Geodesic
The buffer property in a non-classical hyperbolic
Sbornik. Mathematics, Tome 197 (2006) no. 6, pp. 853-885 
A. Yu. Kolesov; N. Kh. Rozov. The buffer property in a non-classical hyperbolic. Sbornik. Mathematics, Tome 197 (2006) no. 6, pp. 853-885. http://geodesic.mathdoc.fr/item/SM_2006_197_6_a3/
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     url = {http://geodesic.mathdoc.fr/item/SM_2006_197_6_a3/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

A mathematical model of a self-excited $RCL$-oscillator with a segment of a solenoid in the feedback loop is considered, which is the following boundary-value problem: \begin{gather*} \frac{\partial^2}{\partial t^2} \biggl(u-\varkappa\frac{\partial^2u}{\partial x^2}\biggr) +\varepsilon\frac{\partial}{\partial t} \biggl(u-\varkappa\frac{\partial^2u}{\partial x^2}\biggr) =\frac{\partial^2u}{\partial x^2}\,, \\ \frac{\partial u}{\partial x}\bigg|_{x=1}=0, \qquad u\big|_{x=0}+(1+\varepsilon^2\gamma)u\big|_{x=1}-u^3\big|_{x=1}=0, \end{gather*} where $0<\varepsilon\ll1$, and $\varkappa$ and $\gamma$ are positive parameters of order 1. For this boundary-value problem with suitably increased $\gamma$ and reduced $\varepsilon$ one proves the existence of an arbitrary prescribed finite number of stable cycles (solutions periodic in $t$). Bibliography: 12 titles.

[1] A. A. Vitt, “Raspredelennye avtokolebatelnye sistemy”, ZhTF, 4:1 (1934), 146–157 | Zbl

[2] A. Yu. Kolesov, E. F. Mischenko, N. Kh. Rozov, Asimptoticheskie metody issledovaniya periodicheskikh reshenii nelineinykh giperbolicheskikh uravnenii, Tr. MIAN, 222, Nauka, M., 1998 | MR | Zbl

[3] A. Yu. Kolesov, N. Kh. Rozov, V. G. Sushko, “Spetsifika avtokolebatelnykh protsessov v rezonansnykh giperbolicheskikh sistemakh”, Fundam. i prikl. matem., 5:2 (1999), 437–473 | MR | Zbl

[4] A. Yu. Kolesov, E. F. Mischenko, N. Kh. Rozov, “Yavlenie bufernosti v rezonansnykh sistemakh giperbolicheskikh uravnenii”, UMN, 55:2 (2000), 95–120 | MR | Zbl

[5] A. Yu. Kolesov, N. Kh. Rozov, “Yavlenie bufernosti v $RCLG$-avtogeneratore: teoreticheskii analiz i rezultaty eksperimenta”, Differentsialnye uravneniya. Nekotorye matematicheskie zadachi optimalnogo upravleniya, Tr. MIAN, 233, Nauka, M., 2001, 153–207 | MR | Zbl

[6] L. V. Postnikov, V. I. Korolev (red.), Sbornik zadach po teorii kolebanii, Nauka, M., 1978

[7] M. I. Rabinovich, D. I. Trubetskov, Vvedenie v teoriyu kolebanii i voln, Nauka, M., 1984 | MR | Zbl

[8] Yu. S. Kolesov, “Matematicheskaya teoriya RC-avtogeneratorov s raspredelennymi parametrami v tsepi obratnoi svyazi”, Differentsialnye uravneniya i ikh primenenie, no. 2, In-t fiziki i matematiki AN LitSSR, Vilnyus, 1971, 1–67 | MR | Zbl

[9] A. Yu. Kolesov, N. Kh. Rozov, Invariantnye tory nelineinykh volnovykh uravnenii, Fizmatlit, M., 2004 | Zbl

[10] L. P. Shilnikov, A. L. Shilnikov, D. V. Turaev, L. Chua, Metody kachestvennoi teorii v nelineinoi dinamike, ch. 1, In-t kompyuternykh issledovanii, M., Izhevsk, 2004

[11] A. B. Vasileva, S. A. Kaschenko, Yu. S. Kolesov, N. Kh. Rozov, “Bifurkatsiya avtokolebanii nelineinykh parabolicheskikh uravnenii s maloi diffuziei”, Matem. sb., 130:4 (1986), 488–499 | MR | Zbl

[12] Yu. S. Kolesov, “Bifurkatsiya invariantnykh torov parabolicheskikh sistem s maloi diffuziei”, Matem. sb., 184:3 (1993), 121–136 | MR | Zbl