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Prologue Some ideas about strange attractors

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Dynamical System and Chaos

Part of the book series: Lecture Notes in Physics ((LNP,volume 179))

Abstract

For two-dimensional diffeomorphisms or flows reducing essentially to them the evolution of S.A. can be described geometrically using bifurcations, homoclinic and heteroclinic points. However, many questions are left open:

  1. 1)

    Prediction of values of the parameters for which a S.A. appears or is suddenly destroyed.

  2. 2)

    Existence of invariant measures on the S.A. Ergodic or mixing properties of the diffeomorphism restricted to the S.A., with respect to this measure.

  3. 3)

    Examination of the geometry of the S.A. for higher dimensions. Mechanisms producing or destroying S.A. in this case: Study of homo/heteroclinic points of normally hyperbolic invariant or periodic objects.

We strongly recommend to look for the geometric structure in physical or numerical experiments. It seems to us that without this knowledge one cannot get a really deep insight in the problem of S.A.

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References

  1. Abraham, R., Shaw, C.:“Dynamics: The Geometry of Behavior”, vol. 1 Aerial Press Inc. 1982 ( following volumes to appear.)

    Google Scholar 

  2. Afraîmoviĉ, V.S., Šil'nikov, L.P.: “On some global bifurcations connected with the disappearance of a fixed point of saddle-node type”, Soviet Math. Doklady 15, 1761–1765 (1974).

    Google Scholar 

  3. Afraîmoviĉ, V.S., Bykov, V.V., Šil'nikov, L.P.: “On the appearance and structure of the Lorenz-attractor”, Dokl. Akad. Nauk. USSR 234, 336–339 (1977).

    Google Scholar 

  4. Aizawa, Y., Shimada, I.: “The Wandering motion on the Lorenz surface”, Prog. Theor. Phys. 57, 2146–2147 (1977)

    Google Scholar 

  5. Allgower, E.L., Glashoff, K., Peitgen, H.O. (ed): “Numerical solution of nonlinear equations”, Lect. Notes in Math. 878, Springer (1981).

    Google Scholar 

  6. Arneodo, A., Coullet, P., Tresser, C.: “Occurrence of strange attractors in 3-dimensional Volterra equations”, Phys. Lett. 79A, 259–263 (1980).

    Google Scholar 

  7. Arneodo, A., Coullet, P., Tresser, C.: “A possible new mechanism for the onset of turbulence”, Phys. Lett. 81A, 197–201 (1981).

    Google Scholar 

  8. Arneodo, A., Coullet, P., Tresser, C.: “Possible new strange a.ttractors with spiral structure”, Commun. Math. Phys. 79, 573–579, (1981).

    Google Scholar 

  9. Arneodo, A., Coullet, P., Tresser, C.: “Oscillators with Chaotic Behavior: An illustration of a theorem of Šil'nikov”, J. Stat. Phys. 27, 171–182 (1982).

    Google Scholar 

  10. Arneodo, A., Coullet, P., Peyraud, J., Tresser, C.: “Strange attractors in Volterra equations for species in competition”, J. Math. Biology 14, 153–157 (1982).

    Google Scholar 

  11. Arneodo, A.,Coullet, P., Tresser, C.: “On the relevance of period-doubling cascades at the onset of turbulence”, to appear in Physica D.

    Google Scholar 

  12. Arneodo, A.,Coullet, P., Libchaber, A., Maurer, J., D'Humières, D., Tresser, C.: “About the observation of the uncompleted cascade in a Rayleigh-Bénard experiment”, to appear in Physica D.

    Google Scholar 

  13. Arnold, V.I.: “Chapitres Supplémentaires de la Théorie des Equations Differentielles Ordinaires”, Editions MIR, Moscow, 1980.

    Google Scholar 

  14. Aronson, D.G., Chory, M.A., Hall, G.R., McGehee, R.P.: “Bifurcations from an invariant circle for two parameter families of maps of the Plane: A computer-assisted study”, Commun. in Math. Phys. 83, 303–354 (1982).

    Google Scholar 

  15. Atten, P., Lacroix, J.C., Malraison, B.: “Chaotic motion in a Coulomb force driven instability: Large aspect ratio experiments”. Phys. Lett. 77A, 255–258 (1980).

    Google Scholar 

  16. Auslander, J., Yorke, J.: “Interval maps, factors of maps and chaos”, Tohoku Math. J. 32, 177–188 (1980).

    Google Scholar 

  17. Bernard, P., Ratiu, T. (ed): “Turbulence Seminar”, Lect. Notes in Math. 615, Springer (1977).

    Google Scholar 

  18. Bountis, T.C.: “Period doubling and universality in conservative systems”, Physica D (to appear).

    Google Scholar 

  19. Bowen, R., Ruelle, D.: “The ergodic theory of axiom A diffeomorphisms”, Inventiones Math. 29, 181–202 (1975).

    Google Scholar 

  20. Bowen, R.: “A horseshoe with positive measure”, Inventiones Math. 29, 203–204 (1975).

    Google Scholar 

  21. Bowen, R.: “On axiom A diffeomorphism”, A.M.S., C.B.M.S. Regional Conference 35 (1978).

    Google Scholar 

  22. Bowen, R.: “Invariant measures for Markov maps of the interval”, Comm. Math. Phys. 69, 1–17 (1979).

    Google Scholar 

  23. Baivé, D., Franceschini, V.: “Symmetry breaking on a model of five-mode truncated Navier-Stokes equations”, J. Stat. Phys. 26, 471–484 (1981).

