The Mathematical Intelligencer

, Volume 22, Issue 3, pp 6–19 | Cite as

What’s new on lorenz strange attractors?

  • Marcelo Viana


Vector Field Equilibrium Point Rayleigh Number Unstable Manifold Mathematical Intelligencer 
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  1. [1]
    V. S. Afraimovich, V. V. Bykov, and L P. Shil’nikov. On the appearance and structure of the Lorenz attractor.Dokl. Acad. Sci. USSR, 234:336–339, 1977.Google Scholar
  2. [2]
    A. Andronov and L. Pontryagin. Systèmes grossiers.Dokl. Akad. Nauk. USSR, 14:247–251, 1937.Google Scholar
  3. [3]
    M. Benedicks and L. Carleson. The dynamics of the Hénon map.Annals of Math., 133:73–169, 1991.CrossRefzbMATHMathSciNetGoogle Scholar
  4. [4]
    C. Bonatti, L. J. Diaz, and E. Pujals. A C1-generic dichotomy for diffeomorphisms: weak forms of hyperbolicity or infinitely many sinks or sources. Preprint, 1999.Google Scholar
  5. [5]
    C. Bonatti, A. Pumarino, and M. Viana. Lorenz attractors with arbitrary expanding dimension. C.R. Acad. Sci. Paris, 325, Série I, Mathématique: 883–888, 1997.CrossRefzbMATHMathSciNetGoogle Scholar
  6. [6]
    R. Bowen.Equilibrium states and the ergodic theory of Anosov diffeomorphisms, volume 470 ofLect. Notes in Math. Springer Verlag, 1975.Google Scholar
  7. [7]
    L. J. Diaz, E. Pujals, and R. Ures. Partial hyperbolicity and robust transitivity.Acta Math., 1999. To appear.Google Scholar
  8. [8]
    J. Guckenheimer and R. F. Williams. Structural stability of Lorenz attractors.Publ. Math. IHES, 50:59–72, 1979.CrossRefzbMATHMathSciNetGoogle Scholar
  9. [9]
    B. Hassard, S. Hastings, W. Troy, and J. Zhang. A computer proof that the Lorenz equations have “chaotic” solutions.Appl. Math. Lett., 7:79–83, 1994.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [10]
    S. Hastings and W. Troy. A shooting approach to the Lorenz equations.Bull. Amer. Math. Soc., (2) 27:298–303, 1992.CrossRefzbMATHMathSciNetGoogle Scholar
  11. [11]
    S. Hayashi. Connecting invariant manifolds and the solution of the C1 stability and Ω-stability conjectures for flows.Annals of Math., 145:81–137, 1997.CrossRefzbMATHGoogle Scholar
  12. [12]
    M. Hénon. A two dimensional mapping with a strange attractor.Comm. Math. Phys. 50:69–77, 1976.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [13]
    M. Hénon and Y. Pomeau. Two strange attractors with a simple structure. InTurbulence and Navier-Stokes equations, volume 565, pages 29–68. Springer Verlag, 1976.Google Scholar
  14. [14]
    M. Jakobson. Absolutely continuous invariant measures for one-parameter families of one-dimensional maps.Comm. Math. Phys., 81:39–88, 1981.CrossRefzbMATHMathSciNetGoogle Scholar
  15. [15]
    Yu. Kifer.Random perturbations of dynamical systems. Birkhäuser, 1988.Google Scholar
  16. [16]
    L. D. Landau and E. M. Lifshitz.Fluid mechanics. Pergamon, 1959.Google Scholar
  17. [17]
    E. N. Lorenz. Deterministic nonperiodic flow.J. Atmosph. Sci., 20:130–141, 1963.CrossRefGoogle Scholar
  18. [18]
    E. N. Lorenz. On the prevalence of aperiodicity in simple systems.Lect. Notes in Math., 755:53–75, 1979.CrossRefMathSciNetGoogle Scholar
  19. [19]
    S. Luzzatto and W. Tucker. Non-uniformly expanding dynamics in maps with singularities and criticalities.Publ. Math. IHES. To appear.Google Scholar
  20. [20]
    S. Luzzatto and M. Viana. Lorenz-like attractors without invariant foliations. In preparation.Google Scholar
  21. [21]
    S. Luzzatto and M. Viana. Positive Lyapunov exponents for Lorenz-like maps with criticalities.Astérisque, 1999.Google Scholar
  22. [22]
