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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

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

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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Differentiable dynamical systems and the problem of turbulence
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by David Ruelle PDF
Bull. Amer. Math. Soc. 5 (1981), 29-42
References
    1. F. Ahlers, Low-temperature studies of the Rayleigh-Bénard instability and turbulence, Phys. Rev. Lett. 33 (1974), 1185-1188.
  • Rufus Bowen, On Axiom A diffeomorphisms, Regional Conference Series in Mathematics, No. 35, American Mathematical Society, Providence, R.I., 1978. MR 0482842
  • Rufus Bowen and David Ruelle, The ergodic theory of Axiom A flows, Invent. Math. 29 (1975), no. 3, 181–202. MR 380889, DOI 10.1007/BF01389848
  • M. Campanino and H. Epstein, On the existence of Feigenbaum’s fixed point, Comm. Math. Phys. 79 (1981), no. 2, 261–302. MR 612250, DOI 10.1007/BF01942063
    • 5. P. Collet, J.-P. Eckmann and H. Koch, Period doubling bifurcations for families of maps on R (to appear).
  • P. Collet, J.-P. Eckmann, and O. E. Lanford III, Universal properties of maps on an interval, Comm. Math. Phys. 76 (1980), no. 3, 211–254. MR 588048, DOI 10.1007/BF02193555
  • James H. Curry, On the Hénon transformation, Comm. Math. Phys. 68 (1979), no. 2, 129–140. MR 543195, DOI 10.1007/BF01418124
  • Mitchell J. Feigenbaum, Quantitative universality for a class of nonlinear transformations, J. Statist. Phys. 19 (1978), no. 1, 25–52. MR 501179, DOI 10.1007/BF01020332
  • Mitchell J. Feigenbaum, The universal metric properties of nonlinear transformations, J. Statist. Phys. 21 (1979), no. 6, 669–706. MR 555919, DOI 10.1007/BF01107909
  • Mitchell J. Feigenbaum, The transition to aperiodic behavior in turbulent systems, Comm. Math. Phys. 77 (1980), no. 1, 65–86. MR 588687, DOI 10.1007/BF01205039
  • Sidnie Dresher Feit, Characteristic exponents and strange attractors, Comm. Math. Phys. 61 (1978), no. 3, 249–260. MR 501172, DOI 10.1007/BF01940767
  • C. Foiaş and G. Prodi, Sur le comportement global des solutions non-stationnaires des équations de Navier-Stokes en dimension $2$, Rend. Sem. Mat. Univ. Padova 39 (1967), 1–34 (French). MR 223716
  • C. Foiaş and R. Temam, Some analytic and geometric properties of the solutions of the evolution Navier-Stokes equations, J. Math. Pures Appl. (9) 58 (1979), no. 3, 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.
  • Hiroshi Fujita and Tosio Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269–315. MR 166499, DOI 10.1007/BF00276188
  • V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, Lecture Notes in Mathematics, vol. 749, Springer-Verlag, Berlin-New York, 1979. MR 548867, DOI 10.1007/BFb0063453
    • 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.
  • John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72. MR 556582, DOI 10.1007/BF02684769
  • M. Hénon, A two-dimensional mapping with a strange attractor, Comm. Math. Phys. 50 (1976), no. 1, 69–77. MR 422932, DOI 10.1007/BF01608556
  • M. W. Hirsch, C. C. Pugh, and M. Shub, Invariant manifolds, Lecture Notes in Mathematics, Vol. 583, Springer-Verlag, Berlin-New York, 1977. MR 0501173, DOI 10.1007/BFb0092042
    • 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.
  • Eberhard Hopf, A mathematical example displaying features of turbulence, Communications on Appl. Math. 1 (1948), 303–322. MR 30113, DOI 10.1002/cpa.3160010401
  • Gérard Iooss, Existence et stabilité de la solution périodique secondaire intervenant dans les problèmes d’evolution du type Navier-Stokes, Arch. Rational Mech. Anal. 47 (1972), 301–329 (French). MR 346350, DOI 10.1007/BF00281637
  • G. Iooss, Sur la deuxième bifurcation d’une solution stationnaire de systèmes du type Navier-Stokes, Arch. Rational Mech. Anal. 64 (1977), no. 4, 339–369 (French). MR 650472, DOI 10.1007/BF00282345
    • 26. A. Katok, unpublished.