    Google Scholar 

  24. Bridges, R., Rowlands, G.: “On the analytic form of some strange attractors”, Physical Lett. 63A, 189–190 (1977).

    Google Scholar 

  25. Campanino, M., Epstein, M.: “On the existence of Feigenbaum's fixed point”, Comm. Math. Phys. 79, 261–302 (1981).

    Google Scholar 

  26. Chang, Shau-Jin, Wortis, M., Wright, J.A.: “Iterative properties of a one dimensional quartic map: Critical lines and tricritical behavior”, Physical Review 24, 2669–2684 (1981).

    Google Scholar 

  27. Chillingworth, D.R.J., Holmes, R.J.: “Dynamical system and the models for reversals of the Earth's magnetic field”, Math. Geology 12, 41–59 (1980).

    Google Scholar 

  28. Chirikov, B.V., Izraelev, F.M.: “Some numerical experiments with a nonlinear mapping: Stochastic component”, in “Transformations ponctuelles et leurs applications”, 409–428, Colloques Internationaux du C.N.R.S. 229, and "Degeneration of turbulence in simple systems”, Physica D 2, 30–37 (1981).

    Google Scholar 

  29. Clerc, R., Hartmann, Ch.: “Bifurcation mechanism of a second order recurrence leading to the appearance of a particular strange attractor”, preprint, Univ. of Toulouse.

    Google Scholar 

  30. Collet, P., Eckmann, J.P.: “On the abundance of aperiodic behavior for maps of the interval”, Comm. Math. Phys. 73, 115–160 (1980).

    Google Scholar 

  31. Collet, P., Eckmann, J.P., Lanford, O.: “Universal properties of maps of an interval”, Comm. Math. Phys. 76, 211–254 (1980).

    Google Scholar 

  32. Collet, P., Eckmann, J.P.: “Iterated maps on the interval as dynamical systems”, Birkhäuser, Boston, 1980.

    Google Scholar 

  33. Collet, P., Eckmann, J.P., Koch, H.: “Period doubling bifurcations for families of maps on Rn”, J. Stat. Phys. 25 (1981).

    Google Scholar 

  34. Collet, P., Crutchfield, J.P., Eckmann, J.P.: “Computing the topological entropy of maps”, to appear in Physica D.

    Google Scholar 

  35. Cook, A., Roberts, D.: “The Rikitake two-disc dynamo system”, Proc. Cambridge Philos. Soc. 68, 547–569 (1970).

    Google Scholar 

  36. Coullet, P., Tresser, C.: “Critical transition to “Stochasticity” for some dynamical systems”, J. de Physique Lett. 41, L 255 (1980).

    Google Scholar 

  37. Crutchfield, J.P.: “Prediction and stability in classical mechanics, University of Santa Cruz Thesis, 1979.

    Google Scholar 

  38. Crutchfield, J.P., Farmer, J.D., Packard, N.H., Shaw, R.S., Jones, G., Donnelly, R.: “Power Spectral Analysis of a Dynamical System”, Phys. Lett. 76A, 1 (1980).

    Google Scholar 

  39. Crutchfield, J.P., Huberman, B.A.: “Fluctuations and the onset of chaos”, Phys. Lett. 77A, 407 (1980).

    Google Scholar 

  40. Crutchfield, J.P., Farmer, J.D., Huberman, B.A.: “Fluctuations and simple chaotic dynamics”, to appear in Physics Reports.

    Google Scholar 

  41. Crutchfield, J.P., Nauenberg, M., Rudnick, J.: “Scaling for external noise at the onset of chaos”, Phys. Rev. Lett. 46, 933 (1981).

    Google Scholar 

  42. Crutchfield, J.P., Packard, N.H.: “Symbolic dynamics of one-dimensional maps: entropies, finite precision and noise”, to appear in Intl. J. Theor. Phys.

    Google Scholar 

  43. Curry, J.: “A generalized Lorenz system”, Comm. Math. Phys. 60, 193–204 (1978).

    Google Scholar 

  44. Curry, J.: “On the Hénon transformation”, Comm. Math. Phys. 68, 129–140 (1979).

    Google Scholar 

  45. Curry, J.: “On computing the, entropy of the Hénon attractor”, J. Stat. Phys. 26, 683–695 (1981).

    Google Scholar 

  46. Daido, H.: “Analytical conditions for the appearance of homoclinic and heteroclinic points of a 2-dimensional mapping”, Prog. Theor. Phys. 63, 1190–1201 (1980).

    Google Scholar 

  47. Daido, N.: “Universal relation of a band-splitting sequence to a preceding period doubling one”, Phys. Lett. 86A, 259–262 (1981).

    Google Scholar 

  48. Dell'Antonio, G., Doplicher, S., Jona-Lasinio, G. (ed.): “Mathematical problems in theoretical physics”, Lect. Notes in Physics 80, Springer (1978).

    Google Scholar 

  49. Derrida, B., Pomeau, Y.: “Feigenbaum's ratios of 2-dimensional area preserving maps”, Phys. Lett. 80A, 217–219 (1980).

