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On a common derivation of the averaging method and the two-timescale method

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Abstract

In this paper it is shown that the well-known averaging method (of Krylov, Bogoliubov-Mitropolski) and the two-timescale method, applied to periodic first-order ordinary differential equations, can be derived from one common principle, as two more or less complementary special cases. The uniformity of this treatment includes the proof of asymptotic convergence of both methods, since a single proof can be given under certain hypotheses, which are verifieda posteriori in both cases.

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References

  • Bogoliubov, N. N. and Mitropolski, J. A.: 1965,Asymoptotische Methoden in der Theorie der nichtlinearen Schwingungen, Akademie Verlag, Berlin.

    Google Scholar 

  • Deprit, A.: 1969,Celest. Mech. 1, 12.

    Google Scholar 

  • Gilbert, A. D.: 1974,Proc. Camb. Phil. Soc. 76, 327.

    Google Scholar 

  • Henard, J. and Roels, J.: 1974,Celest. Mech. 10, 497.

    Google Scholar 

  • Hori, G.: 1966,Publ. Astron. Soc. Japan.18, 287.

    Google Scholar 

  • Hori, G.: 1971,Publ. Astron. Soc. Japan 23, 567.

    Google Scholar 

  • Kamel, A. A.: 1970,Celest. Mech. 3, 90.

    Google Scholar 

  • Kirchgraber, U.: 1973,J. reine angew. Math. 259, 86.

    Google Scholar 

  • Luke, J. C.: 1966,Proc. Roy. Soc. Ser. A 292, 403.

    Google Scholar 

  • Morrison, J. A.: 1966a, in R. L. Duncombe and V. G. Szebehely (eds.),Progress in Astronautics and Aeronautics, Vol. 17, Academic Press, pp. 117–138.

  • Morrison, J. A.: 1966b,SIAM Rev. 8, 66.

    Google Scholar 

  • Nayfeh, A. H.: 1973,Perturbation Methods, Chapter 6, J. Wiley and Sons, New York.

    Google Scholar 

  • Perko, L. M.: 1968,SIAM J. Appl. Math. 17, 698.

    Google Scholar 

  • Rouche, N. and Mawhin, J.: 1973,Equations differentielles ordinaires, T.1, p. 108, Masson et Cie, Paris.

    Google Scholar 

  • Sarlet, W.: 1978,Simon Stevin,52, to be published.

  • Shniad, H.: 1970,Celest. Mech. 2, 114.

    Google Scholar 

  • Stern, D. P.: 1971,Celest. Mech. 3, 241.

    Google Scholar 

  • van der Burgh, A. H. P.: 1974,Studies in the Asymptotic Theory of Nonlinear Resonance, Thesis, Delft.

  • von Zeipel, H.: 1916,Ark. Mat. Astr. Fys. 11, No. 1.

  • Whitham, G. B.: 1965a,Proc. Roy. Soc. Ser. A 283, 238.

    Google Scholar 

  • Whitham, G. B.: 1965b,J. Fluid Mech. 22, 273.

    Google Scholar 

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  1. Instituut voor Theoretische Mechanica, Rijksuniversiteit Gent, Krijgslaan 271, B-9000, Gent, Belgium

    Willy Sarlet

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  1. Willy Sarlet
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Sarlet, W. On a common derivation of the averaging method and the two-timescale method. Celestial Mechanics 17, 299–311 (1978). https://doi.org/10.1007/BF01232834

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