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Global attractors and bifurcations

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Nonlinear Dynamical Systems and Chaos

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 19))

Abstract

We present some recent developments in the study of attractors of smooth dynamical systems, specially attractors whose basin has a global character. A key point in our approach is to explore the relations between this study and that of main bifurcation mechanisms.

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References

  1. V.S. Afraimovich, V.V. Bykov, and L.P. Shil’nikov, On the appearence and structure of the Lorenz attractor, Dokl. Acad, Sci. USSR 234–2 (1977), 336–339.

    Google Scholar 

  2. V.S. Afraimovich, S.-N. Chow, and W. Liu, Lorenz type attractors from codi-mension one bifurcation, J. Dynam. & Diff. Equ. 7–2 (1995), 375–407.

    Article  MathSciNet  Google Scholar 

  3. V.S. Afraimovich and Ya. B. Pesin, The Dimension of Lorenz type attractors, Sov. Math.Phys. Rev., vol. 6, Gordon and Breach Harwood Academic, 1987.

    Google Scholar 

  4. A. Arneodo, P. Coullet, and C. Tresser, Possible new strange attractors with spiral structure, Comm. Math. Phys. 79 (1981), 573–579.

    Article  MathSciNet  MATH  Google Scholar 

  5. V. Baladi and L.-S. Young, On the spectra of randomly perturbed expanding maps, Comm. Math. Phys. 156 (1993), 355–385.

    Article  MathSciNet  MATH  Google Scholar 

  6. V. Baladi and M. Viana Strong stochastic stability and rate of mixing for unimodal maps, preprint IMPA 1995, to appear Annales E.N.S..

    Google Scholar 

  7. M. Benedicks and L. Carleson, On iterations of 1 - ax 2 on (-1. 1), Ann. Math. 122 (1985), 1–25.

    Article  MathSciNet  MATH  Google Scholar 

  8. M. Benedicks and L. Carleson, The dynamics of the Hénon map, Annals of Math. 133 (1991), 73–169.

    Article  MathSciNet  MATH  Google Scholar 

  9. M. Benedicks and L.-S. Young, Absolutely continuous invariant measures and random perturbations for certain one-dimensional maps, Ergod. Th. & Dynam. Sys. 12 (1992). 13–37

    Article  MathSciNet  MATH  Google Scholar 

  10. M. Benedicks and L.-S. Young, SBR-measures for certain Hénon maps, Invent. Math. 112–3 (1993), 541–576.

    Article  MathSciNet  Google Scholar 

  11. R. Bowen, Equilibrium states and the ergodic theory of Anosov diffeomorphisms, Lect. Notes in Math. 470 (1975), Springer Verlag.

    MATH  Google Scholar 

  12. L.A. Bunimovich, Statistical properties of Lorenz attractors, in Nonlinear dynamics and turbulence, 71–92, Longman publ., London, 1983.

    Google Scholar 

  13. L.A. Bunimovich and Ya.G. Sinai, Stochasticity of the attractor in the Lorenz model, in Nonlinear Waves, Proc. Winter School, Moscow. 212–226, Nauka publ., Moscow. 1980.

    Google Scholar 

  14. E. Catsigeras. Cascades of period-doubling of stable codimension one, thesis IMPA 1994 and to appear.

    Google Scholar 

  15. P. Collet, Ergodic properties of some unimodal mappings of the interval, Preprint 11, Institute Mittag Leffler 1984.

    Google Scholar 

  16. L.J. Díaz, J. Rocha, and M. Viana, Strange attractors in saddle-node cycles: prevalence and globality, preprint IMPA 1993 and to appear.

    Google Scholar 

  17. P. Glendinning and C. Sparrow, T-points: A codimension two heteroclinic bifurcation, Jour. Stat. Phys. 43 (1986), 479–488.

    Article  MathSciNet  MATH  Google Scholar 

  18. J. Guckenheimer and R.F. Williams, Structural stability of Lorenz attractors, Publ. Math. IHES 50 (1979), 307–320.

    Google Scholar 

  19. S.P. Hastings and W.C. Troy, A shooting approach to the Lorenz equations, Bull. A.M.S. 27 (1992), 298–303.

    Article  MathSciNet  MATH  Google Scholar 

  20. M. Hénon, A two dimensional mapping with a strange attractor, Comm. Math. Phys. 50 (1976), 69–77.

    Article  MathSciNet  MATH  Google Scholar 

  21. M. Hénon and Y. Pomeau, Two strange attractors with a simple structure. Lect. Notes Math. 565 (1976), 29–68.

    Article  Google Scholar 

  22. M. Jakobson, Absolutely continuous invariant measures for one-parameter families of one-dimensional maps, Comm. Math. Phys. 81 (1981), 39–88.

    Article  MathSciNet  MATH  Google Scholar 

  23. A. Katok and Y. Kifer, Random perturbations of transformations of an interval, J. Analyse Math. 47 (1986), 193–237.

    Article  MathSciNet  MATH  Google Scholar 

  24. G. Keller and T. Nowicki, Spectral theory, zeta functions and the distribution of periodic points for Collet-Eckmann maps, Comm. Math. Phys. 149 (1992), 31–69.