    Google Scholar 

  50. Donnelly, R.J., Park, K., Shaw, R.S., Walden, R.W.: “Early nonperiodic transition in Couette flow”, Phys. Rev. Lett. 44, 987 (1980).

    Google Scholar 

  51. Douady, A., Oesterlé, J.: “Dimension de Hausdorff des attracteurs”, C.R. Acad. Sci. Paris, 290, 1135–1138 (1980).

    Google Scholar 

  52. Easton, R.: “A topological conjugacy invariant involving homoclinic points for diffeomorphisms of two-manifolds”, preprint, Univ. of Boulder, Colorado.

    Google Scholar 

  53. Eckmann, J.P.: “Roads to turbulence in dissipative dynamical systems”, Rev. Mod. Phys. 53, 643–654 (1981).

    Article  Google Scholar 

  54. Eckmann, J.P.: “Renormalization group analysis of some highly bifurcated families”, preprint, Univ. of Genève.

    Google Scholar 

  55. Eckmann, J.P.: “Intermittency in the presence of noise”, J. Phys. A 14, 3153–3168 (1981).

    Google Scholar 

  56. Eckmann, J.P.: “A note on the power spectrum of the iterates of Feigenbaum's function”, Commun. Math. Phys. 81, 261–265 (1981).

    Google Scholar 

  57. Farmer, J.D.: “Spectral broadening of period-doubling bifurcation sequences”, Phys. Rev. Lett. 47, 179 (1981).

    Google Scholar 

  58. Farmer, J.D.: “Order within chaos”, Univ. of Santa Cruz, Ph.D. Thesis (1981).

    Google Scholar 

  59. Farmer, J.D.: “Chaotic attractors of an infinite-dimensional dynamical system”, Physica D 4, 366–393 (1982).

    Google Scholar 

  60. Farmer, J.D. “Information dimension and the probabilistic structure of chaos”, to appear in Z. Naturforschung.

    Google Scholar 

  61. Farmer, J.D., Ott, E., Yorke, J.A.: “The dimension of chaotic attractors”, to appear in Physica D.

    Google Scholar 

  62. Farrell, F.T., Jones, L.E.: “New attractors in hyperbolic dynamics, J. Diff. Geometry 15, 107–133 (1980).

    Google Scholar 

  63. Fatou, P.: “Sur les equations fonctionelles”, Bull. Soc. Math. de France, 47, 161–270 (1919), 48, 33–95 & 208–314 (1920).

    Google Scholar 

  64. Feigenbaum, M.: “Quantitative universality for a class of nonlinear transformations”, J. Stat. Phys. 19, 25–52 (1978).

    Article  Google Scholar 

  65. Feigenbaum, M.: “The onset spectrum of turbulence”, Phys. Lett. 74A, 375 (1979)

    Google Scholar 

  66. Feigenbaum, M.: “The transition to aperiodic behavior in turbulent systems”, Commun. Math. Phys. 77, 65–86 (1980).

    Google Scholar 

  67. Feit, S.: “Characteristic exponents and strange attractors”, Commun. Math. Phys. 61, 249–260 (1978).

    Google Scholar 

  68. Franceschini, V.: “Feigenbaum sequence of bifurcations in the Lorenz model”, J. Stat. Phys. 22, 397–407 (1980).

    Google Scholar 

  69. Franceschini, V., Russo, L.: “Stable and unstable manifolds of the Hénon mapping”, preprint, Univ. of Modena.

    Google Scholar 

  70. Franceschini, V.: “Two models of truncated Navier-Stokes equations on a two-dimensional torus”, Phys. of Fluids (to appear).

    Google Scholar 

  71. Franceschini, V.: “Truncated Navier-Stokes equations on a two-dimensional torus”, preprint, Los Alamos.

    Google Scholar 

  72. Franceschini, V.: “Bifurcation of tori and phase-locking in a dissipative system of differential equations”, preprint, Los Alamos.

    Google Scholar 

  73. Frederickson, P., Kaplan, J.L., Yorke, E.D., Yorke, J.: “The Lyapunov dimension of strange attractors”, J. Diff. Equations (to appear).

    Google Scholar 

  74. Fujisaka, H., Yamada, T.: “Limit cycles and chaos in realistic models of the Belousov-Zhabotinskii reaction system, Z. Physik B, 37, 265–275 (1980).

    Google Scholar 

  75. Garrido, L. (ed.): “Systems far from equilibrium”, Lect. Notes in Physics 132 (1980).

    Google Scholar 

  76. Gollub, J.P. Swinney, H.L.: “Onset of turbulence in a rotating fluid”, Phys. Rev. Lett. 35, 921 (1975).

    Google Scholar 

  77. Grebogi, C., Ott, E., Yorke, J.A.: “Chaotic attractors in crisis”, Phys. Rev. Lett. 48, 1507–1510 (1982).

    Google Scholar 

  78. Grebogi, C., Ott, E., Yorke, J.A.: “Crisis, sudden changes in chaotic attractors and transient chaos”, preprint, Univ. of Maryland.

    Google Scholar 

  79. Grmela, M., Marsden, J.E. (ed.): “Global analysis”, Lect. Notes in Math. 755, Springer (1979).

    Google Scholar 

  80. Grossmann, S., Thomae, S.: “Invariant distributions and stationary correlation functions”, Z. Naturforschung 32A, 1353–1365 (1977).

