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First Order Periodic Averaging

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A Toolbox of Averaging Theorems

Abstract

We have seen averaging results in the Introduction. We will explain now the theory for O(ε) approximations on a long timescale.

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References

  1. N.N. Bogoliubov, Y.A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations (Gordon and Breach, Washington, 1961)

    Google Scholar 

  2. H.W. Broer, M.B. Sevryuk, KAM theory: quasi-periodicity in dynamical systems, in Handbook of Dynamical Systems vol. 3, Chap.6, ed. by H.W, Broer, B. Hasselblatt, F. Takens (Elsevier, Amsterdam, 2010), pp. 251–344

    Google Scholar 

  3. P. Fatou, Sur le mouvement d’un système soumis á des forces á courte période. Bull. Soc. Math. 56, 98–139 (1928)

    MathSciNet  MATH  Google Scholar 

  4. M. Golubitsky, I. Stewart, The Symmetry Perspective (Birkhäuser Verlag, Basel, 2000)

    MATH  Google Scholar 

  5. G. Gorelik, A. Vitt, Oscillations of an elastic pendulum as an example for two parametrically excited vibratory sysstems (Russ.). J. Tech. Phys. USSR col. 3 (1933)

    Google Scholar 

  6. M. Roseau Vibrations Nonlinéaires et Théorie de la Stabilité (Springer, Berlin/Heidelberg, 1966)

    Google Scholar 

  7. J.A. Sanders, F. Verhulst, and J. Murdock, Averaging Methods in Nonlinear Dynamical Systems, Rev edn. (Springer-Verlag, New York, 2007)

    MATH  Google Scholar 

  8. A.H.P. Van der Burgh, On the asymptotic solutions of the differential equations of the elastic pendulum. J. de Mécanique 4, 507–520 (1968)

    MATH  Google Scholar 

  9. F. Verhulst, Discrete symmetric dynamical systems at the main resonances with applications to axi-symmetric galaxies. Philos. Trans. R. Soc. Lond. 290, 435–465 (1979)

    Article  MATH  Google Scholar 

  10. F. Verhulst, Nonlinear Ordinary Differential Equations and Dynamical Systems, 2nd Rev. edn. (Springer, Berlin, 2000)

    Google Scholar 

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Verhulst, F. (2023). First Order Periodic Averaging. In: A Toolbox of Averaging Theorems. Surveys and Tutorials in the Applied Mathematical Sciences, vol 12. Springer, Cham. https://doi.org/10.1007/978-3-031-34515-9_2

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