Chaotic numerics from an integrable Hamiltonian system
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- by Kevin Hockett PDF
- Proc. Amer. Math. Soc. 108 (1990), 271-281 Request permission
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 $E$.References
- V. Arnold, Les méthodes mathématiques de la mécanique classique, Éditions Mir, Moscow, 1976 (French). Traduit du russe par Djilali Embarek. MR 0474391
- Rufus Bowen, On Axiom A diffeomorphisms, Regional Conference Series in Mathematics, No. 35, American Mathematical Society, Providence, R.I., 1978. MR 0482842
- P. J. Channell and C. Scovel, Symplectic integration of Hamiltonian systems, Nonlinearity 3 (1990), no. 2, 231–259. MR 1054575
- James H. Curry, Lucy Garnett, and Dennis Sullivan, On the iteration of a rational function: computer experiments with Newton’s method, Comm. Math. Phys. 91 (1983), no. 2, 267–277. MR 723551 F.R. Gantmacher [1960], The theory of matrices, Vol. II, Chelsea Publishing Co., New York.
- Stephen M. Hammel, James A. Yorke, and Celso Grebogi, Numerical orbits of chaotic processes represent true orbits, Bull. Amer. Math. Soc. (N.S.) 19 (1988), no. 2, 465–469. MR 938160, DOI 10.1090/S0273-0979-1988-15701-1 E. Lorenz [1988], Computational chaos, MIT, preprint. F. Neri [1988], Lie algebras and canonical integration, University of Maryland, preprint. R. Ruth [1983], A canonical integration technique, IEEE Trans. Nucl. Sei., NS30, 2669.
- Donald G. Saari and John B. Urenko, Newton’s method, circle maps, and chaotic motion, Amer. Math. Monthly 91 (1984), no. 1, 3–17. MR 729188, DOI 10.2307/2322163
- Michael Shub, Global stability of dynamical systems, Springer-Verlag, New York, 1987. With the collaboration of Albert Fathi and Rémi Langevin; Translated from the French by Joseph Christy. MR 869255, DOI 10.1007/978-1-4757-1947-5
- Michael Shub and Alphonse Thomas Vasquez, Some linearly induced Morse-Smale systems, the QR algorithm and the Toda lattice, The legacy of Sonya Kovalevskaya (Cambridge, Mass., and Amherst, Mass., 1985) Contemp. Math., vol. 64, Amer. Math. Soc., Providence, RI, 1987, pp. 181–194. MR 881462, DOI 10.1090/conm/064/881462
- Steve Smale, On the efficiency of algorithms of analysis, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 2, 87–121. MR 799791, DOI 10.1090/S0273-0979-1985-15391-1
Additional Information
- © Copyright 1990 American Mathematical Society
- 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