    Google Scholar 

  81. Guckenheimer, J.: “Bifurcations of maps of the interval”, Inventiones math. 39, 165–178 (1977).

    Google Scholar 

  82. Guckenheimer, J.: “Sensitive dependence on initial conditions for one-dimensional maps”, Commun. Math. Phys. 70, 133–160 (1979).

    Google Scholar 

  83. Guckenheimer, J., Moser, J., Newhouse, S.: “Dynamical Systems”, Birkhäuser, Boston, 1980.

    Google Scholar 

  84. Guckenheimer, J., Williams, R.: “Structural stability of Lorenz attractors”, Pub. Math. I.H.E.S. 50, 60–72 (1980).

    Google Scholar 

  85. Gumowski, I., Mira, C.: “Dynamique Chaotique”, Editions Cepadues, Toulouse, 1980.

    Google Scholar 

  86. Gumowski, I., Mira, C.: “Recurrence and discrete dynamical systems”, Lect. Notes in Math. 809, Springer, 1980.

    Google Scholar 

  87. Gurel, O., Rössler, O.E. (ed.): “Bifurcation theory and applications in scientific disciplines”, Annals New York Acad. Sci. 316, (1979).

    Google Scholar 

  88. Haken, H.: “Synergetics, An Introduction”, 2nd ed. Springer series in Synergetics, 1, (1978).

    Google Scholar 

  89. Haken, H. (ed.): “Synergetics, A Workshop”, Springer series in Synergetics 2, (1977).

    Google Scholar 

  90. Haken, H., Wunderlin, A.: “New interpretation and size of strange attractors of the Lorenz model of turbulence”, Phys. Lett. 62A, 133–134 (1977).

    Google Scholar 

  91. Haken, H. (ed.): “Chaos and order in Nature”, Springer (1981).

    Google Scholar 

  92. Haken, H. (ed.): “Evolution of ordered and chaotic patterns in systems treated by the natural sciences and mathematics”, Springer (1982) (to appear).

    Google Scholar 

  93. Helleman, R.H.G.: “Self-generated chaotic behavior in nonlinear mechanics”, in “Fundamental Problems in Statistical Mechanics V”, Ed.: E.G.D. Cohen, 165–233, North Holland (1980).

    Google Scholar 

  94. Helleman, R.H.G. (ed.): “Nonlinear dynamics”, Annals New York Acad. Sci. 357 (1980).

    Google Scholar 

  95. Helleman, R.H.G., Iooss, G. (ed.): “Chaotic behavior in deterministic systems”, North-Holland, (1982) (to appear).

    Google Scholar 

  96. Hénon, M.: “A two-dimensional mapping with a strange attractor”, Commun. Math. Phys. 50, 69–77 (1976).

    Google Scholar 

  97. Herring, C., Huberman, B.A.: “Dislocation motion and solid-state turbulence”, Appl. Phys. Lett. 36, 975–977 (1980).

    Google Scholar 

  98. Hirsch, M.W., Pugh, C.C., Shub, M.: “Invariant manifolds”, Lect. Notes in Math. 583, Springer (1977).

    Google Scholar 

  99. Hitzl, D.L.: “Numerical determination of the capture/escape boundary for the Hénon attractor”, preprint Lockheed, Palo Alto (1981).

    Google Scholar 

  100. Holmes, P.J.: “Strange phenomena in dynamical systems and their physical implications”, Appl. Math. Modelling, 1, 362–366 (1977).

    Google Scholar 

  101. Holmes, P.J.: “A nonlinear oscillator with a stran e attractor”, Phil. Trans. Roy. Soc. London, Ser. A 292, 419–448 (1979).

    Google Scholar 

  102. Holmes, P.J.: “Averaging and chaotic motions in forced oscillations”, SIAM J. Appl. Math. 38, 65–80 (1980).

    Google Scholar 

  103. Holmes, P.J., Moon, F.C.: “A magneto-elastic strange attractor”, J. Sound vibration 65, 275–296 (1979).

    Google Scholar 

  104. Holmes, P.J. (ed.): “New approaches to nonlinear problems in dynamics”, SIAM, Philadelphia (1980).

    Google Scholar 

  105. Hoppenstedt, F.C. (ed.): “Nonlinear oscillations in Biology”, A.M.S. Lectures in Appl. Math. 17, Providence (R.I.) (1979).

    Google Scholar 

  106. Huberman, B.A., Crutchfield, J.P.: “Chaotic states of anharmonic systems in periodic fields”, Phys. Rev. Lett. 43, 1743–1747 (1979).

    Google Scholar 

  107. Huberman, B.A., Crutchfield, J.P., Packard, N.H.: “Noise phenomena in Josephson junctions”, Appl. Phys. Lett. 37, 750–752 (1980).

    Google Scholar 

  108. Huberman, B.A., Rudnick, J.: “Scaling behavior of chaotic flows” Phys. Rev. Lett. 45, 154–156 (1980).

    Google Scholar 

  109. Hurley, M.: “Attractors: Persistence and density of their basins”, Trans. Amer. Math. Soc. 269, 247–271 (1982).

    Google Scholar 

  110. Izraelev, F.M., Rabinovich, M.I., Ugodnikov, A.D.: “Approximate description of three dimensional dissipative systems with stochastic behavior”, Phys. Lett. 86A, 321–325 (1981).

