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Bogolyubov averaging and normalization procedures in nonlinear mechanics. II

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Abstract

By using a new method suggested in the first part of the present work, we study systems which become linear in the zero approximation and have perturbations in the form of polynomials. This class of systems has numerous applications. The following fact is even more important: Our technique demonstrates how to generalize the classical method of Poincaré-Birkhoff normal forms and obtain new results by using group-theoretic methods. After a short exposition of the general theory of the method of asymptotic decomposition, we illustrate the new normalization technique as applied to models based on the Lotka-Volterra equations.

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References

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

    Google Scholar 

  2. N. Jacobson, “Lie algebras,” in:Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience, New York-London (1962).

    Google Scholar 

  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).

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  4. Yu. M. Svirizhev and D. O. Logofet,Stability of Biological Associations [in Russian], Nauka, Moscow (1978).

    Google Scholar 

  5. H. I. Freedman,Deterministic Mathematical Models in Population Ecology, 2nd ed., Hifr Consulting, Edmonton (1987).

    Google Scholar 

  6. N. N. Bogolyubov and Yu. A. Mitropol'skii,Asymptotic Methods in the Theory of Nonlinear Oscillations [in Russian], Nauka, Moscow (1974),English translation: Gordon & Breach, New York (1961).

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Authors and Affiliations

  1. Institute of Mathematics, Ukrainian Academy of Sciences, Kiev

    Yu. A. Mitropol'skii (Academician) & A. K. Lopatin

Authors
  1. Yu. A. Mitropol'skii
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  2. A. K. Lopatin
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Additional information

Published in Ukrainskii Matematicheskii Zhurnal, Vol.46, No. 11, pp. 1509–1526, November, 1994.

The present work was supported by the Ukrainian State Committee on Science and Technology.

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Mitropol'skii, Y.A., Lopatin, A.K. Bogolyubov averaging and normalization procedures in nonlinear mechanics. II. Ukr Math J 46, 1667–1687 (1994). https://doi.org/10.1007/BF01058885

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  • DOI: https://doi.org/10.1007/BF01058885

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