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Roads to turbulence in dissipative dynamical systems

J. -P. Eckmann

J. -P. Eckmann

  • Département de Physique Théorique, Université de Genève, 1211 Genève 4, Switzerland

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Rev. Mod. Phys. 53, 643 – Published 1 October, 1981

DOI: https://doi.org/10.1103/RevModPhys.53.643

Abstract

Three scenarios leading to turbulence in theory and experiment are outlined. The respective mathematical theories are explained and compared.

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References (99)

  1. Arnold, V., 1976 Méthodes mathémaques de la mécanique classique (Editions Mir, Moscow)
  2. Bergé, P., M. Dubois, P. Manneville, and Y. Pomeau, 1980, J. Phys. (Paris) Lett. 41, L341
  3. Bowen, R., 1975, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lecture Notes in Mathematics (Springer, Berlin), Vol. 470
  4. Brayton, R. K., and J. K. Moser, 1964, Q. Appl. Math. 22, 1, 81
  5. Collet, P., and J.-P. Eckmann, 1980, Iterated maps on the interval as dynamical systems, Progress in Physics (Birkhäuser, Basel), Vol. 1 1
  6. Collet, P., J.-P. Eckmann, and H. Koch, 1980, Physica D, in press
  7. Collet, P., J.-P. Eckmann, and H. Koch, 1981, J. Stat. Phys. 25, 1
  8. Collet, P. J.-P. Eckmann, and O. Landford, 1980. Commun. Math. Phys. 76, 211
  9. Collet, P., J.-P. Eckmann, and L. Thomas, 1981, Commun. Math. Phys., in press
  10. Crutchfield, J., M. Nauenberg, and J. Rudnick, 1980, Phys. Rev. Lett. 46, 933
  11. Eckmann, J.-P., 1980, in Bifurcation phenomena in Mathematical Physics and Related Topics, edited by C. Bardos and D. Bessis (Reidel, Dordrecht), p. 115
  12. Eckmann, J.-P., L. Thomas, and P. Wittwer, 1981, J. Phys. A, in press
  13. Feigenbaum, M. J., 1978, J. Stat. Phys. 19, 25
  14. Feigenbaum, M. J., 1979a, J. Stat. Phys. 21, 669
  15. Feigenbaum, M. J., 1979b, Phys. Lett. A 74, 375
  16. Feigenbaum, M. J., 1980, Commun. Math. Phys. 77, 65
  17. Foias, C., and R. Temam, 1979, J. Math. Pures Appl. 58, 339
  18. Greene, J. M., R. S. MacKay, F. Vivaldi, and M. Feigenbaum, 1981, Physica D, in press
  19. Hagedorn, R., 1957 CERN Report No. 57-1
  20. Jakobson, M., in Proceedings of the International Conference on Dynamical Systems, Northeastern University, 1980 (Commun. Math. Phys., in press)
  21. Kifer, J. I., 1974, Math. USSR Izvestija 8, 1083
  22. Lanau, L., and E. Lifshitz, 1959, Fluid Mechanics (Pergamon, Oxford)
  23. Lanford, O. E., 1978, Qualitative and Statistical Theory of Dissipative Systems, CIME Course, 1976 (Liguori Editore, Napoli)
  24. Lanford, O. E., 1981, Strange Attractors and Turbulence, Topics in Applied Physics (Springer, Berlin), Vol. 45, p. 7
  25. Levi, M., Mem. Am. Math. Soc., in press
  26. Libchaber, A., 1981, in Nato Study, Nonlinear Phenomena and Phase Transitions, edited by T. Riste, in press
  27. Libchaber, A., and J. Maurer, 1980, J. Phys. (Paris), Colloq. 41, C3, 51
  28. Lorenz, E. N., 1963, J. Atmos. Sci. 20, 130
  29. Manneville, P., and Y. Pomeau, 1980, Physica D 1, 219
  30. Maurer, J., and A. Libchaber, 1980, J. Phys. (Paris) Lett. 41, L515
  31. Mayer-Kress, G., and H. Haken, 1981, J. Phys. A 82, 151
  32. Misiurewicz, M., 1980, Ann. N.Y. Acad. Sci., Vol. 357
  33. Monin, A. S., and A. M. Yaglom, 1975, Statistical Fluid Mechanics (MIT, Cambridge, Mass.), Vol. 2
  34. Nauenberg, M., and J. Rudnick, 1981, Preprint, University of California, Santa Cruz, UCSC 80/137
  35. Newhouse, S., 1980a, Ann. N.Y. Acad. Sci., Vol. 357
  36. Newhouse, S., 1980b, Lectures on Dynamical Systems, Progress in Mathematics (Birkhäuser, Boston), Vol. 8
  37. Newhouse, S., D. Ruelle, and F. Takens, 1978, Commun. Math. Phys. 64, 35
  38. Nicolis, G., and Y. Prigogine, 1977, Self-Organization in Nonequilibrium Systems (Wiley, New York)
  39. Plykin, R., 1974, Math. USSR Sbornik 23, 233
  40. Pomeau, Y., and P. Manneville, 1980, Commun. Math. Phys. 77, 189
  41. Pomeau, Y., J. C. Roux, A. Rossi, S. Bachelart, and C. Vidal, 1981, J. Phys. Lett. 42, 271
  42. Roux, J. C., A. Rossi, S. Bachelart, and C. Vidal, 1980, Phys. Lett. A 77, 391
  43. Ruelle, D., 1980, La Recherche 108, 132
  44. Ruelle, D., and F. Takens, 1971, Commun. Math. Phys. 20, 167
  45. Shub, M., 1980, Stabilité globale des systèmes dynamiques Astérisque, Vol. 56 (Société Math. Française, Paris)
