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Preservation and destruction of periodic orbits by symplectic integrators

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Abstract

We investigate what happens to periodic orbits and lower-dimensional tori of Hamiltonian systems under discretisation by a symplectic one-step method where the system may have more than one degree of freedom. We use an embedding of a symplectic map in a quasi-periodic non-autonomous flow and a KAM result of Jorba and Villaneuva (J Nonlinear Sci 7:427–473, 1997) to show that periodic orbits persist in the new flow, but with slightly perturbed period and an additional degree of freedom when the map is non-resonant with the periodic orbit. The same result holds for lower-dimensional tori with more degrees of freedom. Numerical experiments with the two degree of freedom Hénon–Heiles system are used to show that in the case where the method is resonant with the periodic orbit, the orbit is destroyed and replaced by two invariant sets of periodic points—analogous to what is understood for one degree of freedom systems.

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References

  1. Arnold, V.I.: Proof of a theorem of A.N. Kolmogorov on the preservation of conditionally periodic motions under a small perturbation of the Hamiltonian. Ups. Mat. Nauk 18(5), 91–192 (1963)

    Google Scholar 

  2. Banyaga, A.: The structure of classical diffeomorphism groups. In: Mathematics and Its Applications, vol. 400. Kluwer Academic, Boston (1997)

    Google Scholar 

  3. Benettin, G., Giorgilli, A.: On the Hamiltonian interpolation of near-to-the-identity symplectic mappings with application to symplectic integration algorithms. J. Stat. Phys. 74(5/6), 1117–1143 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  4. Chierchia, L.: Absolutely continuous spectra of quasi-periodic Schrödinger operators. J. Math. Phys. 28, 2891–2898 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. Eliasson, L.H.: Floquet solutions for the 1-dimensional quasi-periodic Schrödinger equation. Commun. Math. Phys. 146(3), 447–482 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  6. Hairer, E., Lubich, C.: The life-span of backward error analysis for numerical integrators. Numer. Math. 76, 441–462 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  7. Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration: Structure-preserving Algorithms for Ordinary Differential Equations, 2nd edn., pp. 179–236. Springer, Berlin (2006)

    MATH  Google Scholar 

  8. Johnson, R.A., Sell, G.R.: Smoothness of spectral subbundles and reducibility of quasi-periodic linear differential equations. J. Differ. Equ. 41, 262–288 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  9. Jorba, À., Simó, C.: On the reducibility of linear differential equations with quasi-periodic coefficients. J. Differ. Equ. 98(1), 111–124 (1992)

    Article  MATH  Google Scholar 

  10. Jorba, À., Simó, C.: On quasi-periodic perturbations of elliptic equilibrium points. SIAM J. Math. Anal. 27(6), 1704–1737 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Jorba, À., Villanueva, J.: On the persistence of lower dimensional invariant tori under quasi-periodic perturbations. J. Nonlinear Sci. 7, 427–473 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  12. Jorba, À.: A methodology for the numerical computation of normal forms, centre manifolds, and first integrals of Hamiltonian systems. Exp. Math. 8(2), 155–195 (1999)

    MATH  MathSciNet  Google Scholar 

  13. Kolmogorov, A.N.: On conservation of conditionally periodic motions under small perturbations of the Hamiltonian. Dokl. Akad. Nauk SSSR 98, 527–530 (1954)

    MATH  MathSciNet  Google Scholar 

  14. de la Llave, R.: A tutorial on KAM theory. In: Proceedings of the AMS Summer Research Institute on Smooth Ergodic Theory and its Applications, Seattle, 1999, Proc. Sympos. Pure Math., vol. 69, Amer. Math. Soc., vol. 175292 (2001). Expanded version available at ftp.ma.utexas.edu/pub/papers/llave/tutorial.pdf

  15. MacKay, R.S., Meiss, J.D.: Hamiltonian Dynamical Systems: A Reprint Selection. Adam Hilger, Bristol (1987)

    MATH  Google Scholar 

  16. Moan, P.C.: On the KAM and Nekhoroschev theorems for symplectic integrators and implications for error growth. Nonlinearity 17, 67–83 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Moan, P.-C.: On rigorous modified equations for discretizations of ODEs. (2005). Available at http://www.focm.net/gi/gips/2005/3.html

  18. Moan, P.-C.: On modified equations for discretizations of ODEs. J. Phys. A Math. Gen. 39, 5545–5561 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  19. Moser, J.: On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Gott. Math.-Phys. Kl. 2, 1–20 (1962)

    Google Scholar 

  20. Moser, J., Pöschel, J.: An extension of a result by Dinaburg and Sinai on quasiperiodic potentials. Comment. Math. Helv. 59(1), 39–85 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  21. Palis, J.: Vector fields generate few diffeomorphisms. Bull. Am. Math. Soc. 80(3), 503–505 (1974)

    Article  MATH  MathSciNet  Google Scholar 

  22. Poincaré, H.: Les Méthodes Nouvelles de la Mécanique Céleste, pp. 366–401. NASA English translation of the Dover publication (1957)

  23. Pronin, A.V., Treschev, D.V.: On the inclusion of analytic symplectic maps into analytic flows. Regul. Chaotic Dyn. 2(2), 14–24 (1997)

    MATH  MathSciNet  Google Scholar 

  24. Rychlik, M.: Renormalization of cocycles and linear ODE with almost periodic coefficients. Invent. Math. 110(1), 173–206 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  25. Shang, Z.-J.: KAM theorm of symplectic algorithms for Hamiltonian systems. Numer. Math. 83, 477–496 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. Shang, Z.-J.: Resonant and diophantine step sizes in computing invariant tori of Hamiltonian systems. Nonlinearity 13, 299–308 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  27. Shang, Z.-J.: A note on the KAM theorem for symplectic mappings. J. Dyn. Differ. Equ. 12(2), 357–383 (2000)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. I.F.S., Massey University, Palmerston North, New Zealand

    Robert I. McLachlan & Dion R. J. O’Neale

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  1. Robert I. McLachlan
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  2. Dion R. J. O’Neale
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Correspondence to Dion R. J. O’Neale.

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McLachlan, R.I., O’Neale, D.R.J. Preservation and destruction of periodic orbits by symplectic integrators. Numer Algor 53, 343–362 (2010). https://doi.org/10.1007/s11075-009-9352-6

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  • DOI: https://doi.org/10.1007/s11075-009-9352-6

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