    Google Scholar 

  111. Jakobson, M.V.: “Absolutely continuous invariant measures for one-parameter families of one-dimensional maps”. Commun. Math. Phys. 81, 39–88 (1981).

    Google Scholar 

  112. Jeffreis, C., Péerez, J.: “Direct observation of crisis of the chaotic attractor in a nonlinear oscillation”, preprint, Univ. of Berkeley (1982).

    Google Scholar 

  113. Jonker, L., Rand, D.: “Bifurcations in one-dimension, I, II,” Inventiones math. 62, 347–365 and 63, 1–15 (1981).

    Google Scholar 

  114. Jorna, S.(ed.): “Topics in Nonlinear Dynamics”, Amer. Inst. Phys. Conf. Proc. 46, (1978).

    Google Scholar 

  115. Julia, G.: “Mémoire sur l'itération des fonctions rationelles”, J. de Math. sér. 7, 4, 47–245 (1918).

    Google Scholar 

  116. Kai, T., Tomita, K.: “Stroboscopic phase-portrait of a forced nonlinear oscillator”, Prog. Theor. Phys. (to appear).

    Google Scholar 

  117. Kaplan, J.L., Yorke, J.A.: “Preturbulence: A regime observed in a fluid flow model of Lorenz”, Commun. Math. Phys. 67, 93–108, (1979).

    Google Scholar 

  118. Keener, J.P.: “Chaotic cardiac dynamics”, in Lect. in Appl. Math. 19, 299–325, “Mathematical Aspects of Physiology”, ed. F.C. Hoppensteadt, AMS (1981).

    Google Scholar 

  119. Kidachi, H.: “On a chaos as a mode interaction phase”, Prog. Theor. Phys. 65, 1584–1594 (1981).

    Google Scholar 

  120. Knobloch, E.: “Chaos in the segmented disc dynamo”, Phys. Lett. 82A, 439–440 (1981).

    Google Scholar 

  121. Lasota, A., Yorke, J.: “On the existence of invariant measures for piecewise monotonic transformations”, Trans. Amer. Math. Soc. 184, 481–488 (1973).

    Google Scholar 

  122. Laval, G., Gresillon, D. (ed.): “Intrinsic stochasticity in plasmas”, International Workshop, Cargèse, Les Editions de Physique, Orsay (1979).

    Google Scholar 

  123. Leipnick, R.B., Newton, T.A.: “Double strange attractors in rigid body motion with linear feedback control”, Phys. Lett. 86A, 63–67 (1981).

    Google Scholar 

  124. Leven, R.W., Koch, B.D.: “Chaotic behavior of a parametrically excited damped pendulum”, Phys. Lett. 86A, 71–74 (1981).

    Google Scholar 

  125. Levi, M.: “Qualitative analysis of the periodically forced relaxation oscillators”, Mem. A.M.S. 244, (1981).

    Google Scholar 

  126. Li, T.Y., Yorke, J.A.: “Period three implies chaos”, Americ. Math. Monthly, 82, 985–992 (1975).

    Google Scholar 

  127. Li, T.Y., Misiurewicz, M., Pianigiani, G., Yorke, J.: “Odd chaos”, Phys. Lett. 87A, 271–273 (1982).

    Google Scholar 

  128. Libchaber, A., Maurer, J.: “Une expérience de Bénard-Rayleigh de géometrie réduite; multiplication, accrochage et démultiplication de fréquences”, J. de Physique 41, (Coll. C3), 51–56 (1980).

    Google Scholar 

  129. Lopes, A.D.: “An example of interpolation of an attractor”, preprint, Inst. Mat. Porto Alegre.

    Google Scholar 

  130. Lorenz, E.N.: “Deterministic nonperiodic flow”, J. Atmosph. Sci. 20, 130–141 (1963).

    Google Scholar 

  131. Lozi, R.: “Un attracteur étrange (?) du type attracteur de Hénon”, J. de Physique 39 (Coll. C5), 9–10 (1978).

    Google Scholar 

  132. Lozi, R.: “Sur un modéle mathématique de suite de bifurcations de motifs dans la réaction de Belousov-Zhobotinsky”, C.R. Acad. Sci. Paris 294, 21–26 (1982).

    Google Scholar 

  133. L'vov, V.S., Predtechansky, A.A.: “On Landau and stochastic pictures in the problem of transition to turbulence, Physica D, 2 38–51 (1981).

    Google Scholar 

  134. Mackey, M.C., Glass, L.: “Oscillators and chaos in physiological control systems”, Science 197, 287–289 (1977).

    PubMed  Google Scholar 

  135. Manneville, P.: “Intermittency in dissipative dynamical systems”, Phys. Lett. 79A, 33–35 (1980).

    Google Scholar 

  136. Manneville, P., Pomeau, Y.: “Different ways to turbulence in dissipative dynamical systems”, Physica D, 1, 219–226 (1980).

    Google Scholar 

  137. Manneville, P.: “Intermittency, self-similarity and 1/f spectrum in dissipative dynamical systems”, J. Physique 41, 1235–1243 (1980).

    Google Scholar 

  138. Markley, N.G., Martin, J.C., Perrizo, W. (ed.): “The structure of attractors in dynamical systems”, Lect. Notes in Math. 668, Springer, (1978).

