090/S0273-0979-1981-14917-X
MathSciNet review: 614312
Full-text PDF

References | Similar Articles | Additional Information

References [Enhancements On Off] (What's this?)

  • 1. F. Ahlers, Low-temperature studies of the Rayleigh-Bénard instability and turbulence, Phys. Rev. Lett. 33 (1974), 1185-1188.
  • 2. R. Bowen, On Axiom A diffeomorphisms, CBMS Regional Conf. Series, No. 35, Amer. Math. Soc., Providence, R. I., 1978. MR 482842
  • 3. R. Bowen and D. Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), 181-202. MR 380889
  • 4. M. Campanino and H. Epstein, On the existence of Feigenbaum's fixed point, Comm. Math. Phys. (to appear). MR 612250
  • 5. P. Collet, J.-P. Eckmann and H. Koch, Period doubling bifurcations for families of maps on R (to appear).
  • 6. P. Collet, J.-P. Eckmann and O. E. Lanford, Universal properties of maps of an interval, Comm. Math. Phys. 76 (1980), 211-254. MR 588048
  • 7. J. H. Curry, On the Hénon transformation, Comm. Math. Phys. 68 (1979), 129-140. MR 543195
  • 8. M. J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), 25-52. MR 501179
  • 9. M. J. Feigenbaum, The universal metric properties of nonlinear transformations, J. Statist. Phys. 21 (1979), 669-706. MR 555919
  • 10. M. J. Feigenbaum, The transition to aperiodic behavior in turbulent systems, Comm. Math. Phys. (to appear). MR 588687
  • 11. S. D. Feit, Characteristic exponents and strange attractors, Comm. Math. Phys. 61 (1978), 249-260. MR 501172
  • 12. C. Foiaş and G. Prodi, Sur le comportement global des solutions nonstationnaires des équations de Navier-Stokes en dimension 2, Rend. Sem. Mat. Univ. Padova 39 (1967), 1-34. MR 223716
  • 13. C. Foiaş and R. Temam, Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J. Math. Pures Appl. 58 (1979), 339-368. MR 544257
  • 14. U. Frisch, P.-L. Sulem and M. Nelkin, A simple dynamical model of intermittent fully developed turbulence, J. Fluid Mech. 87 (1978), 719-736.
  • 15. H. Fugita and T. Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269-315. MR 166499
  • 16. V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Math., vol. 749, Springer-Verlag, Berlin and New York, 1979. MR 548867
  • 17. J. P. Gollub and H. L. Swinney, Onset of turbulence in a rotating fluid, Phys. Rev. Lett. 35 (1975), 927-930.
  • 18. J. Guckenheimer, A strange, strange attractor, The Hopf Bifurcation Theorem and Its Applications (J. E. Marsden and M. Mc Cracken, eds.) Springer-Verlag, Berlin and New York, 1976, pp. 368-381.
  • 19. J. Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59-72. MR 556582
  • 20. M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys. 50 (1976), 69-77. MR 422932
  • 21. M. Hirsch, C. Pugh and M. Shub, Invariant manifolds, Lecture Notes in Math., vol. 583, Springer-Verlag, Berlin and New York, 1977. MR 501173
  • 22. E. Hopf, Abzweigung einer periodischen Lösung von einer stationaren Lösung eines Differentialsystems, Ber. Math.-Phys. K. Sächs. Akad. Wiss. Leipzig 94 (1942), 1-22.
  • 23. E. Hopf, A mathematical example displaying the features of turbulence, Comm. Pure Appl. Math. 1 (1948), 303-322. MR 30113
  • 24. G. Iooss, Existence et stabilité de la solution périodique secondaire intervenant dans les problèmes d'évolution de type Navier-Stokes, Arch. Rational Mech. Anal. 47 (1972), 301-329. MR 346350
  • 25. G. Iooss, Sur la deuxième bifurcation d'une solution stationnaire de systèmes du type Navier-Stokes, Arch. Rational Mech. Anal. 64 (1977), 339-369. MR 650472
  • 26. A. Katok, unpublished.
  • 27. Ju. I. Kifer, On the limiting behavior of invariant measures of small random perturbations for some smooth dynamical systems, Dokl. Akad. Nauk SSSR 216 (5) (1974) = Soviet Math. Dokl. 15 (1941), 918-921. MR 356255
  • 28. Ju. I. Kifer, On small random perturbations of some smooth dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 38 (5) (1974), 1091-1115 = Math. USSR-Izv. 8 (1974), 1083-1107. MR 388452
  • 29. A. N. Kolmogorov, Local structure of turbulence in an incompressible fluid at very high Reynolds numbers, Dokl. Akad. Nauk SSSR 30 (1941), 299-303.
  • 30. O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, 2nd ed., Nauka, Moscow, 1970; 2nd English ed., Gordon and Breach, New York, 1969. MR 254401
  • 31. O. A. Ladyzhenskaya, A dynamical system generated by the Navier-Stokes equations, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov 27 (1972), 91-115 = J. Soviet Math. 3 (1976), 458-479. MR 328378
  • 32. L. D. Landau, Turbulence, Dokl. Akad. Nauk SSSR 44 (1944), 339-342.
  • 33. O. E. Lanford, Remarks on the accumulation of period-doubling bifurcations, Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, vol. 116, Springer-Verlag, Berlin and New York, 1980, pp. 340-342. MR 582642
  • 34. J. Leray, Sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Math. 63 (1934), 193-248. MR 1555394
  • 35. A. Libchaber et J. Maurer, Une expérience de Rayleigh-Bénard de géométrie réduite; multiplication, accrochage et démultiplication de fréquences, J. Physique (to appear).