  • Ju. I. Kifer, The limit behavior of invariant measures of small random perturbations of certain smooth dynamical systems, Dokl. Akad. Nauk SSSR 216 (1974), 979–981 (Russian). MR 0356255
  • Ju. I. Kifer, Small random perturbations of certain smooth dynamical systems, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 1091–1115 (Russian). MR 0388452
    • 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.
  • O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Mathematics and its Applications, Vol. 2, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Second English edition, revised and enlarged; Translated from the Russian by Richard A. Silverman and John Chu. MR 0254401
  • O. A. Ladyženskaja, The dynamical system that is generated by the Navier-Stokes equations, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 27 (1972), 91–115 (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions, 6. MR 0328378
    • 32. L. D. Landau, Turbulence, Dokl. Akad. Nauk SSSR 44 (1944), 339-342.
  • Oscar E. Lanford III, Remarks on the accumulation of period-doubling bifurcations, Mathematical problems in theoretical physics (Proc. Internat. Conf. Math. Phys., Lausanne, 1979) Lecture Notes in Phys., vol. 116, Springer, Berlin-New York, 1980, pp. 340–342. MR 582642
  • Jean Leray, Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math. 63 (1934), no. 1, 193–248 (French). MR 1555394, DOI 10.1007/BF02547354
    • 35. A. Libchaber et J. Maurer, Une expérience de Rayleigh-Bénard de géométrie réduite; multiplication, accrochage et démultiplication de fréquences, J. Physique (to appear).
  • J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod, Paris; Gauthier-Villars, Paris, 1969 (French). MR 0259693
    • 37. E. N. Lorenz, Deterministic non-periodic flow, J. Atmosphere Sci. 20 (1963), 130-141.
  • John Mallet-Paret, Negatively invariant sets of compact maps and an extension of a theorem of Cartwright, J. Differential Equations 22 (1976), no. 2, 331–348. MR 423399, DOI 10.1016/0022-0396(76)90032-2
  • Ricardo Mañé, On the dimension of the compact invariant sets of certain nonlinear maps, Dynamical systems and turbulence, Warwick 1980 (Coventry, 1979/1980), Lecture Notes in Math., vol. 898, Springer, Berlin-New York, 1981, pp. 230–242. MR 654892
  • Ricardo Mañé, Lyapounov exponents and stable manifolds for compact transformations, Geometric dynamics (Rio de Janeiro, 1981) Lecture Notes in Math., vol. 1007, Springer, Berlin, 1983, pp. 522–577. MR 730286, DOI 10.1007/BFb0061433
  • J. E. Marsden and M. McCracken, The Hopf bifurcation and its applications, Applied Mathematical Sciences, Vol. 19, Springer-Verlag, New York, 1976. With contributions by P. Chernoff, G. Childs, S. Chow, J. R. Dorroh, J. Guckenheimer, L. Howard, N. Kopell, O. Lanford, J. Mallet-Paret, G. Oster, O. Ruiz, S. Schecter, D. Schmidt and S. Smale. MR 0494309, DOI 10.1007/978-1-4612-6374-6
  • A. S. Monin, On the nature of turbulence; Russian transl., Soviet Phys. Uspekhi 125 (1978), no. 1, 97–122. MR 545760
  • Ju. I. Neĭmark, Some cases of the dependence of periodic motions on parameters, Dokl. Akad. Nauk SSSR 129 (1959), 736–739 (Russian). MR 0132256
  • Sheldon E. Newhouse, Diffeomorphisms with infinitely many sinks, Topology 13 (1974), 9–18. MR 339291, DOI 10.1016/0040-9383(74)90034-2
  • Sheldon E. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 101–151. MR 556584, DOI 10.1007/BF02684771
  • V. I. Oseledec, A multiplicative ergodic theorem. Characteristic Ljapunov, exponents of dynamical systems, Trudy Moskov. Mat. Obšč. 19 (1968), 179–210 (Russian). MR 0240280
  • Ja. B. Pesin, Characteristic Ljapunov exponents, and ergodic properties of smooth dynamical systems with invariant measure, Dokl. Akad. Nauk SSSR 226 (1976), no. 4, 774–777 (Russian). MR 0410804
  • Ja. B. Pesin, Families of invariant manifolds that correspond to nonzero characteristic exponents, Izv. Akad. Nauk SSSR Ser. Mat. 40 (1976), no. 6, 1332–1379, 1440 (Russian). MR 0458490
  • Ja. B. Pesin, Characteristic Ljapunov exponents, and smooth ergodic theory, Uspehi Mat. Nauk 32 (1977), no. 4 (196), 55–112, 287 (Russian). MR 0466791
  • R. V. Plykin, Sources and sinks of $\textrm {A}$-diffeomorphisms of surfaces, Mat. Sb. (N.S.) 94(136) (1974), 243–264, 336 (Russian). MR 0356137
    • 51. H. Poincaré, Théorie des tourbillons, Georges Carré, Paris, 1893.
