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Proceedings of the American Mathematical Society

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Chaotic numerics from an integrable Hamiltonian system


Author: Kevin Hockett
Journal: Proc. Amer. Math. Soc. 108 (1990), 271-281
MSC: Primary 58F13; Secondary 58F05, 65D99, 65L99, 70F05, 70H05
DOI: https://doi.org/10.1090/S0002-9939-1990-0993752-7
MathSciNet review: 993752
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the dynamics of the map $ E$ obtained by applying Euler's method with stepsize $ h$ to the central force problem. We prove that, for any $ h > 0$, the nonwandering set of $ E$ contains a subset on which the dynamics of $ E$ are topologically semiconjugate to a subshift of finite type. The subshift has positive topological entropy, hence so does $ E$.


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  • [1] V.I. Arnold [1978], Mathematical methods of classical mechanics, Springer-Verlag, New York. MR 0690288 (57:14033b)
  • [2] R. Bowen [1978], On axiom A diffeomorphisms, CBMS Reg. Conf. Ser. in Math., No. 35, Amer. Math. Soc., Providence, Rhode Island. MR 0482842 (58:2888)
  • [3] P.J. Channell and J.C. Scovel [1988], Symplectic integration of Hamiltonian systems, LANL, preprint. MR 1054575 (91g:58073)
  • [4] J. Curry, L. Garnett and D. Sullivan [1983], On the iteration of a rational function: computer experiments with Newton's method, Comm. Math. Phys., 91, 267-277. MR 723551 (85e:30040)
  • [5] F.R. Gantmacher [1960], The theory of matrices, Vol. II, Chelsea Publishing Co., New York.
  • [6] S. Hammel, J. Yorke and C. Grebogi [1988], Numerical orbits of chaotic processes represent true orbits, Bull. Amer. Math. Soc. (New Series), 19, 465-469. MR 938160 (89m:58180)
  • [7] E. Lorenz [1988], Computational chaos, MIT, preprint.
  • [8] F. Neri [1988], Lie algebras and canonical integration, University of Maryland, preprint.
  • [9] R. Ruth [1983], A canonical integration technique, IEEE Trans. Nucl. Sei., NS30, 2669.
  • [10] D. Saari and J. Urenko [1984], Newton's method, circle maps and chaotic motion, Amer. Math. Monthly, 91, 3-17. MR 729188 (85a:58060)
  • [11] M. Shub [1987], Global stability of dynamical systems, Springer-Verlag, New York. MR 869255 (87m:58086)
  • [12] M. Shub and A. Vasquez [1987], Some linearly induced Morse-Smale systems, the QR algorithm and the Toda lattice, in Contemporary Mathematics, 64, The Legacy of Sonya Kovalevskaya, (Linda Keen, ed.), Amer. Math. Soc., Providence, Rhode Island. MR 881462 (88c:58034)
  • [13] S. Smale [1985], On the efficiency of algorithms of analysis, Bull. Amer. Math. Soc. (New Series), 13, 87-121. MR 799791 (86m:65061)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1990-0993752-7
Article copyright: © Copyright 1990 American Mathematical Society

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