    Google Scholar 

  139. Marotto, F.R.: “Chaotic behavior in the Hénon mapping”, Commun. Math. Phys. 68, 187–194 (1979).

    Google Scholar 

  140. Marsden, J.E., McCracken, M.: “The Hopf bifurcation and its applications”, Appl. Math. Sci. 19, Springer (1976).

    Google Scholar 

  141. Marzec, C.J., Spiegel, E.A.: “Ordinary differential equations with strange attractors”, SIAM J. Appl. Math. 38, 387–421 (1980).

    Google Scholar 

  142. May, R., Oster, G.: “Bifurcations and dynamic complexity in simple ecological models”, The Amer. Natur. 110, 573–599 (1976).

    Google Scholar 

  143. Mayer-Kress, G., Haken, H.: “Intermittent behavior of the logistic system”, Phys. Lett. 82A, 151–155 (1981).

    Google Scholar 

  144. McLaughlin, J., Martin, P.: “Transition to turbulence in a statically stressed fluid system”, Phys. Rev. A 12, 186–203 (1975).

    Google Scholar 

  145. Metropolis, M., Stein, M.L., Stein, P.R.: “On finite limit sets for transformations of the unit interval”, J. Combinatorial Theory 15, 25–44 (1973).

    Google Scholar 

  146. Milnor, J., Thurston, W.: “On iterated maps of the interval”, preprint, Univ. of Princeton.

    Google Scholar 

  147. Mira, C.: “Accumulation de bifurcations et “structures boîtes emboîtées dans les recurrences et transformations ponctuelles”, VII ICNO, Berlin (1975).

    Google Scholar 

  148. Misiurewicz, M., Swecz, B.: “Existence of a homoclinic point for the Hénon map”, Commun. Math. Phys. 75, 285–291 (1980).

    Google Scholar 

  149. Myrberg, P.J.: “Iteration der reellen Polynome zweiten Grades, I, II, II”, Ann. Acad. Sci. Fenn. 256A, 1–10 (1958); 268A, 1–10 (1959), 336A, 1–10 (1963).

    Google Scholar 

  150. Myrberg, P.J.: “Iteration der Polynome mit reellen Koeffizienten” Ann. Acad. Sci. Fenn. 374A, 1–18 (1965).

    Google Scholar 

  151. Newhouse, S.: “Diffeomorphisms with infinitely many sinks”, Topology 12, 9–18 (1974).

    Google Scholar 

  152. Nitecki, Z., Robinson, C. (ed.): “Global theory of dynamical systems”, Lect. Notes in Math. 819, Springer (1980).

    Google Scholar 

  153. Ogura, H., Ueda, Y., Yoshida, Y.: “Periodic stationarity of a chaotic motion in the system governed by Duffing's equation”, Prog. Theor. Phys. 66, 2280–2283 (1981).

    Google Scholar 

  154. Oono, Y., Osikawa, M.: “Chaos in nonlinear differential equations, I”, Prog. Theor. Phys. 64, 54–67 (1980).

    Google Scholar 

  155. Oseledec, V.I.: “A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems”, Trans. Moscow Math. Soc. 19, 197–231 (1968).

    Google Scholar 

  156. Packard, N.H., Crutchfield, J.P., Farmer, J.D., Shaw, R.S.: “Geometry from a time series”, Phys. Rev. Lett. 45, 712 (1980).

    Google Scholar 

  157. Peitgen, H.O., Walther, H.O. (ed.): “Functional differential equations and approximation of fixed points”, Lect. Notes in Math. 730, Springer (1979).

    Google Scholar 

  158. Pesin, Ya. B.: “Characteristic Lyapunov exponents and smooth ergodic theory”, Russ. Math. Surveys, 32, 55–115 (1977).

    Google Scholar 

  159. Peters, H.: “Chaotic behavior of nonlinear differential-delay equations”, preprint, Univ. of Bremen.

    Google Scholar 

  160. Pikowsky, A.S., Rabinovich, M.I.: “Stochastic oscillations in dissipative systems”. Physica D 2, 8–24 (1981).

    Google Scholar 

  161. Pixton, D.: “Planar homoclinic points”, J. of Diff. Eq. 44, 365–382 (1982).

    Google Scholar 

  162. Plykin, R.: “Sources and sinks for A-diffeomorphisms”, Math. Sb. 23, 233–253 (1974).

    Google Scholar 

  163. Pomeau, Y., Manneville, P.: “Intermittent transition to turbulence in dissipative dynamical systems”, Commun. Math. Phys. 74, 189–197 (1980).

    Article  Google Scholar 

  164. Pounder, J.R., Rogers, T.D.: “The geometry of chaos:dynamics of a nonlinear second order difference equations”, Bull. Math. Biol. 42, 551–597 (1980).

    Google Scholar 

  165. Rand, D.A., Young, L.S. (ed.): “Dynamical systems and turbulence, Warwick 1980”, Lect. Notes in Math. 898, Springer (1981).

    Google Scholar 

  166. Roux, J.C., Rossi, A., Bachelart, S., Vidal, C.: “Representation of a strange attractor from an experimental study of chemical turbulence”, Phys. Lett. 77A, 391–393 (1980).

