Geodesic
Autoresonant asymptotics in an oscillating system with weak dissipation
Teoretičeskaâ i matematičeskaâ fizika, Tome 160 (2009) no. 1, pp. 102-111 Cet article a éte moissonné depuis la source Math-Net.Ru

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We study a system of two first-order differential equations arising in averaging nonlinear systems over fast single-frequency oscillations. We consider the situation where the original system contains weak dissipative terms. We construct the asymptotic form of a two-parameter solution with an unbounded increasing amplitude. This result gives a key for understanding autoresonance in weak dissipative systems as a phenomenon of significant increase in the forced nonlinear oscillation initiated by a small external pumping.
Keywords: nonlinear equation, small parameter, asymptotic behavior, resonance, dissipation.
Mots-clés : perturbation 
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L. A. Kalyakin; M. A. Shamsutdinov. Autoresonant asymptotics in an oscillating system with weak dissipation. Teoretičeskaâ i matematičeskaâ fizika, Tome 160 (2009) no. 1, pp. 102-111. http://geodesic.mathdoc.fr/item/TMF_2009_160_1_a9/

[1] B. Meerson, L. Friedland, Phys. Rev. A, 41:9 (1990), 5233–5236 ; I. Fajans, L. Friedland, Amer. J. Phys., 69:10 (2001), 1096–1102 | DOI | DOI

[2] V. Dering, “Inertsiya granits mezhdu ferromagnitnymi oblastyami”, Ferromagnitnyi rezonans i povedenie ferromagnetikov v peremennykh polyakh, eds. S. V. Vonsovskii, Inostrannaya literatura, M., 1952, 312–320; М. А. Шамсутдинов, И. Ю. Ломакина, В. Н. Назаров, А. Т. Харисов, Д. М. Шамсутдинов, Ферро- и антиферромагнитодинамика, Нелинейные колебания, волны и солитоны, Гилем, Уфа, 2007

[3] L. A. Kalyakin, Dokl. RAN, 378:5 (2001), 594–597 ; ТМФ, 137:1 (2003), 142–152 ; “Авторезонанс в динамической системе”, Асимптотические методы функционального анализа, Соврем. матем. и ее прил., 5, АН Грузии, Институт кибернетики, Тбилиси, 2003, 79–109 | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl

[4] L. A. Kalyakin, Matem. zametki, 73:3 (2003), 449–452 | DOI | MR | Zbl

[5] N. N. Bogolyubov, Yu. A. Mitropolskii, Asimptoticheskie metody v teorii nelineinykh kolebanii, Nauka, M., 1974 | MR | MR | Zbl

[6] L. A. Kalyakin, “Obosnovanie asimptotik dlya uravnenii glavnogo rezonansa”, Tr. IMM UrO RAN, 9, no. 1, 2003, 88–101 | MR | Zbl

[7] A. T. Sinclair, Mon. Not. Roy. Astron. Soc., 160 (1972), 169–187 | DOI

[8] A. I. Neishtadt, PMM, 39:4 (1975), 621–632 | DOI | MR | Zbl

[9] R. Haberman, E. K. Ho, J. Appl. Mech., 62:4 (1995), 941–946 | DOI | MR | Zbl

[10] A. P. Itin, A. I. Neishtadt, A. A. Vasiliev, Phys. D, 141:3–4 (2000), 281–296 | DOI | MR | Zbl

[11] A. A. Kolomenskii, A. N. Lebedev, Teoriya tsiklicheskikh uskoritelei, Fizmatgiz, M., 1962

[12] L. A. Kalyakin, R. N. Garifullin, M. A. Shamsutdinov, ZhVM i MF, 47:7 (2007), 1208–1220 | DOI | MR

[13] V. V. Kozlov, S. D. Furta, Asimptotiki reshenii silno nelineinykh sistem differentsialnykh uravnenii, Izd-vo MGU, M., 1996 | MR | Zbl

[14] G. E. Kuzmak, PMM, 23:3 (1959), 515–526 ; В. И. Арнольд, В. В. Козлов, А. И. Нейштадт, “Математические аспекты классической и небесной механики”, Динамические системы – 3, Итоги науки и техн. Сер. Соврем. пробл. мат. Фундам. направления, ВИНИТИ, М., 1985, 5–290 | DOI | MR | Zbl | DOI | MR | MR | Zbl | Zbl

[15] A. I. Neishtadt, PMM, 48:2 (1984), 197–204 | DOI | MR | Zbl

[16] M. V. Fedoryuk, Asimptoticheskie metody dlya lineinykh obyknovennykh differentsialnykh uravnenii, Nauka, M., 1983 | MR | MR | Zbl

[17] P. Boutroux, Ann. Sci. École Norm. Sup., 30 (1913), 255–375 ; 31 (1914), 99–159 ; N. Joshi, “Asymptotic Studies of the Painlevé Equations”, The Painlevé Property. One Centure Later, CRM Ser. Math. Phys., ed. R. Conte, Springer, New York, 1999, 181–227 | DOI | MR | Zbl | DOI | MR | Zbl | MR | Zbl

[18] L. A. Kalyakin, “Asimptotika na beskonechnosti reshenii uravnenii, blizkikh k gamiltonovym”, Itogi nauki i tekhniki. Sovrem. matem. i ee pril., 53, 2008, 138–160 | DOI

[19] L. A. Kalyakin, Dokl. RAN, 360:3 (1998), 328–331 | MR | Zbl