    J. C. Maxwell.Matter and motion. Dover Publ., 1952. First edition in 1876.Google Scholar
  23. [23]
    K. Mischaikow and M. Mrozek. Chaos in the Lorenz equations: a computer assisted proof (I).Bull. Amer. Math. Soc, (2) 32:66–72, 1995.CrossRefzbMATHMathSciNetGoogle Scholar
  24. [24]
    K. Mischaikow and M. Mrozek. Chaos in the Lorenz equations: a computer assisted proof (II).Math. Comp., 67:1023–1046, 1998.CrossRefzbMATHMathSciNetGoogle Scholar
  25. [25]
    C. Morales, M. J. Pacifico, and E. Pujals. Partial hyperbolicity and persistence of singular attractors. Preprint 1999.Google Scholar
  26. [26]
    C. Morales, M. J. Pacifico, and E. Pujals. On C1 robust singular transitive sets for three-dimensional flows. C. ftAcad. Sci. Paris, 326, Série 1:81–86, 1998.CrossRefzbMATHMathSciNetGoogle Scholar
  27. [27]
    J. Palis. A global view of Dynamics and a conjecture on the denseness of finitude of attractors.Astérisque, 261:339–351, 1999.Google Scholar
  28. [28]
    J. Palis and F. Takens.Hyperbolicity and sensitive-chaotic dynamics at homoclinic bifurcations. Cambridge University Press, 1993.Google Scholar
  29. [29]
    Ya. Pesin. Families of invariant manifolds corresponding to non-zero characteristic exponents.Math. USSR. Izv., 10:1261–1302, 1976.CrossRefGoogle Scholar
  30. [30]
    H. Poincaré.Science and method. Dover Publ., 1952. First edition in 1909.Google Scholar
  31. [31]
    Lord Rayleigh. On convective currents in a horizontal layer of fluid when the higher temperature is on the under side.Phil. Mag., 32:529–546, 1916.CrossRefGoogle Scholar
  32. [32]
    C. Robinson. Homoclinic bifurcation to a transitive attractor of Lorenz type.Nonlinearity, 2:495–518, 1989.CrossRefzbMATHMathSciNetGoogle Scholar
  33. [33]
    D. Ruelle and F. Takens. On the nature of turbulence.Comm. Math. Phys., 20:167–192, 1971.CrossRefzbMATHMathSciNetGoogle Scholar
  34. [34]
    M. Rychlik. Lorenz attractors through Shil’nikov-type bifurcation. Part 1.Erg. Th. & Dynam. Syst., 10:793–821, 1989.MathSciNetGoogle Scholar
  35. [35]
    B. Saltzmann. Finite amplitude free convection as an initial value problem.J. Atmos. Sci., 19:329–341, 1962.CrossRefGoogle Scholar
  36. [36]
    M. Shub.Global stability of dynamical systems. Springer Verlag, 1987.Google Scholar
  37. [37]
    Ya. Sinai. Gibbs measure in ergodic theory.Russian Math. Surveys, 27:21–69, 1972.CrossRefzbMATHMathSciNetGoogle Scholar
  38. [38]
    S. Smale. Differentiable dynamical systems.Bull. Am. Math. Soc., 73:747–817, 1967.CrossRefzbMATHMathSciNetGoogle Scholar
  39. [39]
    C. Sparrow.The Lorenz equations: bifurcations, chaos and strange attractors, volume 41 ofApplied Mathematical Sciences. Springer Verlag, 1982.Google Scholar
  40. [40]
    S. Sternberg. On the structure of local homeomorphisms of eu-clideann-space-II.Amer. J. Math., 80:623–631, 1958.CrossRefzbMATHMathSciNetGoogle Scholar
  41. [41]
    W. Tucker.The Lorenz attractor exists. PhD thesis, Univ. Uppsala, 1998. Text and program codes available at www. Scholar
  42. [42]
    W. Tucker. The Lorenz attractor exists.C. R. Acad. Sci. Paris, 328, Série I, Mathématique: 1197–1202, 1999.CrossRefzbMATHGoogle Scholar
  43. [43]
    M. Viana. Dynamics: a probabilistic and geometric perspective. InProcs. International Congress of Mathematicians ICM98-Berlin, Documenta Mathematica, vol I, pages 557–578. DMV, 1998.MathSciNetGoogle Scholar
  44. [44]
    R.F. Williams. The structure of the Lorenz attractor.Publ. Math. IHES, 50:73–99, 1979.CrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, Inc. 2000

Authors and Affiliations

  1. 1.IMPARio de JaneiroBrazil

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