    Article  MathSciNet  MATH  Google Scholar 

  25. Y. Kifer, Random Perturbations of Dynamical Systems, Birkhäuser, 1988.

    Book  MATH  Google Scholar 

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

    Article  Google Scholar 

  27. S. Luzzatto and M. Viana, Positive Lyapunov exponents for Lorenz-like maps with criticalities, preprint 1995.

    Google Scholar 

  28. S. Luzzatto and M. Viana, Lorenz-like attractors without invariant foliations, in preparation.

    Google Scholar 

  29. R. Ma né, Hyperbolicity, sinks and measure in one-dimensional dynamics, Comm. Math. Phys. 100 (1985), 495–524.

    Article  MathSciNet  Google Scholar 

  30. L. Mora and M. Viana, Abundance of strange attractors, Acta Mathematica 171 (1993), 1–71.

    Article  MathSciNet  MATH  Google Scholar 

  31. C.A. Morales, Lorenz attractors through saddle-node bifurcations, thesis IMPA 1995, to appear Annales I.H.P., Analyse non linéaire.

    Google Scholar 

  32. C.A. Morales and E. Pujals, Singular attractors on the boundary of Morse-Smale systems, preprint 1995.

    Google Scholar 

  33. T. Nowicki, A positive Lyapunov exponent for the critical value of an S-unimodal

    Google Scholar 

  34. mapping implies uniform hyperbolicity, Ergod. Th. & Dynam. Sys. 8 (1988), 425–435.

    Google Scholar 

  35. M.J. Pacifico, A. Rovella, and M. Viana, Global attractors in saddle-focus bifurcations: a one-dimensional model, preprint 1995.

    Google Scholar 

  36. M.J. Pacifico, A. Rovella, and M. Viana, Persistence of global spiraling attractors, in preparation.

    Google Scholar 

  37. J. Palis and F. Takens, Hyperbolicity and sensitive-chaotic dynamics at homo-clinic bifurcations, Cambridge University Press, 1993.

    Google Scholar 

  38. Ya.B. Pesin, Ergodic properties and dimensionlike characteristics of strange attractors that are close to hyperbolic, in Procs. ICM 1986.

    Google Scholar 

  39. Ya.B. Pesin, Dynamical systems with generalized hyperbolic attractors; hyperbolic, ergodic and topological properties, Erg. Th. & Dyn. Syst. 12 (1992), 123–151.

    MathSciNet  MATH  Google Scholar 

  40. A. Pumariño, Coexistence with positive probability of strange attractors in homoclinic connections of saddle-focus type, thesis Department of Mathematics, University of Oviedo, Spain, 1994.

    Google Scholar 

  41. C. Robinson, Homoclinic bifurcation to a transitive attractor of Lorenz type, Nonlinearity 2 (1989), 495–518.

    Article  MathSciNet  MATH  Google Scholar 

  42. A. Rovella, The dynamics of perturbations of the contracting Lorenz attractor, Bull. Braz. Math. Soc. 24 (1993), 233–259.

    Article  MathSciNet  MATH  Google Scholar 

  43. M. Rychlik, Lorenz attractors through Shil’nikov-type bifurcation. Part 1, Erg. h. & Dyn. Syst. 10 (1989), 793–821.

    MathSciNet  Google Scholar 

  44. E.A. Sataev, Invariant measures for hyperbolic maps with singularities, Russ. Math. Surveys 471 (1992), 191–251.

    Article  MathSciNet  Google Scholar 

  45. M. Shub, Global stability of dynamical systems, Springer Verlag, 1987.

    Book  MATH  Google Scholar 

  46. L.P. Shil’nikov, A case of the existence of a denumerable set of periodic motions, Sov. Math. Dokl. 6 (1965), 163–166.

    Google Scholar 

  47. D. Singer, Stable orbits and bifurcations of maps of the interval. SIAM J. Appl. Math. 35 (1978), 260–267.

    Article  MathSciNet  MATH  Google Scholar 

  48. S. Smale, Differentiable dynamical systems, Bull. Am. Math. Soc. 73 (1967), 747–817.

    Article  MathSciNet  MATH  Google Scholar 

  49. C. Sparrow, The Lorenz equations: bifurcations, chaos and strange attractors, Applied Mathematical Sciences, vol. 41, Springer-Verlag, 1982.

    Book  MATH  Google Scholar 

  50. R. Ures, On the approximation of Hénon-like attractors by homoclinic tangencies, thesis IMPA 1993, to appear Ergod. Th. & Dynam. Sys..

    Google Scholar 

  51. M. Viana, Strange attractors in higher dimensions, Bull. Braz. Math. Soc. 24 (1993), 13–62.

    Article  MathSciNet  MATH  Google Scholar 

  52. L.-S. Young, Decay of correlations for certain quadratic maps. Comm. Math. Phys. 146 (1992), 123–138.

    Article  MathSciNet  MATH  Google Scholar 

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Viana, M. (1996). Global attractors and bifurcations. In: Broer, H.W., van Gils, S.A., Hoveijn, I., Takens, F. (eds) Nonlinear Dynamical Systems and Chaos. Progress in Nonlinear Differential Equations and Their Applications, vol 19. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7518-9_14

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  • DOI: https://doi.org/10.1007/978-3-0348-7518-9_14

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-7520-2

  • Online ISBN: 978-3-0348-7518-9

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