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Homoclinic Bifurcations and Strange Attractors

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Real and Complex Dynamical Systems

Part of the book series: NATO ASI Series ((ASIC,volume 464))

Abstract

We present an overview of the theory of homoclinic bifurcations, with particular emphasis on recent developments exploring its links to the study of chaotic dynamics and strange attractors.

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References

  • R. Bamón, R. Labarca, R. Marié, M. J. Pacifico, Bifurcation of simple vector fields through singular cycles,to appear in Publ. Math. IHES.

    Google Scholar 

  • M. Benedicks, L. Carleson, On iterations of 1 — aæ2 on (-1, 1), Ann. Math. 122 (1985), 1–25.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  Google Scholar 

  • L. J. Diaz, J. Rocha, M. Viana, Saddle-node cycles and prevalence of strange attractors,preprint IMPA, to appear.

    Google Scholar 

  • L. J. Diaz, J. Rocha, M. Viana, Global strange attractors for dissipative maps of the annulus,in preparation.

    Google Scholar 

  • P. Duarte, Many elliptic islands in the Standard Family of diffeomorphisms,thesis IMPA 1992, to appear.

    Google Scholar 

  • J. Guckenheimer, R. Williams, Structural stability of Lorenz attractors, Publ. Math. IHES 50 (1979), 59–72.

    MathSciNet  MATH  Google Scholar 

  • M. Hirsch, C. Pugh, M. Shub, Invariant manifolds, Lecture Notes in Mathematics 583, Springer Verlag (1977).

    Google Scholar 

  • 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 

  • A. Katok, Liapounov exponents, entropy and periodic orbits for diffeomorphisms, Publ. Math. IHES 51 (1980), 137–174.

    MathSciNet  MATH  Google Scholar 

  • J. M. Marstrand, Some fundamental properties of plane sets of fractal dimensions, Proc. London Math. Soc. 4 (1954), 257–302.

    MathSciNet  MATH  Google Scholar 

  • L. Mora, M. Viana, Abundance of strange attractors, Acta Math., 1993.

    Google Scholar 

  • F. J. Moreira, Chaotic dynamics of quadratic maps, Master’s thesis, Univ. of Porto, 1992.

    Google Scholar 

  • S. Newhouse, Non-density of Axiom A(a) on S 2, Proc. A.M.S. Symp. Pure Math. 14 (1970), 191–202.

    MathSciNet  Google Scholar 

  • S. Newhouse, The abundance of wild hyperbolic sets and nonsmooth stable sets for diffeomorphisms, Publ. Math. I.H.E.S. 50 (1979), 101–151.

    MathSciNet  MATH  Google Scholar 

  • S. Newhouse, J. Palis, F. Takens, Bifurcations and stability of families of diffeomorphisms, Publ. Math. I.H.E.S. 57 (1983), 7–71.

    Google Scholar 

  • J. Palis, F. Takens, Hyperbolicity and the creation of homoclinic orbits, Annals of Math. 125 (1987), 337–374.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Palis, F. Takens, Hyperbolicity and Sensitive-Chaotic Dynamics at Homoclinic Bifurcations, Cambridge University Press, 1993.

    Google Scholar 

  • J. Palis, M. Viana, High dimension diffeomorphisms displaying infinitely many sinks,preprint IMPA, to appear.

    Google Scholar 

  • J. Palis, J.-C. Yoccoz, Large Hausdorff dimension and non-prevalence of hyperbolicity in homoclinic bifurcations,to appear in Acta Math.

    Google Scholar 

  • N. Romero, Persistence of tangencies in arbitrary dimension,thesis IMPA 1992, to appear.

    Google Scholar 

  • C. Robinson, Bifurcation to infinitely many sinks, Comm. Math. Phys. 90 (1983), 433–459.

    Article  MATH  Google Scholar 

  • A. Rovella, Other strange attractors for vector fields of the Lorenz type,thesis IMPA 1991, to appear.

    Google Scholar 

  • M. Shub, Stabilité Globale des Systèmes Dynamiques, Astérisque 56 (1978).

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • C. Sparrow, The Lorenz equations: bifurcations, chaos and strange attractors, Appli. Math. Sci. 41, Springer Verlag, (1982).

    Google Scholar 

  • F. Takens, Homoclinic bifurcations, Proc. Int. Congress of Math., Berkeley (1986), 1229–1236.

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  • L. Tedeschini-Lalli, J.A. Yorke, How often do simple dynamical processes have infinitely many coexisting sinks ? Comm. Math. Phys. 106 (1986), 635–657.

    Article  MathSciNet  MATH  Google Scholar 

  • J. A. Yorke, K. T. Alligood, Cascades of period doubling bifurcations: a prerequisite for horseshoes, Bull. A.M.S. 9 (1983), 319–322.

    Article  MathSciNet  MATH  Google Scholar 

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Viana, M. (1995). Homoclinic Bifurcations and Strange Attractors. In: Branner, B., Hjorth, P. (eds) Real and Complex Dynamical Systems. NATO ASI Series, vol 464. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-8439-5_10

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  • DOI: https://doi.org/10.1007/978-94-015-8439-5_10

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-4565-2

  • Online ISBN: 978-94-015-8439-5

  • eBook Packages: Springer Book Archive

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