  46. Siegel, C. L., and J. K. Moser, 1971, Lectures on Celestial Mechanics (Springer, Berlin)
  47. Smale, S., 1967, Bull. Am. Math. Soc. 73, 747
  48. Swinney, H. L., and H. P. Gollub, 1978, Phys. Today 31 (8), 41
  49. Takens, F., 1980, Preprint No. 7907, Dept. of Mathematics University, Groningen
  50. Thom, R., 1972, Stabilité Structurelle et Morphogenèse (Benjamin, Reading)
  51. Ventsel, A. D., and M. I. Freidlin, 1970, Russ. Math. Surveys 25, 1
  52. Poston, T., and I. Stewart, 1978, Catastrophe theory and its applications (Pitman, London)
  53. Thom, R., 1977, Stabilité structurelle et morphogenèse (Inter Editions, Paris)
  54. Zeeman, E. C., 1977, Catastrophe theory: selected papers, 1972-1977 (Addison-Wesley, London)
  55. Abraham, R. H., 1978, Foundations of mechanics (Benjamin, Reading)
  56. Abraham, R. H., and C. D. Shaw, 1980, University of California, Santa Cruz
  57. Helleman, R. H. G., 1980, in Fundamental problems in statistical mechanics, edited by E. G. D. Cohen (North-Holland, Amsterdam), p. 165
  58. Shaw, R., 1978, Z. Naturforschung, in press
  59. Smale, S., 1967, Bull. Am. Math. Soc. 73, 747
  60. Bowen, R., 1979, Commun. Math. Phys. 69, 1
  61. Guckenheimer, J., 1979, Commun. Math. Phys. 70, 133
  62. Hofbauer, F., 1979, Preprint, Inst. f. Mathematik, University, Wien
  63. Jakobson, M. V., Commun. Math. Phys., in press
  64. Jonker, L., 1979, Proc. London Math. Soc. 39, 428
  65. Lasota, A., and J. A. Yorke, 1973, Am. Math. Soc. Transl. 183, 481
  66. Li, T., and J. A. Yorke, 1972, Amer. Math. Monthly 82, 985
  67. Lorenz, E. N., 1979, On the Prevalence of Aperiodicity in Simple Systems, Lecture Notes in Mathematics (Springer, Berlin), Vol. 755
  68. Matsumoto, S., 1980
  69. May, R. M., 1976, Nature 261, 459
  70. Metropolis, M., M. L. Stein, and P. R. Stein, 1973, J. Comb. Theory 15, 25
  71. Milnor, J., and P. Thurston, 1977, Preprint, Dept. of Mathematics, Princeton University, Princeton
  72. Misiurewicz, M., Publ. Math. IHES, Vol. 53, in press
  73. Nitecki, Z., to appear, Topological Dynamics on the Interval. Ergodic Theory and Dynamical Systems, II, Progress in Mathematics (Birkhäuser, Boston)
  74. Ruelle, D., 1977, Commun. Math. Phys. 55, 47
  75. Šarkovskii, A., 1964, Ukr. Mat. Zh. 16, 61
  76. Singer, D., 1978, SIAM J. Appl. Math. 35, 260
  77. Stef̌an, P., 1977, Commun. Math. Phys. 54, 237
  78. Ulam, S. M., and J. v. Neumann, 1947, Bull. Am. Math. Soc. 53, 1120
  79. Benettin, G., C. Cercignani, L. Galgani, and A. Giorgilli, 1980, Lett. Nuovo Cimento 28, 1
  80. Benettin, G., L. Galgani, and A. Giorgilli, 1980, Lett. Nuovo Cimento 29, 163
  81. Bountis, T., Physica D, in press
  82. Campanino, M., H. Epstein, and D. Ruelle, 1980, Topology, in press Commun. Math. Phys. 79, 261
  83. Coullet, P., and J. Tresser, 1978, C. R. Acad. Sci. 287, 577 J. Phys. (Paris) C 5, 25
  84. Derrida, B., A. Gervois, and Y. Pomeau, 1979, J. Phys. (Paris) A 12, 269
  85. Derrida, B., and Y. Pomeau, Phys. Lett. 80A, 217
  86. Lanford, O. E., III, 1980, in Mathematical Problems in Theoretical Physics, Lecture Notes in Physics (Springer, Berlin), Vol. 116, p. 103
  87. McLaughlin, J. B., 1981, J. Stat. Phys. 24, 375
  88. Mayer-Kress, G., and H. Haken, 1981, Phys. Lett. A 82, 151
  89. Crutchfield, J. P., J. D. Farmer, and B. A. Huberman, 1980, Preprint, University of California, Santa Cruz
  90. Shraiman, B., C. E. Wayne, and P. C. Martin, 1981, Phys. Rev. Lett. 46, 935
  91. Feit, S., 1978, Commun. Math. Phys. 61, 249
  92. Francheschini, V., 1980, J. Stat. Phys. 22, 397
  93. Francheschini, V., and C. Tebaldi, 1979, J. Stat. Phys. 21, 707
  94. Gollub, J. P., and S. V. Benson, 1980, J. Fluid Mech. 100, 449
  95. Maurer, J., and A. Libchaber, 1979, J. Phys. (Paris) Lett. 40, L419
  96. Maurer, J.A. LibchaberNonlinear Dynamics, edited by R. Helleman Sci., Vol. 357
  97. Ruelle, D., 1980, Ann. N.Y. Acad. Sci., Vol. 357, p. 1
  98. Vidal, C., and J. C. Roux, 1980, La Recherche
  99. Vidal, C., J. C. Roux, S. Bachelart, and A. Rossi, 1980, Ann. N.Y. Acad. Sci., Vol. 357, p. 377

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Rev. Mod. Phys. 53, 643– Published 1 October, 1981

    © 1981 American Physical Society

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