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Bogolyubov averaging and normalization procedures in nonlinear mechanics. IV

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Abstract

In this paper, we apply the theory developed in parts I-III [Ukr. Math. Zh.,46, No. 9, 1171–1188; No. 11, 1509–1526; No. 12, 1627–1646 (1994)] to some classes of problems. We consider linear systems in zero approximation and investigate the problem of invariance of integral manifolds under perturbations. Unlike nonlinear systems, linear ones have centralized systems, which are always decomposable. Moreover, restrictions connected with the impossibility of diagonalization of the coefficient matrix in zero approximation are removed. In conclusion, we apply the method of local asymptotic decomposition to some mechanical problems.

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References

  1. 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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  2. 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).

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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. 47, No. 8, pp. 1044–1068, August, 1995.

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

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Mitropol'skii, Y.A., Lopatin, A.K. Bogolyubov averaging and normalization procedures in nonlinear mechanics. IV. Ukr Math J 47, 1195–1221 (1995). https://doi.org/10.1007/BF01057710

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  • DOI: https://doi.org/10.1007/BF01057710

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