Geodesic
Resonance capture in a system of two oscillators near equilibrium
Teoretičeskaâ i matematičeskaâ fizika, Tome 194 (2018) no. 3, pp. 385-402 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a system of differential equations describing the interaction of two weakly coupled nonlinear oscillators. We assume that one oscillator is initially far from equilibrium, the other is near equilibrium, and their frequencies are close. We study the effect of resonance capture, when the frequencies of the coupled oscillators remain close while the oscillation energies change in time significantly; in particular, the second oscillator goes far from equilibrium. We find that the initial stage of resonance capture is described by the second Painlevé equation. We obtain such a description in the asymptotic approximation in a small parameter corresponding to the coupling constant.
Keywords: nonlinear oscillation, small parameter, asymptotic approximation, resonance capture. 
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L. A. Kalyakin. Resonance capture in a system of two oscillators near equilibrium. Teoretičeskaâ i matematičeskaâ fizika, Tome 194 (2018) no. 3, pp. 385-402. http://geodesic.mathdoc.fr/item/TMF_2018_194_3_a1/

[1] A. I. Neishtadt, “Usrednenie, prokhozhdenie cherez rezonansy i zakhvat v rezonans v dvukhchastotnykh sistemakh”, UMN, 69:5 (2014), 3–80 | DOI | DOI | MR | Zbl

[2] L. A. Kalyakin, “Asimptoticheskii analiz modelei avtorezonansa”, UMN, 63:5 (2008), 3–72 | DOI | DOI | Zbl

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

[4] A. A. Kapaev, “Asimptoticheskie formuly dlya funktsii Penleve vtorogo”, TMF, 77:3 (1988), 323–332 | DOI | MR | Zbl

[5] V. I. Arnold, “Malye znamenateli i problemy ustoichivosti dvizheniya v klassicheskoi i nebesnoi mekhanike”, UMN, 18:6 (1963), 91–192 | DOI | MR | Zbl

[6] A. I. Neishtadt, “Zakhvat v rezonans i rasseyanie na rezonansakh v dvukhchastotnykh sistemakh”, Tr. MIAN, 250 (2005), 198–218 | MR | Zbl

[7] I. Bryuning, S. Yu. Dobrokhotov, M. A. Poteryakhin, “Ob usrednenii dlya gamiltonovykh sistem s odnoi bystroi fazoi i malymi amplitudami”, Matem. zametki, 70:5 (2001), 660–669 | DOI | DOI

[8] L. A. Kalyakin, “Rezonansnyi zakhvat v nelineinoi sisteme”, TMF, 144:1 (2005), 74–82 | DOI | DOI | MR | Zbl

[9] L. A. Kalyakin, “Avtorezonans v dinamicheskoi sisteme”, Asimptoticheskie metody funktsionalnogo analiza, Sovremennaya matematika i ee prilozheniya, 5, In-t kibernetiki AN Gruzii, Tbilisi, 2003, 79–109 | MR | Zbl

[10] N. A. Kudryashov, “The second Painlevé equation as a model for the electric field in a semiconductor”, Phys. Lett. A., 233:4–6 (1997), 397–400 | DOI | MR | Zbl

[11] A. R. Its, A. A. Kapaev, V. Yu. Novokshenov, A. S. Fokas, Transendenty Penleve. Metod zadachi Rimana, NITs “RKhD”, M.–Izhevsk, 2005

[12] L. A. Kalyakin, “Metod usredneniya v zadachakh ob asimptotike na beskonechnosti”, Ufimskii matem. zhurn., 1:2 (2009), 29–52 | Zbl

[13] A. M. Ilin, Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989 | MR | MR | Zbl

[14] L. A. Kalyakin, “Usrednenie v modeli avtorezonansa”, Matem. zametki, 73:3 (2003), 449–452 | DOI | DOI | MR | Zbl

[15] L. A. Kalyakin, “Uravnenie Penleve-II kak model rezonansnogo vzaimodeistviya ostsillyatorov”, Tr. IMM UrO RAN, 23:2 (2017), 104–116 | DOI

[16] R. N. Garifullin, “Asimptoticheskoe reshenie zadachi ob avtorezonanse. Vneshnee razlozhenie”, Zhurn. vychisl. matem. i matem. fiz., 46:9 (2006), 1605–1616 | DOI | MR