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Bogolyubov averaging and normalization procedures in nonlinear mechanics. III

Abstract

We describe the technique of normalization based on the method of asymptotic decomposition in the space of representation of a finite-dimensional Lie group. The main topics of the theory necessary for understanding the method are outlined. Models based on the Van der Pol equation are investigated by the method of asymptotic decomposition in the space of homogeneous polynomials (the space of representation of a general linear group in a plane) and in the space of representation of a rotation group on a plane (ordinary Fourier series). The comparison made shows a dramatic decrease in the necessary algebraic manipulations in the second case. We also discuss other details of the technique of normalization based on the method of asymptotic decomposition.

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References

  1. 1.

    A. Barut and R. Raczka,Theory of Group Representations and Applications, PWN, Warsaw (1977).

  2. 2.

    Yu. A. Mitropol'skii and A. K. Lopatin,Group-Theoretic Approach in Asymptotic Methods of Nonlinear Mechanics [in Russian], Naukova Dumka, Kiev (1988).

  3. 3.

    Yu. A. Mitropol'skii and A. K. Lopatin, “Bogolyubov averaging and normalization procedures in nonlinear mechanics. I,”Ukr. Mat. Zh.,46, No. 9, 1171–1188 (1994).

  4. 4.

    Yu. A. Mitropol'skii and A. K. Lopatin, “Bogolyubov averaging and normalization procedures in nonlinear mechanics. II,”Ukr. Mat. Zh.,46, No. 11, 1509–1526 (1994).

  5. 5.

    N. Ya. Vilenkin,Special Functions and the Group-Theoretic Representations [in Russian], Nauka, Moscow (1992).

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Additional information

Published in Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 12, pp. 1627–1646, December, 1994.

This research was partially supported by the International Scientific Foundation, grant No. UB2000.

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Mitropol'skii, Y.A., Lopatin, A.K. Bogolyubov averaging and normalization procedures in nonlinear mechanics. III. Ukr Math J 46, 1803–1826 (1994). https://doi.org/10.1007/BF01063169

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Keywords

  • Fourier Series
  • Main Topic
  • Normalization Procedure
  • Dramatic Decrease
  • Linear Group