Geodesic
Asymptotic analysis of the parametric instability of nonlinear hyperbolic equations
Sbornik. Mathematics, Tome 208 (2017) no. 8, pp. 1088-1112 Cet article a éte moissonné depuis la source Math-Net.Ru

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This paper is concerned with parametric resonance under nonlinear periodic perturbations of differential equations which are abstract analogues of hyperbolic systems. A modification of the Krylov-Bogolyubov averaging method capable of circumventing the well-known small divisor problem is applied to reduce the description of solutions of perturbed equations at resonance to the study of autonomous dynamical systems in finite-dimensional spaces. Bibliography: 28 titles.
Keywords: hyperbolic equations, parametric resonance, averaging method. 
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V. S. Belonosov. Asymptotic analysis of the parametric instability of nonlinear hyperbolic equations. Sbornik. Mathematics, Tome 208 (2017) no. 8, pp. 1088-1112. http://geodesic.mathdoc.fr/item/SM_2017_208_8_a1/

[1] M. V. Kurlenya, S. V. Serdyukov, “Intensifikatsiya dobychi nefti pri nizkochastotnom vibroseismicheskom vozdeistvii”, Gornyi informatsionno-analiticheskii byulleten, 2004, no. 5, 29–34

[2] U. B. Bazaron, V. B. Deryagin, A. V. Bulgadaev, “Izmerenie sdvigovoi uprugosti zhidkostei i ikh granichnykh sloev rezonansnym metodom”, ZhETF, 51:4(10) (1966), 969–981

[3] V. N. Dorovsky, S. V. Dorovsky, “A hydrodynamic model of water-oil layered systems containing gas”, Math. Comput. Modelling, 35:7-8 (2002), 751–757 | DOI | Zbl

[4] V. N. Dorovsky, V. S. Belonosov, A. S. Belonosov, “Numerical investigation of parametric resonance in water-oil structures containing gas”, Math. Comput. Modelling, 36:1-2 (2002), 203–209 | DOI | MR | Zbl

[5] V. S. Belonosov, V. N. Dorovskii, A. S. Belonosov, S. V. Dorovskii, “Gidrodinamika gazosoderzhaschikh sloistykh sistem”, Uspekhi mekhaniki, 3:2 (2005), 37–70

[6] V. A. Yakubovich, V. M. Starzhinskii, Parametricheskii rezonans v lineinykh sistemakh, Nauka, M., 1987, 328 pp. | MR | Zbl

[7] G. M. Zaslavskii, R. Z. Sagdeev, Vvedenie v nelineinuyu fiziku. Ot mayatnika do turbulentnosti i khaosa, Nauka, M., 1988, 368 pp. | MR | Zbl

[8] N. A. Lyulko, N. A. Kudryavtseva, A. N. Kudryavtsev, “Asimptoticheskii i chislennyi analiz parametricheskogo rezonansa v nelineinoi sisteme dvukh ostsillyatorov”, Sib. elektron. matem. izv., 11 (2014), 675–694 | MR | Zbl

[9] N. A. Lyul'ko, “Instability of a nonlinear system of two oscillators under main and combination resonances”, Comput. Math. Math. Phys., 55:1 (2015), 53–70 | DOI | DOI | MR | Zbl

[10] S. G. Kreĭn, Linear differential equations in Banach space, Transl. Math. Monogr., 29, Amer. Math. Soc., Providence, RI, 1971, v+390 pp. | MR | MR | Zbl | Zbl

[11] T. Kato, Perturbation theory for linear operators, Grundlehren Math. Wiss., 132, Springer-Verlag New York, Inc., New York, 1966, xix+592 pp. | MR | MR | Zbl | Zbl

[12] Yu. L. Daletskii, M. G. Krein, Stability of solutions of differential equations in Banach space, Transl. Math. Monogr., 43, Amer. Math. Soc., Providence, RI, 1974, vi+386 pp. | MR | MR | Zbl | Zbl

[13] B. M. Levitan, Pochti-periodicheskie funktsii, Gostekhizdat, M., 1953, 396 pp. | MR | Zbl

[14] B. M. Levitan, V. V. Zhikov, Almost periodic functions and differential equations, Cambridge Univ. Press, Cambridge–New York, 1982, xi+211 pp. | MR | MR | Zbl | Zbl

[15] C. Corduneanu, Almost periodic functions, 2nd engl. ed., Chelsea Publishing Company, New York, 1989, x+257 pp. | MR | Zbl

[16] J. Favard, “Sur les équations différentielles linéaires à coefficients presque-périodiques”, Acta Math., 51:1 (1928), 31–81 | DOI | MR | Zbl

[17] J. Dieudonné, Foundations of modern analysis, Pure Appl. Math., 10, Academic Press, New York–London, 1960, xiv+361 pp. | MR | Zbl | Zbl

[18] V. I. Derguzov, “Ob ustoichivosti reshenii uravnenii Gamiltona s neogranichennymi periodicheskimi operatornymi koeffitsientami”, Matem. sb., 63(105):4 (1964), 591–619 | MR | Zbl

[19] V. I. Derguzov, “Dostatochnye uslooviya ustoichivosti gamiltonovykh uravnenii s neogranichennymi periodicheskimi operatornymi koeffitsientami”, Matem. sb., 64(106):3 (1964), 419–435 | MR | Zbl

[20] V. N. Fomin, Matematicheskaya teoriya parametricheskogo rezonansa v lineinykh raspredelennykh sistemakh, Izd-vo LGU, L., 1972, 240 pp. | MR | Zbl

[21] N. Krylov, N. Bogoliubov, Introduction to non-linear mechanics, Ann. of Math. Stud., 11, Princeton Univ. Press, Princeton, NJ, 1943, iii+105 pp. | MR | Zbl

[22] N. N. Bogolyubov, Yu. A. Mitropolśkii, Asymptotic methods in the theory of non-linear oscillations, Hindustan Publishing Corp., Delhi; Gordon and Breach Science Publishers, New York, 1961, v+537 pp. | MR | MR | Zbl | Zbl

[23] Yu. A. Mitropolskii, Metod usredneniya v nelineinoi mekhanike, Naukova dumka, Kiev, 1971, 440 pp. | MR | Zbl

[24] V. I. Arnold, Mathematical methods of classical mechanics, Grad. Texts in Math., 60, Springer-Verlag, New York–Heidelberg, 1978, xvi+462 pp. | MR | MR | Zbl | Zbl

[25] V. I. Arnol'd, V. V. Kozlov, A. I. Neĭshtadt, “Mathematical aspects of classical and celestial mechanics”, Dynamical systems, III, Encyclopaedia Math. Sci., 3, Springer-Verlag, Berlin, 1988, 1–291 | MR | MR | Zbl

[26] V. S. Belonosov, “The spectral properties of distributions and asymptotic methods in perturbation theory”, Sb. Math., 203:3 (2012), 307–325 | DOI | DOI | MR | Zbl

[27] B. M. Levitan, I. S. Sargsjan, Sturm–Liouville and Dirac operators, Math. Appl. (Soviet Ser.), 59, Kluwer Academic Publishers Group, Dordrecht, 1991, xii+350 pp. | DOI | MR | MR | Zbl | Zbl

[28] M. A. Krasnosel'skii, P. P. Zabreiko, E. I. Pustyl'nik, P. E. Sobolevskii, Integral operators in spaces of summable functions, Monographs and Textbooks on Mechanics of Solids and Fluids, Mechanics: Analysis, Noordhoff International Publishing, Leiden, 1976, xv+520 pp. | MR | MR | Zbl | Zbl