Bogolyubov averaging and normalization procedures in nonlinear mechanics. I
We suggest a new method for asymptotic analysis of nonlinear dynamical systems based on group-the-oretic methods. On the basis of the Bogolyubov averaging method, we develop a new normalization procedure — “asymptotic decomposition.” We clarify the contribution of this procedure to the interpretatio...
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| Datum: | 1994 |
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| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
1994
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/5634 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860511863763107840 |
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| author | Lopatin, A. K. Mitropolskiy, Yu. A. Лопатін, А. К. Митропольський, Ю. О. |
| author_facet | Lopatin, A. K. Mitropolskiy, Yu. A. Лопатін, А. К. Митропольський, Ю. О. |
| author_sort | Lopatin, A. K. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-19T09:14:58Z |
| description | We suggest a new method for asymptotic analysis of nonlinear dynamical systems based on group-the-oretic methods. On the basis of the Bogolyubov averaging method, we develop a new normalization procedure — “asymptotic decomposition.” We clarify the contribution of this procedure to the interpretation and development of the averaging method for systems in the standard form and systems with several fast variables. According to this method, the centralized system is regarded as a direct analog of the system averaged in Bogolyubov's sense. The operation of averaging is interpreted as the Bogolyubov projector, i.e., the operation of projection of an operator onto the algebra of centralizer. |
| first_indexed | 2026-03-24T03:19:39Z |
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| id | umjimathkievua-article-5634 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:19:39Z |
| publishDate | 1994 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/ec/39f979b3d0de7df5e94d53c158db32ec.pdf |
| spelling | umjimathkievua-article-56342020-03-19T09:14:58Z Bogolyubov averaging and normalization procedures in nonlinear mechanics. I Bogolyubov averaging and normalization procedures in nonlinear mechanics. I Lopatin, A. K. Mitropolskiy, Yu. A. Лопатін, А. К. Митропольський, Ю. О. We suggest a new method for asymptotic analysis of nonlinear dynamical systems based on group-the-oretic methods. On the basis of the Bogolyubov averaging method, we develop a new normalization procedure — “asymptotic decomposition.” We clarify the contribution of this procedure to the interpretation and development of the averaging method for systems in the standard form and systems with several fast variables. According to this method, the centralized system is regarded as a direct analog of the system averaged in Bogolyubov's sense. The operation of averaging is interpreted as the Bogolyubov projector, i.e., the operation of projection of an operator onto the algebra of centralizer. A new method of asymptotic analysis of nonlinear dynamical systems is developed with extensive use of group-theoretical methods. The technique of normalization, which is named an "asymptotic decomposition" by the authors, is developed in the context of Bogolyubov averaging. In this paper, we also discuss how this technique helps to understand and develop the averaging method for systems in standard form and systems with several fast variables. The new method treats a centralized system as a direct analog of an averaged system according to Bogolyubov. The operation of averaging is interpreted as a Bogolyubov projector in the construction of the projection of an operator onto the algebra of the centralizer. Institute of Mathematics, NAS of Ukraine 1994-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5634 Ukrains’kyi Matematychnyi Zhurnal; Vol. 46 No. 9 (1994); 1171–1188 Український математичний журнал; Том 46 № 9 (1994); 1171–1188 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/5634/7953 https://umj.imath.kiev.ua/index.php/umj/article/view/5634/7954 Copyright (c) 1994 Lopatin A. K.; Mitropolskiy Yu. A. |
| spellingShingle | Lopatin, A. K. Mitropolskiy, Yu. A. Лопатін, А. К. Митропольський, Ю. О. Bogolyubov averaging and normalization procedures in nonlinear mechanics. I |
| title | Bogolyubov averaging and normalization procedures in nonlinear mechanics. I |
| title_alt | Bogolyubov averaging and normalization procedures in nonlinear mechanics. I |
| title_full | Bogolyubov averaging and normalization procedures in nonlinear mechanics. I |
| title_fullStr | Bogolyubov averaging and normalization procedures in nonlinear mechanics. I |
| title_full_unstemmed | Bogolyubov averaging and normalization procedures in nonlinear mechanics. I |
| title_short | Bogolyubov averaging and normalization procedures in nonlinear mechanics. I |
| title_sort | bogolyubov averaging and normalization procedures in nonlinear mechanics. i |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5634 |
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