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Prog. Theor. Phys. Vol. 125 No. 3 (2011) pp. 435-471

[ Full Text PDF : FREE ACCESS (617K) ]

A New Interpretation of the Symbolic Codes for the Hénon Map. II

Yoshihiro Yamaguchi1 and Kiyotaka Tanikawa2

1Teikyo Heisei University, Tokyo 170-8445, Japan
2National Astronomical Observatory, Mitaka 181-8588, Japan

(Received December 20, 2010)

Abstract:

A certain region in the phase space for the area- and orientation-preserving Hénon map is filled with the resonance chains. Taking this fact into account, the concept of block word has been introduced in the preceding paper. The concept of resonance chains in the symbol plane is also introduced and the marquetry structure of the symbol plane is investigated. If the orbit in the horseshoe travels over the resonance chains, the orbit is represented by a block word sequence. There are four purposes discussed in this paper. The first purpose is to introduce a new concept of parity to characterize the periodic orbits. The second is to study the detailed structure of the symbol plane. The third is to clarify the behaviors of symmetric periodic orbits in the resonance chain and determine their block word sequences. The fourth is to propose an improved construction method to make a braid for a non-Birkhoff periodic orbit in a resonance chain.

Subject Index : 030
URL : http://ptp.ipap.jp/link?PTP/125/435/
DOI : 10.1143/PTP.125.435

[ Full Text PDF : FREE ACCESS (617K) ] Citation:

References:

  1. M. Hénon, Quart. Appl. Math. 27 (1969), 291.
  2. Y. Yamaguchi and K. Tanikawa, Prog. Theor. Phys. 122 (2009), 569[PTP].
  3. H. R. Dullin, J. D. Meiss and D. G. Sterling, SIAM J. Appl. Dyn. Sys. 4 (2005), 515.
  4. T. Hall, Nonlinearity 7 (1994), 861[CrossRef].
  5. R. Brown, Ergod. Th. & Dynam. Sys. 15 (1995), 1045.
  6. C. Robinson, Dynamical Systems, Second Edition (CRC Press LLC, 1999). See Exercise 2.7.
  7. H. To-yama, Shoto-Seisuron (in Japanese), (Nippon Hyoronsha, Tokyo, 1972).
  8. S. Bullet, Commun. Math. Phys. 107 (1986), 241[CrossRef].
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  10. W. P. Thurston, Bull. Amer. Math. Soc. 19 (1988), 417.

Citing Article(s) :

  1. Progress of Theoretical Physics Vol. 126 No. 5 (2011) pp. 811-839 :
    Forcing Relations for Homoclinic Orbits of the Reversible Horseshoe Map
    Yoshihiro Yamaguchi and Kiyotaka Tanikawa
  2. Progress of Theoretical Physics Vol. 128 No. 1 (2012) pp. 15-30 :
    Nonsymmetric Saddle-Node Pairs for the Reversible Smale Horseshoe Map
    Yoshihiro Yamaguchi and Kiyotaka Tanikawa
  3. Progress of Theoretical Physics Vol. 128 No. 5 (2012) pp. 845-871 :
    New Period-Doubling and Equiperiod Bifurcations of the Reversible Area-Preserving Map
    Yoshihiro Yamaguchi and Kiyotaka Tanikawa

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