    Google Scholar 

  167. Ruelle, D., Takens, F.: “On the nature of turbulence”, Commun. Math. Phys. 20, 167–192 (1971).

    Article  Google Scholar 

  168. Ruelle, D.: “A measure associated to axiom A attractors”, Amer. J. of Math. 98, 619–654 (1976).

    Google Scholar 

  169. Ruelle, D.: “Applications conservant une mésure absolument continue par rapport â dx sur [o,1]”, Comm. Math. Phys. 55, 47–51 (1977).

    Google Scholar 

  170. Ruelle, D.: “Strange attractors”, The Mathematical Intelligenter 2, 126–137 (1980).

    Google Scholar 

  171. Ruelle, D.: “Small random perturbations of dynamical systems and the definition of attractors”, Commun. Math. Phys. 82, 137–151 (1981).

    Google Scholar 

  172. Ruelle, D.: “Do there exist turbulent crystals?”, preprint, I.H.E.S.

    Google Scholar 

  173. Šarkovskii, A.N.: “Coexistence of cycles of a continuous map of a line into itself”, Ukr. Mat. Z. 16, 61–71 (1964).

    Google Scholar 

  174. Scholz, H.J., Yamada, T., Brand, H., Graham, R.: “Intermittency and chaos in a laser system with modulated inversion”, Phys. Lett. 82A, 321–323 (1981).

    Google Scholar 

  175. Shaw, R.S.: “Strange attractors, chaotic behavior, and information flow”, Z. Naturforschung 36A, 80 (1981).

    Google Scholar 

  176. Shaw, R.S.: “On the predictabilit of mechanical systems”, Univ. of Santa Cruz, Ph.D. Thesis (1980).

    Google Scholar 

  177. Shaw, R.S., Anderek, C.D., Reith, L.A., Swinney, M.L.: “Nonlinear superposition of traveling waves in circular Couette flow”, Phys. Rev. Lett. (to appear).

    Google Scholar 

  178. Shimada, I., Nagashima, T.: “The iterative transition phenomenon between periodic and turbulent states in a dissipative dynamical system”, Prog. Theor. Phys. 59, 1033–1035 (1978).

    Google Scholar 

  179. Shimada, I.: “Gibbsian distribution on the Lorenz attractor”, Prog. Theor. Phys. 62, 61–69 (1979).

    Google Scholar 

  180. Shimizu, T. ”Asymptotic form of a strange attractor”, Phys. Lett. 84A, 85 (1981).

    Google Scholar 

  181. Simó, C.: “On the Hénon-Pomeau attractor”, J. Stat. Phys. 21, 465–493 21 (1979)

    Google Scholar 

  182. Simonov, A.A.: “An investigation of bifurcations in some dynamical systems by the methods of symbolic dynamics”, Soviet Math. Dok. 19, 759–763 (1978).

    Google Scholar 

  183. Sinai, J.G. Vul, E.B.: “H perbolicity conditions for the Lorenz model”, Physica D, 2, 3–7 (1981).

    Google Scholar 

  184. Smale, S.: “Diffeomorphisms with many periodical points”, in Differential and Combinatorial Topology, 63–80, Princeton Univ. Press (1965).

    Google Scholar 

  185. Smale, S.: “Differentiable dynamical systems”, Bull. Amer. Math. Soc. 73, 747–817 (1967).

    Google Scholar 

  186. Smale, S. (ed.): “Global analysis”, Proc. Sympos. Pure Math. 14, Amer. Math. Soc. (1970).

    Google Scholar 

  187. Steeb, W.H., Kunick, A.: “Lagrange functions of a class of dynamical systems with limit cycle and chaotic behavior”, Phys. Rev. A 25, 2889–2892 (1982).

    Google Scholar 

  188. Steeb, W.H.: “Continuous symmetries of the Lorenz and the Rikitake two-disc dynamo system”, J. Phys. A 15, L 389–390 (1982).

    Google Scholar 

  189. Steeb, W.H., Erig, W., Kunick, A.: “Chaotic behavior and limit cycle behavior of anharmonic systems with periodic external perturbations”, preprint, Univ. of Paderborn.

    Google Scholar 

  190. Steeb, W.H., Kunick, A.: “On the Painlevé property of anharmonic systems with an external period field”, preprint, Univ. of Paderborn.

    Google Scholar 

  191. Stefan, P.: “A theorem of Šarkovskii on the existence of periodic orbits of continuous endomorphisms of the real line”, Commun. Math. Phys. 54, 237–248 (1977).

    Google Scholar 

  192. Swinney, H.L., Gollub, J.P.: “The transition to turbulence”, Physics Today 31, 41–49 (1978).

    Google Scholar 

  193. Takeyama, K.: “Dynamics of the Lorenz model of convective instabilities II”, Prog. Theor. Phys. 63, 91–105 (1980).

    Google Scholar 

  194. Tél, T.: “On the construction of stable and unstable manifolds of two-dimensional invertible maps”, preprint, Eötvös Univ.

    Google Scholar 

  195. Temam, R. (ed.): “Turbulence and Navier-Stokes Equation”, Lect. Notes in Math. 565, Springer (1976).

    Google Scholar 

  196. Testa, J., Held, G.A.: “Period doubling, bifurcations, chaos and periodic windows of the cubic map”, preprint, Univ. of Berkeley.

    Google Scholar 

  197. Thomae, S., Grossmann, S.: “Correlations and spectra of periodic chaos generated by the logistic parabola”, J. Stat. Phys. 26, 485–504 (1981).

