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Rotation numbers of periodic orbits in the Hénon map

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Abstract

For invertible, area-contracting maps of the plane, it is common for a basin of attraction to have a fractal basin boundary. Certain periodic orbits on the basin boundary are distinguished by being accessible (by a path) from the interior of the basin. A numerical study is made of the accessible periodic orbits for the Hénon family of maps. Theoretical results on rotary homoclinic tangencies are given, which describe the appearance of the accessible saddles, and organize them in a natural way according to the continued fractions expansions of their rotation numbers.

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References

  1. Alligood, K., Tedeschini-Lalli, L., Yorke, J.: Metamporphoses: Sudden jumps in basin boundaries. Commun. Math. Phys. (to appear)

  2. Alligood, K., Yorke, J.: Accessible saddles on fractal basin boundaries. Preprint

  3. Aronson, D., Chory, M., Hall, G., McGehee, R.: Bifurcations from an invariant circle for two-parameter families of maps of the plane: a computer assisted study. Commun. Math. Phys.83, 303–354 (1982)

    Google Scholar 

  4. Birkhoff, G.D.: Sur quelques courbes fermees remarquables. Bull. Soc. Math. France60, 1–26 (1932)

    Google Scholar 

  5. Cartwright, M.L., Littlewood, J.E.: Some fixed point theorems. Ann. Math.54, 1–37 (1951)

    Google Scholar 

  6. Gavrilov, N., Silnikov, L.: On the three dimensional dynamical systems close to a system with a structurally unstable homoclinic curve. I. Math. USSR Sbornik17, 467–485 (1972); II. Math. USSR Sbornik19, 139–156 (1973)

    Google Scholar 

  7. Grebogi, C., Ott, E., Yorke, J.: Basin boundary metamorphoses: changes in accessible boundary orbits. Physica24 D, 243–262 (1987)

    Google Scholar 

  8. Guckenheimer, J., Holmes, P.: Nonlinear oscillations, dynamical systems, and bifurcation of vector fields. Berlin, Heidelberg, New York: Springer 1983

    Google Scholar 

  9. Hammel, S., Jones, C.: Jumping stable manifolds for dissipative maps of the plane. Preprint

  10. Hénon, M.: A two-dimensional mapping with a strange attractor. Commun. Math. Phys.50, 69–78 (1976)

    Google Scholar 

  11. Hockett, K., Holmes, P.: Josephson's junction, annulus maps, Birkhoff attractors, horseshoes and rotation sets. Ergod. Th. Dynam. Syst.6, 205–239 (1986)

    Google Scholar 

  12. Newhouse, S.: Diffeomorphisms with infinitely many sinks. Topology13, 9–18 (1974)

    Google Scholar 

  13. Robinson, C.: Bifurcation to infinitely many sinks. Commun. Math. Phys.90, 433–459 (1983)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, George Mason University, 22030, Fairfax, VA, USA

    K. T. Alligood & T. Sauer

Authors
  1. K. T. Alligood
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  2. T. Sauer
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Additional information

Communicated by J. N. Mather

Partially supported by the National Science Foundation

Partially supported by a contract from the Applied and Computational Mathematics Program of DARPA

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Alligood, K.T., Sauer, T. Rotation numbers of periodic orbits in the Hénon map. Commun.Math. Phys. 120, 105–119 (1988). https://doi.org/10.1007/BF01223208

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  • DOI: https://doi.org/10.1007/BF01223208

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