  • Charles Pugh and Michael Shub, Stable ergodicity and julienne quasi-conformality, J. Eur. Math. Soc. (JEMS) 2 (2000), no. 1, 1–52. MR 1750453, DOI 10.1007/s100970050013
  • M. S. Raghunathan, A proof of Oseledec’s multiplicative ergodic theorem, Israel J. Math. 32 (1979), no. 4, 356–362. MR 571089, DOI 10.1007/BF02760464
  • David Ruelle, A measure associated with axiom-A attractors, Amer. J. Math. 98 (1976), no. 3, 619–654. MR 415683, DOI 10.2307/2373810
  • David Ruelle, An inequality for the entropy of differentiable maps, Bol. Soc. Brasil. Mat. 9 (1978), no. 1, 83–87. MR 516310, DOI 10.1007/BF02584795
  • David Ruelle, Dynamical systems with turbulent behavior, Mathematical problems in theoretical physics (Proc. Internat. Conf., Univ. Rome, Rome, 1977) Lecture Notes in Phys., vol. 80, Springer, Berlin-New York, 1978, pp. 341–360. MR 518445
  • David Ruelle, Sensitive dependence on initial condition and turbulent behavior of dynamical systems, Bifurcation theory and applications in scientific disciplines (Papers, Conf., New York, 1977) Ann. New York Acad. Sci., vol. 316, New York Acad. Sci., New York, 1979, pp. 408–416. MR 556846
  • David Ruelle, Ergodic theory of differentiable dynamical systems, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 27–58. MR 556581, DOI 10.1007/BF02684768
  • David Ruelle, Characteristic exponents and invariant manifolds in Hilbert space, Ann. of Math. (2) 115 (1982), no. 2, 243–290. MR 647807, DOI 10.2307/1971392
  • David Ruelle, Sensitive dependence on initial condition and turbulent behavior of dynamical systems, Bifurcation theory and applications in scientific disciplines (Papers, Conf., New York, 1977) Ann. New York Acad. Sci., vol. 316, New York Acad. Sci., New York, 1979, pp. 408–416. MR 556846
  • David Ruelle and Michael Shub, Stable manifolds for maps, Global theory of dynamical systems (Proc. Internat. Conf., Northwestern Univ., Evanston, Ill., 1979) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 389–392. MR 591198
  • David Ruelle and Floris Takens, On the nature of turbulence, Comm. Math. Phys. 20 (1971), 167–192. MR 284067, DOI 10.1007/BF01646553
    • 63. R. J. Sacker, On invariant surfaces and bifurcation of periodic solutions of ordinary differential equations, Thesis, New York University, 1964.
  • Vladimir Scheffer, Turbulence and Hausdorff dimension, Turbulence and Navier-Stokes equations (Proc. Conf., Univ. Paris-Sud, Orsay, 1975) Lecture Notes in Math., Vol. 565, Springer, Berlin, 1976, pp. 174–183. MR 0452123
  • Vladimir Scheffer, The Navier-Stokes equations on a bounded domain, Comm. Math. Phys. 73 (1980), no. 1, 1–42. MR 573611, DOI 10.1007/BF01942692
  • Ja. G. Sinaĭ, Gibbs measures in ergodic theory, Uspehi Mat. Nauk 27 (1972), no. 4(166), 21–64 (Russian). MR 0399421
  • S. Smale, Differentiable dynamical systems, Bull. Amer. Math. Soc. 73 (1967), 747–817. MR 228014, DOI 10.1090/S0002-9904-1967-11798-1
    • 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.
  • Roger Temam, Navier-Stokes equations, Revised edition, Studies in Mathematics and its Applications, vol. 2, North-Holland Publishing Co., Amsterdam-New York, 1979. Theory and numerical analysis; With an appendix by F. Thomasset. MR 603444
  • Peter Walters, Anosov diffeomorphisms are topologically stable, Topology 9 (1970), 71–78. MR 254862, DOI 10.1016/0040-9383(70)90051-0
  • John Guckenheimer and R. F. Williams, Structural stability of Lorenz attractors, Inst. Hautes Études Sci. Publ. Math. 50 (1979), 59–72. MR 556582, DOI 10.1007/BF02684769
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Additional Information
  • 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