    Google Scholar 

  198. Thurston, W.: “On the geometry and dynamics of diffeomorphisms of surfaces”, preprint, Univ. of Princeton.

    Google Scholar 

  199. Tomita, K., Kai, T.: “Stroboscopic phase-portrait and strange attractors”, Phys. Lett. 66A, 91–93 (1978).

    Google Scholar 

  200. Tomita, K., Daido, H.: “Possibility of chaotic behavior and multi-basins in forced glycolytic oscillations”, Phys. Lett. 79A, 133–137 (1980).

    Google Scholar 

  201. Tomita, K., Kai, T.: “Chaotic response of a nonlinear oscillator” J. Stat. Phys. (to appear).

    Google Scholar 

  202. Tomita, K., Tsuda, I.: “Towards an interpretation of Hudson's experiment on the Belousov-Zhabotinskii Reaction: chaos due to delocalization”, Prog. Theor. Phys. 64, 1138–1160 (1980).

    Google Scholar 

  203. Tomita, K., Tsuda, I.: “Chaos in the Belousov-Zhabotinskii reaction in a flow system”, Phys. Lett. 71A, 489–492 (1979).

    Google Scholar 

  204. Tomita, K.: “Chaotic behavior of deterministic orbits. The problem of turbulent phase”, preprint, Univ. of Kyoto.

    Google Scholar 

  205. Tresser, C., Coullet, P., Arneodo, A.: “On the existence of hysteresis in a transition to chaos after a single bifurcation”, J. de Physique Lettres 41, L 243–246 (1980).

    Google Scholar 

  206. Tresser, C., Coullet, P., Arneodo, A.: “Topological horseshoe and numerically observed chaotic behavior in the Hénon mapping”, Lett. to the Editor, J. Phys. A13, L 123–127 (1980).

    Google Scholar 

  207. Tsuda, I. “Self-similarity in the Belusov-Zhabotinsky reaction” Phys. Lett. 85A, 4 (1981).

    Google Scholar 

  208. Ueda, Y., Hayashi, C., Akamatsu, N.: “Computer simulation of nonlinear ordinary differential equations and nonperiodic oscillations”, Electronics and Communications in Japan 56A, 27–34 (1973).

    Google Scholar 

  209. Ueda, Y.: “Randomly transitional phenomena in the system governed by Duffing]"s equation”, J. Stat. Phys. 20, 181–196 (1979).

    Google Scholar 

  210. Ueda, Y., Akamatsu, N.: “Chaoticaly transitional phenomena in the forced negative-resistance oscillator”, IEEE, Trans. on Circuits and Systems 28, 217–224 (1981).

    Google Scholar 

  211. Ulam, S., Von Neumann, J.: “On combinations of stochastic and deterministic processes”, Bull. Amer. Math. Soc. 53, 1120 (1947).

    Google Scholar 

  212. Velsen, R.V., Oberman, C.R.: “Statistical properties of chaotic dynamical systems which exhibit strange attractors”, Physica D, 4, 183–196 (1982).

    Google Scholar 

  213. Vidal, Ch. et al.: “Étude de la transition vers la turbulence chimique dans la réaction de Belousov-Zhabotinskii”, C.R. Acad. Sci. Paris 289 C, 73–77 (1979).

    Google Scholar 

  214. Walther, H.O.: “Homoclinic solutions and chaos in x(t)=f(x(t-1))”, Nonlinear Analysis: Theory, Methods and Appl. 5, 775–788 (1981).

    Google Scholar 

  215. Wegmann, K., Rössler, D.: “Different kinds of chaotic oscillations in the Belousov-Zhabotinskii reaction”, Z. Naturforschung 33A, 1179–1183 (1978).

    Google Scholar 

  216. Wersinger, J.M., Finn, J.H. Ott, E.: “Bifurcation and “strange” behavior in instability saturation by nonlinear three-wave mode coupling”, Phys. Fluids 23, 1142–1154 (1980).

    Article  Google Scholar 

  217. Willamowski, K.D., Rössler, O.E.: “Irregular oscillations in a realistic abstract quadratic mass action system”, Z. Naturforschung 35A, 317–318 (1980).

    Google Scholar 

  218. Williams, R.: “One dimensional nonwandering sets”, Topology 6, 473–487 (1967).

    Article  Google Scholar 

  219. Williams, R.: “The structure of Lorenz attractors”, Pub. Math. I.H.E.S. 50, 73–99 (1980).

    Google Scholar 

  220. Young, L.S.: “Capacity of attractors”,Ergod. Th. and Dynam. Syst. 1, 381–388 (1981).

    Google Scholar 

  221. Zaslavsky, G.M.: “The simplest case of a strange attractor”, Phys. Letters 69A, 145–147 (1978).

    Google Scholar 

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  1. Facultad de Física, Universidad de Barcelona, Diagonal 657, Barcelona-28, Spain

    L. Garrido

  2. Facultat de Matemàtiques, Universitat de Barcelona, Gran Via 585, Barcelona-7, Spain

    C. Simó

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Garrido, L., Simó, C. (1983). Prologue Some ideas about strange attractors. In: Garrido, L. (eds) Dynamical System and Chaos. Lecture Notes in Physics, vol 179. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-12276-1_1

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