  • 36. J. L. Lions, Quelques méthodes de résolution des problèmes aux limites non-linéaires, Dunod, Paris, 1969. MR 259693
  • 37. E. N. Lorenz, Deterministic non-periodic flow, J. Atmosphere Sci. 20 (1963), 130-141.
  • 38. J. Mallet-Paret, Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, J. Differential Equations 22 (1976), 331-348. MR 423399
  • 39. R. Mañé, On the dimension of the compact invariant sets of certain nonlinear maps, preprint. MR 654892
  • 40. R. Mañé, Lyapunov exponents and stable manifolds for compact transformations, preprint. MR 730286
  • 41. J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Applied Math. Sci., vol. 19, Springer-Verlag, Berlin and New York, 1976. MR 494309
  • 42. A. S. Monin, On the nature of turbulence, Uspehi Fiz. Nauk 125 (1) (1978), 97-122; English transl., Soviet Physiks-Uspekhi 21 (1978), 429-442. MR 545760
  • 43. Ju. I. Naimark, On some cases of dependence of periodic solutions on parameters, Dokl. Akad. Nauk SSSR 129 (4) (1959), 736-739. MR 132256
  • 44. S. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 9-18. MR 339291
  • 45. S. Newhouse, The abundance of wild hyperbolic sets and non-smooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 102-151. MR 556584
  • 46. V. I. Oseledec, A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 197-221. MR 240280
  • 47. Ya. B. Pesin, Lyapunov characteristic exponents and ergodic properties of smooth dynamical systems with an inveriant measure, Dokl. Akad. Nauk SSSR 226 (4) (1976), 774-777 = Soviet Math. Dokl. 17 (1) (1976), 196-199. MR 410804
  • 48. Ya. B. Pesin, Invariant manifold families which correspond to nonvanishing characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat. 40 (6) (1976), 1332-1379 = Math. USSR-Izv. 10 (6) (1976), 1261-1305. MR 458490
  • 49. Ya. B. Pesin, Lyapunov characteristic exponents and smooth ergodic theory, Uspehi Mat. Nauk 32 (4) (196) (1977), 55-112 = Russian Math. Surveys 32 (4) (1977), 55-114. MR 466791
  • 50. R. V. Plykin, Sources and currents of A-diffeomorphisms of surfaces, Mat. Sb. 94 2(6) (1974), 243-264. MR 356137
  • 51. H. Poincaré, Théorie des tourbillons, Georges Carré, Paris, 1893.
  • 52. C. C. Pugh and M. Shub, Differentiability and continuity of invariant manifolds, preprint. MR 1750453
  • 53. M. S. Raghunathan, A proof of Oseledec' multiplicative ergodic theorem, Israel J. Math. 32 (1979), 356-362. MR 571089
  • 54. D. Ruelle, A measure associated with Axiom A attractors, Amer. J. Math. 98 (1976), 619-654. MR 415683
  • 55. D. Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat. 9 (1978), 83-87. MR 516310
  • 56. D. Ruelle, Dynamical systems with turbulent behavior, Mathematical Problems in Theoretical Physics, Lecture Notes in Physics, vol. 80, Springer-Verlag, Berlin and New York, 1978, pp. 341-360. MR 518445
  • 57. D. Ruelle, Sensitive dependence on initial condition and turbulent behavior of dynamical systems, Ann. N. Y. Acad. Sci. 316 (1978), 408-416. MR 556846
  • 58. D. Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 275-306. MR 556581
  • 59. D. Ruelle, Characteristic exponents and invariant manifolds in Hilbert space (to appear). MR 647807
  • 60. D. Ruelle, Measures describing a turbulent flow, Ann. N. Y. Acad. Sci. (to appear). MR 556846
  • 61. D. Ruelle and M. Shub, Stable manifolds for maps, Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 389-392. MR 591198
  • 62. D. Ruelle and F. Takens, On the nature of turbulence, Comm. Math. Phys. 20 (1971), 167-192; note 23 (1971), 343-344. MR 284067
  • 63. R. J. Sacker, On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations, Thesis, New York University, 1964.
  • 64. V. Scheffer, Turbulence and Hausdorff dimension, Turbulence and Navier-Stokes Equation, Lecture Notes in Math., vol. 565, Springer-Verlag, Berlin and New York, 1979, pp. 94-112. MR 452123
  • 65. V. Scheffer, The Navier-Stokes equations on a bounded domain, Comm. Math. Phys. 73 (1980), 1-42. MR 573611
  • 66. Ia. G. Sinai, Gibbs measure in ergodic theory, Uspehi Mat. Nauk 27 (4) (1972), 21-64 = Russian Math. Surveys 27 (4) (1972), 21-69. MR 399421
  • 67. S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747-817. MR 228014
  • 68. H. L. Swinney and J. P. Gollub, The transition to turbulence, Physics Today 31 (1978), 41-49.
  • 69. G. I. Taylor, Stability of a viscous liquid contained between two rotating cylinders, Philos. Trans. Roy. Soc. London Ser. A 223 (1923), 289-343.
  • 70. R. Temam, Navier-Stokes equations, rev. ed., North-Holland, Amsterdam, 1979. MR 603444
  • 71. P. Walters, Characteristic exponents and the entropy of diffeomorphisms, preprint. MR 254862
  • 72. R. F. Williams, The structure of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 73-99. MR 556583

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (1980): 76A, 58F

Retrieve articles in all journals with MSC (1980): 76A, 58F


Additional Information

DOI: https://doi.org/10.1090/S0273-0979-1981-14917-X

American Mathematical Society